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Resonance-based schemes for dispersive equations via decorated trees

Published online by Cambridge University Press:  13 January 2022

Yvain Bruned*
Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD,United Kingdom
Katharina Schratz
Affiliation:
Laboratoire Jacques-Louis Lions (UMR 7598), Sorbonne Université, 4 place Jussieu, 75252 Paris cedex 05, France; E-mail: Katharina.Schratz@sorbonne-universite.fr

Abstract

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

Type
Applied Analysis
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

We consider nonlinear dispersive equations

(1) $$ \begin{align} \begin{split} & i \partial_t u(t,x) + \mathcal{L}\left(\nabla, \tfrac{1}{\varepsilon}\right) u(t,x) =\vert \nabla\vert^{\alpha} p\left(u(t,x), \overline{u}(t,x)\right)\\ & u(0,x) = v(x), \end{split} \end{align} $$

where we assume a polynomial nonlinearity p and that the structure of (1) implies at least local well-posedness of the problem on a finite time interval $]0,T]$ , $T<\infty $ in an appropriate functional space. Here, u is the complex-valued solution that we want to approximate. Concrete examples are discussed in Section 5, including the cubic nonlinear Schrödinger (NLS) equation

(2) $$ \begin{align} i \partial_t u + \mathcal{L}\left(\nabla\right) u = \vert u\vert^2 u, \quad \mathcal{L}\left(\nabla\right) = \Delta, \end{align} $$

the Korteweg–de Vries (KdV) equation

(3) $$ \begin{align} \partial_t u +\mathcal{L}\left(\nabla\right) u = \frac12 \partial_x u^2, \quad \mathcal{L}\left(\nabla\right) = i\partial_x^3, \end{align} $$

as well as highly oscillatory Klein–Gordon type systems

(4) $$ \begin{align} i \partial_t u = -\mathcal{L}\left(\nabla, \tfrac{1}{\varepsilon}\right) u + \frac{1}{\varepsilon^2}\mathcal{L}\left(\nabla, \tfrac{1}{\varepsilon}\right)^{-1} \textstyle p(u,\overline{u}), \quad \mathcal{L}\left(\nabla, \tfrac{1}{\varepsilon}\right) = \frac{1}{\varepsilon}\sqrt{\frac{1}{\varepsilon^2}-\Delta}. \end{align} $$

In the last decades, Strichartz and Bourgain space estimates allowed establishing well-posedness results for dispersive equations in low regularity spaces [Reference Burq, Gérard and Tzvetkov15Reference Bourgain9Reference Keel and Tao58Reference Strichartz77Reference Tao78]. Numerical theory for dispersive partial differential equations (PDEs), on the other hand, is in general still restricted to smooth solutions. This is due to the fact that most classical approximation techniques were originally developed for linear problems and thus, in general, neglect nonlinear frequency interactions in a system. In the dispersive setting (1) the interaction of the differential operator $\mathcal {L}$ with the nonlinearity p, however, triggers oscillations both in space and in time and, unlike for parabolic problems, no smoothing can be expected. At low regularity and high oscillations, these nonlinear frequency interactions play an essential role: Note that while the influence of $i\mathcal {L}$ can be small, the influence of the interaction of $+i\mathcal {L}$ and $-i\mathcal {L}$ can be huge and vice versa. Classical linearised frequency approximations, used, for example, in splitting methods or exponential integrators (see Table 1) are therefore restricted to smooth solutions. The latter is not only a technical formality: The severe order reduction in case of nonsmooth solutions is also observed numerically (see, e.g., [Reference Jahnke and Lubich57Reference Ostermann and Schratz72] and Figure 2), and only very little is known on how to overcome this issue. For an extensive overview on numerical methods for Hamiltonian systems, geometric numerical analysis, structure preserving algorithms and highly oscillatory problems we refer to the books Butcher [Reference Butcher17], Engquist et al. [Reference Engquist, Fokas, Hairer and Iserles36], Faou [Reference Faou37], E. Hairer et al. [Reference Hairer, Nørsett and Wanner46Reference Hairer, Lubich and Wanner45], Holden et al. [Reference Holden, Karlsen, Lie and Risebro51], Leimkuhler and Reich [Reference Leimkuhler and Reich61], McLachlan and Quispel [Reference McLachlan and Quispel67], Sanz-Serna and Calvo [Reference Sanz-Serna and Calvo75] and the references therein.

Table 1 Classical frequency approximations of the principal oscillations (7).

In this work, we establish a new framework of resonance-based approximations for dispersive equations which will allow us to approximate with high-order accuracy a large class of equations under (much) lower regularity assumptions than classical techniques require. The key in the construction of the new methods lies in analysing the underlying oscillatory structure of the system (1). We look at the corresponding mild solution given by Duhamel’s formula

(5) $$ \begin{align} u(t) = e^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} v - ie^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)}\vert \nabla\vert^\alpha \int_0^t e^{ -i\xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} p\left(u(\xi), \overline{u}(\xi)\right) d\xi \end{align} $$

and its iterations

(6) $$ \begin{align} u(t) = e^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} v - ie^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} \vert \nabla\vert^\alpha\mathcal{I}_1( t, \mathcal{L},v,p) +\vert \nabla\vert^{2\alpha} \int_0^t \int_0^\xi \ldots d\xi_1 d \xi. \end{align} $$

The principal oscillatory integral $\mathcal {I}_1( t, \mathcal {L},v,p)$ thereby takes the form

$$ \begin{align*} \mathcal{I}_1( t, \mathcal{L},v,p) = \int_0^t \mathcal{O}\mathcal{s}\mathcal{c}(\xi, \mathcal{L}, v,p) d\xi \end{align*} $$

with the central oscillations

(7) $$ \begin{align} \mathcal{O}\mathcal{s}\mathcal{c}(\xi, \mathcal{L}, v,p) = e^{ -i\xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} p\left(e^{ i \xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} v , e^{ - i \xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} \overline{v} \right) \end{align} $$

driven by the nonlinear frequency interactions between the differential operator $\mathcal {L}$ and the nonlinearity p. In order to obtain a suitable approximation at low regularity, it is central to resolve these oscillations – characterised by the underlying structure of resonances – numerically. Classical linearised frequency approximations, however, neglect the nonlinear interactions in (7). This linearisation is illustrated in Table 1 for splitting and exponential integrator methods ([Reference Hairer, Lubich and Wanner45Reference Hochbruck and Ostermann49]).

The aim of this article is to introduce a framework which allows us to embed the underlying nonlinear oscillations (7) and their higher-order counterparts into the numerical discretisation. The main idea for tackling this problem is to introduce a decorated tree formalism that optimises the structure of the local error by mapping it to the particular regularity of the solution.

While first-order resonance-based discretisations have been presented for particular examples – for example, the Nonlinear Schrödinger (NLS), Korteweg–de Vries (KdV), Boussinesq, Dirac and Klein–Gordon equation; see [Reference Hofmanová and Schratz50Reference Baumstark, Faou and Schratz4Reference Baumstark and Schratz5Reference Ostermann and Schratz72Reference Ostermann and Su73Reference Schratz, Wang and Zhao76] – no general framework could be established so far. Each and every equation had to be targeted carefully one at a time based on a sophisticated resonance analysis. This is due to the fact that the structure of the underlying oscillations (7) strongly depends on the form of the leading operator $\mathcal {L}$ , the nonlinearity p and, in particular, their nonlinear interactions.

In addition to the lack of a general framework, very little is known about the higher-order counterpart of resonance-based discretisations. Indeed, some attempts have been made for second-order schemes (see, e.g., [Reference Hofmanová and Schratz50] for KdV and [Reference Knöller, Ostermann and Schratz60] for NLS), but they are not optimal. This is due to the fact that the leading differential operator $\mathcal {L} $ triggers a full spectrum of frequencies $k_{\kern-1.2pt j} \in {\mathbf {Z}}^{d}$ . Up to now it was an unresolved issue on how to control their nonlinear interactions up to higher order, in particular, in higher spatial dimensions where stability poses a key problem. Even in case of a simple NLS equation it is an open question whether stable low regularity approximations of order higher than one can be achieved in spatial dimensions $d\geq 2$ . In particular, previous works suggest a severe order reduction ([Reference Knöller, Ostermann and Schratz60]).

To overcome this, we introduce a new tailored decorated tree formalism. Thereby the decorated trees encode the Fourier coefficients in the iteration of Duhamel’s formula, where the node decoration encodes the frequencies which is in the spirit close to [Reference Christ27Reference Guo, Kwon and Oh44Reference Gubinelli43]. The main difficulty then lies in controlling the nonlinear frequency interactions within these iterated integrals up to the desired order with the constraint of a given a priori regularity of the solution. The latter is achieved by embedding the underlying oscillations, and their higher-order iterations, via well-chosen Taylor series expansions into our formalism: The dominant interactions will be embedded exactly, whereas only the lower-order parts are approximated within the discretisation.

We base our algebraic structures on the ones developed for stochastic partial differential equations (SPDEs) with regularity structure [Reference Hairer47] which is a generalisation of rough paths [Reference Lyons63Reference Lyons64Reference Gubinelli41Reference Gubinelli42]. Part of the formalism is inspired by [Reference Bruned, Hairer and Zambotti13] and the recentring map used for giving a local description of the solution of singular SPDEs. We adapt it to the context of dispersive PDEs by using a new class of decorated trees encoding the underlying dominant frequencies.

The framework of decorated trees and the underlying Hopf algebras have allowed the resolution of a large class of singular SPDEs [Reference Hairer47Reference Bruned, Hairer and Zambotti13Reference Chandra and Hairer22Reference Bruned, Chandra, Chevyrev and Hairer11] which include a natural random dynamic on the space of loops in a Riemannian manifold in [Reference Bruned, Gabriel, Hairer and Zambotti12]; see [Reference Bruned, Hairer and Zambotti14] for a very brief survey on these developments. With this general framework, one can study properties of singular SPDEs solutions in full subcritical regimes [Reference Chandra, Hairer and Shen23Reference Berglund and Bruned6Reference Hairer and Schönbauer48Reference Chandra, Moinat and Weber24]. The formalism of decorated trees together with the description of the renormalised equation in this context (see [Reference Bruned, Chandra, Chevyrev and Hairer11]) was directly inspired from numerical analysis of ordinary differential equations (ODEs), more precisely, from the characterisation of Runge–Kutta methods via B-series. Indeed, B-series are numerical (multi-)step methods for ODEs represented by a tree expansion; see, for example, [Reference Butcher16Reference Berland, Owren and Skaflestad7Reference Chartier, Hairer and Vilmart26Reference Hairer, Lubich and Wanner45Reference Iserles, Quispel and Tse56Reference Calaque, Ebrahimi-Fard and Manchon19]. We also refer to [Reference Munthe-Kaas and Føllesdal68] for a review of B-series on Lie groups and homogeneous manifolds as well as to [Reference Murua and Sanz-Serna69] providing an alternative structure via word series. The field of singular SPDEs took advantage of the B-series formalism and extended their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and enlarges their scope.

This work proposes a new application of the Butcher–Connes–Kreimer Hopf algebra [Reference Butcher16Reference Connes and Kreimer31] to dispersive PDEs. It gives a new light on structures that have been used in various fields such as numerical analysis, renormalisation in quantum field theory, singular SPDEs and dynamical systems for classifying singularities via resurgent functions introduced by Jean Ecalle (see [Reference Ecalle34Reference Fauvet and Menous38]). This is another testimony of the universality of this structure and adds a new object to this landscape. Our construction is motivated by two main features: Taylor expansions that are at the foundation of the numerical scheme (added at the level of the algebra as for singular SPDEs) and the frequency interaction (encoded in a tree structure for dispersive PDEs). The combination of the two together with the Butcher–Connes–Kreimer Hopf algebra allows us to design a novel class of schemes at low regularity. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to perform the local error analysis.

Our main result is the new general resonance-based scheme presented in Definition 4.4 with its error structure given in Theorem 4.8. Our general framework is illustrated on concrete examples in Section 5 and simulations show the efficacy of the scheme. The algebraic structure in Section 2 has its own interest where the main objective is to understand the frequency interactions. The Birkhoff factorisation given in Subsection 3.2 is designed for this purpose and is helpful in proving Theorem 4.8. This factorisation seems new in comparison to the literature.

Assumptions. We impose a periodic boundary condition, $x \in {\mathbf {T}}^d$ . However, our theory can be extended to the full space ${\mathbf {R}}^d$ . We assume that the differential operator $\mathcal {L}$ is real and consider two types of structures of the system (1) which will allow us to handle dispersive equations at low regularity (such as NLS and KdV) and highly oscillatory Klein–Gordon type systems; see also (2)–(4).

  • The differential operators $\mathcal {L}\left (\nabla , \frac {1}{\varepsilon }\right ) = \mathcal {L}\left (\nabla \right ) $ and $\vert \nabla \vert ^\alpha $ cast in Fourier space into the form

    (8) $$ \begin{align} \mathcal{L}\left(\nabla \right)(k) = k^\sigma + \sum_{\gamma : |\gamma| < \sigma} a_{\gamma} \prod_{j} k_j^{\gamma_j} ,\qquad \vert \nabla\vert^\alpha(k) = \prod_{ \gamma : |\gamma| {\leq \alpha}} k_j^{\gamma_j} \end{align} $$
    for some $ \alpha \in {\mathbf {R}} $ , $ \gamma \in {\mathbf {Z}}^d $ and $ |\gamma | = \sum _i \gamma _i $ , where for $k = (k_1,\ldots ,k_d)\in {\mathbf {Z}}^d$ and $m = (m_1, \ldots , m_d)\in {\mathbf {Z}}^d$ we set
    $$ \begin{align*} k^\sigma = k_1^\sigma + \ldots + k_d^\sigma, \qquad k \cdot m = k_1 m_1 + \ldots + k_d m_d. \end{align*} $$
  • We also consider the setting of a given high frequency $\frac {1}{\vert \varepsilon \vert } \gg 1$ . In this case we assume that the operators $\mathcal {L}\left (\nabla , \frac {1}{\varepsilon }\right ) $ and $\vert \nabla \vert ^\alpha $ take the form

    (9) $$ \begin{align} \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right) = \frac{1}{\varepsilon^{\sigma}} + \mathcal{B}\left(\nabla, \frac{1}{\varepsilon}\right), \qquad \vert \nabla\vert^\alpha = \mathcal{C}\left(\nabla, \frac{1}{\varepsilon}\right) \end{align} $$
    for some differential operators $\mathcal {B}\left (\nabla , \frac {1}{\varepsilon }\right )$ and $\mathcal {C}\left (\nabla , \frac {1}{\varepsilon }\right )$ which can be bounded uniformly in $ \vert \varepsilon \vert $ and are relatively bounded by differential operators of degree $\sigma $ and degree $\alpha < \sigma $ , respectively. This allows us to include, for instance, highly oscillatory Klein–Gordon type equations (4) (see also Subsection 5.3).

Figure 1 Initial values for Figure 2: $u_0 \in H^1$ (left) and $u_0 \in \mathcal {C}^\infty $ (right).

Figure 2 Order reduction of classical schemes based on linearised frequency approximations (cf. Table 1) in case of low regularity data (error versus step size for the cubic Schrödinger equation). For smooth solutions, classical methods reach their full order of convergence (right). In contrast, for less smooth solutions they suffer from severe order reduction (left). The initial values in $H^1$ and $\mathcal {C}^{\infty }$ are plotted in Figure 1. The slope of the reference solutions (dashed lines) is one and two, respectively.

In the next section we introduce the resonance-based techniques to solve the dispersive PDE (1) and illustrate our approach on the example of cubic nonlinear Schrödinger equation (2); see Example 1.

1.1 Resonances as a computational tool

Instead of employing classical linearised frequency approximations (cf. Table 1), we want to embed the underlying nonlinear oscillations

(10) $$ \begin{align} \mathcal{O}\mathcal{s}\mathcal{c}(\xi, \mathcal{L}, v,p) = e^{ -i\xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} p\left(e^{ i \xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} v , e^{ - i \xi \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} \overline{v} \right) \end{align} $$

(and their higher-order counterparts) into the numerical discretisation. In case of the 1-dimensional cubic Schrödinger equation (2) the central oscillations (10), for instance, take in Fourier the form (see Example 1 for details)

$$ \begin{align*}\mathcal{O}\mathcal{s}\mathcal{c}(\xi, \Delta, v,\text{cub}) =\sum_{\substack{k_1,k_2,k_3 \in {\mathbf{Z}}\\-k_1+k_2+k_3 = k} } e^{i k x } \overline{\hat{v}}_{k_1} \hat{v}_{k_2} \hat{v}_{k_3} \int_0^\tau e^{i s \mathscr{F}(k) } ds \end{align*} $$

with the underlying resonance structure

(11) $$ \begin{align} \mathscr{F}(k) = 2 k_1^2 - 2 k_1 (k_2+k_3) + 2 k_2 k_3. \end{align} $$

Ideally we would like to resolve all nonlinear frequency interactions (11) exactly in our scheme. However, these result in a generalised convolution (of Coifman–Meyer type [Reference Coifman and Meyer30]) which cannot be converted as a product into the physical space. Thus, the iteration would need to be carried out fully in Fourier space which does not yield a scheme which can be practically implemented in higher spatial dimensions; see also Remark 1.3. The latter in general also holds true in the abstract setting (10).

In order to obtain an efficient and practical resonance-based discretisation, we extract the dominant and lower-order parts from the resonance structure (10). More precisely, we filter out the dominant parts $ \mathcal {L}_{\text {dom}}$ and treat them exactly while only approximating the lower-order terms in the spirit of

(12) $$ \begin{align} \mathcal{O}\mathcal{s}\mathcal{c}(\xi, \mathcal{L}, v,p) = \left[e^{i \xi \mathcal{L}_{\text{dom}} \left(\nabla, \frac{1}{\varepsilon}\right)} p_{\text{dom}}\left(v,\overline{v}\right) \right] p_{\text{low}}(v,\overline{v}) + \mathcal{O}\Big(\xi\mathcal{L}_{\text{low}}\left(\nabla\right)v\Big). \end{align} $$

Here, $\mathcal {L}_{\text {dom}}$ denotes a suitable dominant part of the high frequency interactions and

(13) $$ \begin{align} \mathcal{L}_{\text{low}} = \mathcal{L} - \mathcal{L}_{\text{dom}} \end{align} $$

the corresponding nonoscillatory parts (details will be given in Definition 2.6). The crucial issue is to determine $\mathcal {L}_{\text {dom}}$ , $p_{\text {dom}}$ and $\mathcal {L}_{\text {low}}, p_{\text {low}}$ in (12) with an interplay between keeping the underlying structure of PDE and allowing a practical implementation at a reasonable cost. We refer to Example 1 for the concrete characterisation in case of cubic NLS, where $\mathcal {L}_{\text {low}} = \nabla $ and $\mathcal {L}_{\text {dom}} = \Delta $ .

Thanks to the resonance-based ansatz (12), the principal oscillatory integral

$$ \begin{align*} \mathcal{I}_1( t, \mathcal{L},v,p) = \int_0^t \mathcal{O}\mathcal{s}\mathcal{c}(\xi, \mathcal{L}, v,p) d\xi \end{align*} $$

in the expansion of the exact solution (6)

(14) $$ \begin{align} u(t) = e^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} v - ie^{ it \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)} \vert \nabla\vert^\alpha \mathcal{I}_1( t, \mathcal{L},v,p) + \mathcal{O}\left( t^2\vert \nabla\vert^{2 \alpha}{ q_1( v)} \right) \end{align} $$

(for some polynomial $q_1$ ) then takes the form

(15) $$ \begin{align} \begin{split} \mathcal{I}_1( t, \mathcal{L},v,p) & = \int_0^t \left[e^{i \xi \mathcal{L}_{\text{dom}}} p_{\text{dom}}\left(v,\overline{v}\right) \right] p_{\text{low}}(v,\overline{v}) + \mathcal{O}\Big(\xi\mathcal{L}_{\text{low}}\left(\nabla\right){ q_2(v)}\Big)d \xi\\ & = t p_{\text{low}}(v,\overline{v}) \varphi_1\left(i t \mathcal{L}_{\text{dom}} \right) p_{\text{dom}}\left(v,\overline{v}\right) + \mathcal{O}\Big(t^2\mathcal{L}_{\text{low}}\left(\nabla\right){ q_2(v)}\Big) \end{split} \end{align} $$

(for some polynomial $q_2$ ) where for shortness we write $\mathcal {L} = \mathcal {L}\left (\nabla , \frac {1}{\varepsilon }\right )$ and define $\varphi _1(\gamma ) = \gamma ^{-1}\left (e^\gamma -1\right )$ for $\gamma \in \mathbf {C}$ . Plugging (15) into (14) yields for a small time step $\tau $ that

(16) $$ \begin{align} u(\tau) = e^{ i\tau \mathcal{L}} v - \tau ie^{ i\tau \mathcal{L}} \vert \nabla\vert^\alpha &\Big[p_{\text{low}}(v,\overline{v}) \varphi_1\left(i \tau \mathcal{L}_{\text{dom}} \left(\nabla, \frac{1}{\varepsilon}\right)\right) p_{\text{dom}}\left(v,\overline{v}\right) \Big] \notag \\ &\qquad\qquad \qquad + \mathcal{O}\left( \tau^2\vert \nabla\vert^{2 \alpha} q_ 1(v) \right) + \mathcal{O}\Big(\tau^2\vert \nabla\vert^{ \alpha}\mathcal{L}_{\text{low}} \left(\nabla\right)q_2({ v})\Big) \end{align} $$

for some polynomials $q_1, q_2$ . The expansion of the exact solution (16) builds the foundation of the first-order resonance-based discretisation

(17) $$ \begin{align} u^{n+1} = e^{ i\tau \mathcal{L}} u^n - \tau ie^{ i\tau \mathcal{L}} { \vert \nabla\vert^\alpha} \Big[p_{\text{low}}(u^n,\overline{u}^n) \varphi_1\left(i \tau\mathcal{L}_{\text{dom}} \left(\nabla, \frac{1}{\varepsilon}\right) \right) p_{\text{dom}}\left(u^n,\overline{u}^n\right) \Big]. \end{align} $$

Compared to classical linear frequency approximations (cf. Table 1), the main gain of the more involved resonance-based approach (17) is the following: All dominant parts $\mathcal {L}_{\text {dom}}$ are captured exactly in the discretisation, while only the lower-order/nonoscillatory parts $\mathcal {L}_{\text {low}}$ are approximated. Henceforth, within the resonance-based approach (17) the local error only depends on the lower-order, nonoscillatory operator $\mathcal {L}_{\text {low}}$ , while the local error of classical methods involves the full operator $\mathcal {L}$ and, in particular, its dominant part $\mathcal {L}_{\text {dom}}$ . Thus, the resonance-based approach (17) allows us to approximate a more general class of solutions

(18) $$ \begin{align} &u \in \underbrace{ \mathcal{D}\left(\vert \nabla\vert^\alpha \mathcal{L}_{\text{low}}\left(\nabla, \frac{1}{\varepsilon}\right)\right) }_{\text{resonance domain}} \cap \mathcal{D}\left(\vert \nabla\vert^{2\alpha}\right) \notag \\ &\qquad \qquad \supset { \mathcal{D}\left(\vert \nabla\vert^\alpha \mathcal{L}\left(\nabla, \frac{1}{\varepsilon}\right)\right) }\cap \mathcal{D}\left(\vert \nabla\vert^{2\alpha}\right) =\underbrace{ \mathcal{D}\left(\vert \nabla\vert^\alpha \mathcal{L}_{\text{dom}}\left(\nabla, \frac{1}{\varepsilon}\right)\right) }_{\text{classical domain}}\cap \mathcal{D}\left(\vert \nabla\vert^{2\alpha}\right). \end{align} $$

Higher-order resonance-based methods. Classical approximation techniques, such as splitting or exponential integrator methods, can easily be extended to higher order; see, for example, [Reference Hairer, Lubich and Wanner45Reference Hochbruck and Ostermann49Reference Thalhammer80]. The step from a first- to higher-order approximation lies in subsequently employing a higher-order Taylor series expansion to the exact solution

$$ \begin{align*} u(t) = u(0) + t\partial_t u(0) + \ldots + \frac{t^{r}}{r!} \partial_t^{r} u(0) + \mathcal{O} \left( t^{r+1} \partial_{t}^{r+1} u\right). \end{align*} $$

Within this expansion, the higher-order iterations of the oscillations (7) in the exact solution are, however, not resolved but subsequently linearised. Therefore, classical high-order methods are restricted to smooth solutions as their local approximation error in general involves high-order derivatives

(19) $$ \begin{align} \mathcal{O} \left( t^{r+1} \partial_{t}^{r+1} u\right)=\mathcal{O} \left( t^{r+1} \mathcal{L}^{r+1}\left(\nabla, \tfrac{1}{\varepsilon}\right) u\right). \end{align} $$

This phenomenon is also illustrated in Figure 2 where we numerically observe the order reduction of the Strang splitting method (of classical order 2) down to the order of the Lie splitting method (of classical order 1) in case of rough solutions. In particular, we observe that classical high-order methods do not pay off at low regularity as their error behaviour reduces to the one of lower-order methods.

At first glance our resonance-based approach can also be straightforwardly extended to higher order. Instead of considering only the first-order iteration (6) the natural idea is to iterate Duhamel’s formula (5) up to the desired order r; that is, for initial value $u(0)=v$ ,

(20) $$ \begin{align} \begin{split} u(t) &= e^{i t \mathcal{L}} v -i e^{i t \mathcal{L}}\nabla^{\alpha} \int_0^te^{ -i \xi_1 \mathcal{L}} p\left( e^{ i \xi_1 \mathcal{L}} v,e^{- i \xi_1 \mathcal{L}} \overline{v}\right) d\xi_1 \\ &\quad{}-e^{i t \mathcal{L}}\nabla^{\alpha} \int_0^te^{ -i \xi_1 \mathcal{L}} \Big[D_1 p \left( e^{ i \xi_1 \mathcal{L}} v,e^{- i \xi_1 \mathcal{L}} \overline{v}\right) \\ &\quad \cdot e^{ i \xi_1\mathcal{L}}\nabla^{\alpha} \int_0^{\xi_1} e^{ -i \xi_2 \mathcal{L}} p\left( e^{ i \xi_1 \mathcal{L}} v,e^{- i \xi_1 \mathcal{L}} \overline{v}\right) d\xi_2 \Big]d\xi_1 \\ &\quad{}+e^{i t \mathcal{L}}\nabla^{\alpha} \int_0^te^{ -i \xi_1 \mathcal{L}} \Big[D_2p \left( e^{ i \xi_1 \mathcal{L}} v,e^{- i \xi_1 \mathcal{L}} \overline{v}\right) \\ & \quad{} \cdot e^{ -i \xi_1\mathcal{L}}\nabla^{\alpha} \int_0^{\xi_1} e^{ i \xi_2 \mathcal{L}} \overline{p\left( e^{ i \xi_1 \mathcal{L}} v,e^{- i \xi_1 \mathcal{L}} \overline{v}\right) }d\xi_2 \Big]d\xi_1 \\ & \quad{}+ \ldots+\nabla^{\alpha} \int_0^t\nabla^{\alpha} \int_0^\xi \ldots \nabla^{\alpha}\int_0^{\xi_r} d\xi_{r} \ldots d\xi_1 d \xi \end{split} \end{align} $$

where $ D_1 $ (respectively $ D_2 $ ) corresponds to the derivative in the first (respectively second) component of $ p $ . The key idea will then be the following: Instead of linearising the frequency interactions by a simple Taylor series expansion of the oscillatory terms $e^{\pm i \xi _\ell \mathcal {L}}$ (as classical methods would do), we want to embed the dominant frequency interactions of (20) exactly into our numerical discretisation. By neglecting the last term involving the iterated integral of order r, we will then introduce the desired local error $\mathcal {O}\Big (\nabla ^{(r+1)\alpha } t^{r+1}q(u)\Big )$ for some polynomial q.

Compared to the first-order approximation (12), this is much more involved as high-order iterations of the nonlinear frequency interactions need to be controlled. The control of these iterated oscillations is not only a delicate problem on the discrete (numerical) level, concerning accuracy, stability, etc., but already on the continuous level: We have to encode the structure (which strongly depends on the underlying structure of the PDE; that is, the form of operator $\mathcal {L}$ and the shape of nonlinearity p) and at the same time keep track of the regularity assumptions. In order to achieve this in the general setting (1), we will introduce the decorated tree formalism in Subsection 1.2. First, let us first illustrate the main ideas on the example of the cubic periodic Schrödinger equation.

Example 1 (cubic periodic Schrödinger equation). We consider the 1-dimensional cubic Schrödinger equation

(21) $$ \begin{align} i \partial_t u + \partial_x^2 u = \vert u\vert^2 u \end{align} $$

equipped with periodic boundary conditions; that is, $x \in {\mathbf {T}}$ . The latter casts into the general form (1) with

(22) $$ \begin{align} \mathcal{L}\left(\nabla, \tfrac{1}{\varepsilon}\right) = \partial_x^2, \quad \alpha = 0 \quad \text{and}\quad p(u,\overline{u}) =u^2 \overline{u}. \end{align} $$

In the case of cubic NLS, the central oscillatory integral (at first order) takes the form (cf. (7))

(23) $$ \begin{align} \mathcal{I}_1(\tau, \partial_x^2,v) = \int_0^\tau e^{-i s \partial_x^2}\left[ \left( e^{- i s \partial_x^2} \overline{v} \right) \left ( e^{ i s \partial_x^2} v \right)^2\right] d s. \end{align} $$

Assuming that $v\in L^2$ , the Fourier transform $ v(x) = \sum _{k \in {\mathbf {Z}}}\hat {v}_k e^{i k x} $ allows us to express the action of the free Schrödinger group as a Fourier multiplier; that is,

$$ \begin{align*} e^{\pm i t \partial_x^2}v(x) = \sum_{k \in {\mathbf{Z}}} e^{\mp i t k^2} \hat{v}_k e^{i k x}. \end{align*} $$

With this at hand, we can express the oscillatory integral (23) as follows:

(24) $$ \begin{align} \mathcal{I}_1(\tau, \partial_x^2,v) = \sum_{\substack{k_1,k_2,k_3 \in {\mathbf{Z}} \\ -k_1+k_2+k_3 = k} } e^{i k x } \overline{\hat{v}}_{k_1} \hat{v}_{k_2} \hat{v}_{k_3} \int_0^\tau e^{i s \mathscr{F}(k) } ds \end{align} $$

with the underlying resonance structure

(25) $$ \begin{align}{ \mathscr{F}(k) = 2 k_1^2 - 2 k_1 (k_2+k_3) + 2 k_2 k_3}. \end{align} $$

In the spirit of (12) we need to extract the dominant and lower-order parts from the resonance structure (25). The choice is based on the following observation. Note that $2k_1^2$ corresponds to a second-order derivative; that is, with the inverse Fourier transform $\mathcal {F}^{\,-1}$ , we have

$$ \begin{align*} \mathcal{F}^{-1}\left(2k_1^2 \overline{\hat v}_{k_1} \hat v_{k_2} \hat v_{k_3}\right) = \left(- 2\partial_x^2 \overline{v}\right) v^2 \end{align*} $$

while the terms $k_\ell \cdot k_m$ with $\ell \neq m$ correspond only to first-order derivatives; that is,

$$ \begin{align*} \mathcal{F}^{-1}\left(k_1 \overline{\hat v}_{k_1} k_2 \hat v_{k_2} \hat v_{k_3}\right) = -\vert\partial_x v\vert^2 v, \quad \mathcal{F}^{-1}\left( \overline{\hat v}_{k_1} k_2 \hat v_{k_2} k_3 \hat v_{k_3}\right) = - (\partial_x v)^2\overline{v}. \end{align*} $$

This motivates the choice

$$ \begin{align*} \mathscr{F}(k) = \mathcal{L}_{\text{dom}}(k_1) + \mathcal{L}_{\text{low}}(k_1,k_2,k_3) \end{align*} $$

with

(26) $$ \begin{align} {\mathcal{L}_{\text{dom}}(k_1) = 2k_1^2 \quad \text{and}\quad \mathcal{L}_{\text{low}}(k_1,k_2,k_3) = - 2 k_1 (k_2+k_3) + 2 k_2 k_3}. \end{align} $$

In terms of (17) we thus have

(27) $$ \begin{align} \mathcal{L}_{\text{dom}} = - 2\partial_x^2, \quad \quad p_{\text{dom}}(v,\overline{v}) = \overline{v} \quad \text{and}\quad p_{\text{low}}(v,\overline{v}) = v^2 \end{align} $$

and the first-order NLS resonance-based discretisation (17) takes the form

(28) $$ \begin{align} u^{n+1} = e^{ i\tau \partial_x^2} u^n - \tau ie^{ i\tau \partial_x^2} \Big[(u^n)^2 \varphi_1\left(-2 i \tau \partial_x^2 \right) \overline{u}^n \Big]. \end{align} $$

Thanks to (16), we readily see by (26) that the NLS scheme (28) introduces the approximation error

(29) $$ \begin{align} \mathcal{O}\left(\tau^2 \mathcal{L}_{\text{low}}q(u)\right)= \mathcal{O}\left(\tau^2 \partial_xq(u)\right) \end{align} $$

for some polynomial q in u. Compared to the error structure of classical discretisation techniques, which involve the full and thus dominant operator $\mathcal {L}_{\text {dom}} = \partial _x^2$ , we thus gain one derivative with the resonance-based scheme (28). This favourable error at low regularity is underlined in Figure 3.

Figure 3 Error versus step size (double logarithmic plot). Comparison of classical and resonance-based schemes for the cubic Schrödinger equation (21) with $H^2$ initial data.

In Example 1 we illustrated the idea of the resonance-based discretisation on the cubic periodic Schrödinger equation in one spatial dimension. In order to control frequency interactions in the general setting (1) in arbitrary dimensions $d\geq 1$ up to arbitrary high order, we next introduce our decorated tree formalism.

1.2 Main idea of decorated trees for high-order resonance-based schemes

The iteration of Duhamel’s formulation (20) can be expressed using decorated trees. We are interested in computing the iterated frequency interactions in (20). This motivates us to express the latter in Fourier space. Let $ r $ be the order of the scheme and let us assume that we truncate (20) at this order. Its kth Fourier coefficient at order $ r $ is given by

(30) $$ \begin{align} U_{k}^{r}(\tau, v) = \sum_{T\kern-1pt\in\kern0.5pt {\cal V}^r_k} \frac{\Upsilon^{p}(T)(v)}{S(T)} \left( \Pi T \right)(\tau), \end{align} $$

where $ {\cal V}^r_k $ is a set of decorated trees which incorporate the frequency k, $ S(T) $ is the symmetry factor associated to the tree $ T $ , $ \Upsilon ^{p}(T) $ is the coefficient appearing in the iteration of Duhamel’s formulation and $ (\Pi T)(t) $ represents a Fourier iterated integral. The exponent r in $ {\cal V}^r_k $ means that we consider only trees of size $ r +1 $ which are the trees producing an iterated integral with $ r + 1$ integrals. The decorations that need to be put on the trees are illustrated in Example 2.

The main difficulty then lies in developing for every $T \in {\cal V}^r_k$ a suitable approximation to the iterated integrals $ (\Pi T)(t) $ with the aim of minimising the local error structure (in the sense of regularity). In order to achieve this, the key idea is to embed – in the spirit of (12) – the underlying resonance structure of the iterated integrals $ (\Pi T)(t) $ into the discretisation.

Example 2 (cubic periodic Schrödinger equation with decorated trees). When $r=2$ , decorated trees for cubic NLS are given by

(31)

where on the nodes we encode the frequencies such that they add up depending on the edge decorations. The root has no decoration. For example, in $T_1$ the two extremities of the blue edge have the same decoration given by $ -k_1 + k_2 + k_3 $ where the minus sign comes from the dashed edge. Therefore, $ {\cal V}^r_k $ contains infinitely many trees (finitely many shapes but infinitely many ways of splitting up the frequency $ k $ among the branches). An edge of type encodes a multiplication by $ e^{-i \tau k^2} $ where k is the frequency on the nodes adjacent to this edge. An edge of type encodes an integration in time of the form

$$ \begin{align*} \int_0^{\tau} e^{i s k^2} \cdots d s. \end{align*} $$

In fact, $r +1$ , the truncation parameter, corresponds to the maximum number of integration in time; that is, the number of edges with type . The dashed dots on the edges correspond to a conjugate and a multiplication by $(-1)$ applied to the frequency at the top of this edge. Then, if we apply the map $ \Pi $ (which encodes the oscillatory integrals in Fourier space; see Subsection 3.1) to these trees, we obtain

(32) $$ \begin{align} \begin{split} (\Pi T_0)(\tau) & = e^{-i \tau k^2}, \\ (\Pi T_1)(\tau) & = - i e^{-i \tau k^2} \int_0^\tau e^{i s k^2}\left[ \left( e^{ i s k_1^2} \right) \left ( e^{ -i s k_2^2} \right) \left ( e^{ -i s k_3^2} \right) \right] d s \\ & = - i e^{-i \tau k^2} \int_0^{\tau} e^{is \mathscr{F}(k)} ds \\ (\Pi T_2)(\tau) & = -i e^{-i \tau k^2} \int_0^\tau e^{i s k^2}\left[ \left( e^{ i s k_4^2} \right) \Big( (\Pi T_1)(s) \Big) \left ( e^{ - i s k_5^2} \right) \right] d s \\ (\Pi T_3)(\tau) & = -i e^{-i \tau k^2} \int_0^\tau e^{i s k^2}\left[ \Big( \overline{(\Pi T_1)(s)} \Big)\left ( e^{ -i s k_4^2} \right) \left ( e^{ -i s k_5^2} \right) \right] d s \end{split} \end{align} $$

where the resonance structure $\mathscr {F}(k)$ is given in (25). One has the constraints $k= -k_1 +k_2 +k_3$ for $T_1$ , $k= -k_1 + k_2 + k_3 -k_4 + k_5$ for $ T_2 $ and $k= k_1 - k_2 - k_3 +k_4 + k_5$ for $ T_3 $ . Using the definitions in Section 4, one can compute the following coefficients:

$$ \begin{align*} \begin{split} \frac{\Upsilon^p(T_0)(v)}{S(T_0)} & = \hat v_k, \quad \frac{\Upsilon^p(T_1)(v)}{S(T_1)} = \bar{\hat{v}}_{k_1} \hat{v}_{k_2} \hat{v}_{k_3} \\ \frac{\Upsilon^p(T_2)(v)}{S(T_2)} & = 2 \overline{\hat{v}}_{k_1} \hat v_{k_2} \hat v_{k_3} \overline{\hat{v}}_{k_4} \hat v_{k_5}, \quad \frac{\Upsilon^p(T_3)(v)}{S(T_3)} = \hat v_{k_1} \overline{\hat{v}}_{k_2} \overline{\hat{v}}_{k_3} \hat v_{k_4} \hat v_{k_5} \end{split} \end{align*} $$

which together with the character $ \Pi $ encode fully the identity (30).

Our general scheme is based on the approximation of $(\Pi T)(t)$ for every tree in ${\cal V}_k^r$ . This approximation is given by a new map of decorated trees denoted by $\Pi ^{n,r}$ where r is the order of the scheme and n corresponds to the a priori assumed regularity of the initial value v. This new character $\Pi ^{n,r}$ will embed the dominant frequency interactions and neglect the lower-order terms in the spirit of (12). Our general scheme will thus take the form

(33) $$ \begin{align} U_{k}^{n,r}(\tau, v) = \sum_{T\kern-1pt\in\kern0.5pt {\cal V}^r_k} \frac{\Upsilon^{p}(T)(v)}{S(T)} \left( \Pi^{n,r} T \right)(\tau) \end{align} $$

where the map $ \Pi ^{n,r} T $ is a low regularity approximation of order $ r $ of the map $ \Pi T $ in the sense that

(34) $$ \begin{align} \left(\Pi T - \Pi^{n,r} T \right)(\tau) = \mathcal{O}\left( \tau^{r+2} \mathcal{L}^{r}_{\tiny{\text{low}}}(T,n) \right). \end{align} $$

Here $\mathcal {L}^{r}_{\tiny {\text {low}}}(T,n)$ involves all lower-order frequency interactions that we neglect in our resonance-based discretisation. At first order this approximation is illustrated in (15). The scheme (33) and the local error approximations (34) are the main results of this work (see Theorem 4.8). Let us give the main ideas on how to obtain them.

The approximation $ \Pi ^{n,r} $ is constructed from a character $ \Pi ^n $ defined on the vector space $ {\mathcal {H}} $ spanned by decorated forests taking values in a space $ {{\cal C}} $ which depends on the frequencies of the decorated trees (see, e.g., (31) in case of NLS). However, we will add at the root the additional decoration r which stresses that this tree will be an approximation of order r. For this purpose we will introduce the symbol ${\cal D}^r$ (see, e.g., (35) for $T_1$ of NLS). Indeed, we disregard trees which have more integrals in time than the order of the scheme. In particular, we note that $ \Pi ^n {\cal D}^r(T) = \Pi ^{n,r} T$ .

The map $ \Pi ^n $ is defined recursively from an operator $ {{\cal K}} $ which will compute a suitable approximation (matching the regularity of the solution) of the integrals introduced by the iteration of Duhamel’s formula. This map $ {{\cal K}} $ corresponds to the high-order counterpart of the approach described in Subsection 1.1: It embeds the idea of singling out the dominant parts and integrating them exactly while only approximating the lower-order terms, allowing for an improved local error structure compared to classical approaches. The character $ \Pi ^n $ is the main map for computing the numerical scheme in Fourier space.

Example 3 (cubic periodic Schrödinger equation: computation of  $\Pi ^n$ ). We consider the decorated trees ${\cal D}^r(\bar T_1)$ and $\bar T_1$ given by

(35)

One can observe that $ (\Pi T_1)(t) = e^{-i k^2 t} (\Pi \bar T_1)(t). $ We will define recursively two maps $\mathscr {F}_{\tiny {\text {dom}}} $ and $ \mathscr {F}_{\tiny {\text {low}}} $ (see Definition 2.6) on decorated trees that compute the dominant and the lower part of the nonlinear frequency interactions within the oscillatory integral $ (\Pi \bar T_1)(t) $ . In this example, one gets back the values already computed in (26); that is,

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) = \mathcal{L}_{\tiny{\text{dom}}}(k_1), \quad \mathscr{F}_{\tiny{\text{low}}}(\bar T_1) =\mathcal{L}_{\tiny{\text{low}}}(k_1,k_2,k_3). \end{align*} $$

Moreover, the dominant part of $T_1$ is due to the observation that $(\Pi T_1)(t) = e^{-i k^2 t} (\Pi \bar T_1)(t)$ given by

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}( T_1) = -k^2 +\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) , \end{align*} $$

because the tree $T_1$ does not start with an intregral in time. Then, one can write

$$ \begin{align*} (\Pi \bar T_1)(t) = -i \int_0^\tau e^{i s\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } e^{i s\mathscr{F}_{\tiny{\text{low}}}(\bar T_1) } ds \end{align*} $$

and Taylor expand around $0$ the lower-order term; that is, the factor containing $ \mathscr {F}_{\tiny {\text {low}}}(\bar T_1)$ . The term $\Pi ^{n,1} \bar T_1 = \Pi ^n {\cal D}^1(\bar T_1)$ is then given by

(36) $$ \begin{align} ( \Pi^{n,1} \bar T_1)(t) = - i \int_0^\tau e^{i s\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } ds + \mathscr{F}_{\tiny{\text{low}}}(\bar T_1) \int_0^\tau s e^{i s \mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } ds. \end{align} $$

One observes that we obtain terms of the form $ \frac {P}{Q} e^{i R} $ where $P,Q, R$ are polynomials in the frequencies $ k_1, k_2, k_3 $ . Linear combinations of these terms are actually the definition of the space $ {{\cal C}} $ . For the local error, one gets

(37) $$ \begin{align} ( \Pi^{n,1} \bar T_1)(t) - ( \Pi \bar T_1)(t) = {{\cal O}}( t^3 \mathscr{F}_{\tiny{\text{low}}} (\bar T_1)^2 ). \end{align} $$

Here the term $ \mathscr {F}_{\tiny {\text {low}}} (\bar T_1)^2$ corresponds to the regularity that one has to impose on the solution. One can check by hand that the expression of $ \Pi ^{n,1} \bar T_1$ can be mapped back to the physical space. Such a statement will in general hold true for the character $ \Pi ^n $ ; see Proposition 3.18. This will be important for the practical implementation of the new schemes; see also Remark 1.3. We have not used n in the description of the scheme yet. In fact, it plays a role in the expression of $ \Pi ^{n,1} \bar T_1 $ . One has to compare $ n $ with the regularity required by the local error (37) introduced by the polynomial $ \mathscr {F}_{\tiny {\text {low}}} (\bar T_1)^2 $ but also with the term $\mathscr {F}_{\tiny {\text {dom}}}(\bar T_1)^2 $ . Indeed, if the initial value is regular enough, we may want to Taylor expand all of the frequencies – that is, even the dominant parts – in order to get a simpler scheme; see also Remark 1.1.

In order to obtain a better understanding of the error introduced by the character $ \Pi ^n $ , one needs to isolate each interaction. Therefore, we will introduce two characters $ \hat \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}} $ and $ A^n : {\mathcal {H}} \rightarrow \mathbf {C} $ such that

(38) $$ \begin{align} \Pi^n = \left( \hat \Pi^n \otimes A^n \right) \Delta \end{align} $$

where $ \Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+ $ is a coaction and $ ({\mathcal {H}},\Delta ) $ is a right comodule for a Hopf algebra $ {\mathcal {H}}_+ $ equipped with a coproduct ${\Delta ^{\!+}} $ and an antipode $ {\mathcal {A}} $ . In fact, on can show that

(39) $$ \begin{align} \hat \Pi^n = \left( \Pi^n \otimes \left( {\mathcal{Q}} \circ \Pi^n {\mathcal{A}} \cdot \right)(0) \right) \Delta, \quad A^n = ({\mathcal{Q}} \circ \Pi^n \cdot)(0) \end{align} $$

where $ \Pi ^n $ is extended to a character on $ {\mathcal {H}}_+ $ and $ {\mathcal {Q}} $ is a projection defined on $ {{\cal C}} $ which keeps only the terms with no oscillations. The identity (39) can be understood as a Birkhoff type factorisation of $ \hat \Pi ^n $ using the character $ \Pi ^n $ . This identity is also reminiscent in the main results obtained for singular SPDEs [Reference Bruned, Hairer and Zambotti13] where two twisted antipodes play a fundamental role providing a variant of the algebraic Birkhoff factorisation.

Example 4 (cubic periodic Schrödinger equation: Birkhoff factorisation). Integrating the first term in (36) exactly yields two contributions:

$$ \begin{align*} \int_0^\tau e^{i s\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } ds = \frac{e^{i\tau\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) }}{i\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } - \frac{1}{i\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) }. \end{align*} $$

Plugging these two terms into $(\Pi T_2)(\tau )$ defined in (32), we see that we have to control the following two terms:

(40) $$ \begin{align} \begin{split} - e^{-i \tau k^2} & \int_0^\tau e^{i s k^2}\left[ \left( e^{ i s k_4^2} \right) \Big( \frac{e^{i s\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) - is \bar k^2 } }{i\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } \Big) \left ( e^{- i s k_5^2} \right) \right] d s \\ - e^{-i \tau k^2} & \int_0^\tau e^{-i s k^2}\left[ \left( e^{ i s k_4^2} \right) \Big( - \frac{e^{-i s \bar k^2 }}{i\mathscr{F}_{\tiny{\text{dom}}}(\bar T_1) } \Big) \left ( e^{ -i s k_5^2} \right) \right] d s \end{split} \end{align} $$

where $ \bar k = -k_1 + k_2 + k_3$ . The frequency analysis is needed again for approximating the time integral and defining an approximation of $(\Pi T_2)(\tau ) $ . One can see that the dominant part of these two terms may differ. This implies that one can get two different local errors for the approximation of these two terms; the final local error is the maximum between the two. At this point, we need an efficient algebraic structure for dealing with all of these frequency interactions in the iterated integrals. We first consider a character $ \hat \Pi ^n $ that keeps only the main contribution; that is, the second term of (40). For any decorated tree $ T $ , one expects $ \hat \Pi ^n $ to be of the form

$$ \begin{align*} (\hat \Pi^n {\cal D}^r(T))(t) = B^n({\cal D}^r(T))(t) e^{it\mathscr{F}_{\tiny{\text{dom}}}( T) } \end{align*} $$

where $ B^n({\cal D}^r(T))(t) $ is a polynomial in $ t $ depending on the decorated tree $ T $ . The character $ \hat \Pi ^n $ singles out oscillations by keeping at each iteration only the nonzero one. This separation between the various oscillations can be encoded via the Butcher–Connes–Kreimer coaction $ \Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+ $ . An example of computation is given below:

where $ \ell = - k_1 + k_2 + k_3$ . The space ${\mathcal {H}}_+$ corresponds to the forest of planted trees where for each planted tree the edge connecting the root to the rest of the tree must be blue (an integration in time). Only blue edges are cut (located on the right-hand side of the tensor product) and the trunk is on the left-hand side. The extra terms missing in the computation correspond to the higher-order terms introduced by the Taylor approximation. Indeed, one plays with decorations introducing an mth derivative on the blue edges cut denoted by $ \hat {\cal D}^{(r,m)} $ and decorations on the nodes where the edges were previously attached. A node of the form

in the example above corresponds to the frequency $ \ell $ and the monomial $ \lambda $ . The length of the Taylor extension is dictated by the order of the scheme $ r $ . The operator $ \hat {\cal D}^{(r,m)} $ is nonzero only if $ m \leq r+1 $ . The formulae (38) and (3.8) give the relation between $ \Pi ^n $ and $ \hat \Pi ^n $ which can be interpreted as a Birkhoff factorisation with the explicit formula (3.8) for $ A^n $ . Such a factorisation is new and does not seem to have an equivalent in the literature. It is natural to observe this factorisation in this context: The integration in time $\int ^{t}_0 \ldots ds$ gives two different frequency interactions that can be controlled via a projection $ {\mathcal {Q}} $ which needs to be iterated deeper in the tree. This will be the equivalent of the Rota–Baxter map used for such a type of factorisation.

The coproduct $ {\Delta ^{\!+}} $ and the coaction $ \Delta $ are extremely close in spirit to the ones defined for the recentring in [Reference Hairer47Reference Bruned, Hairer and Zambotti13]. Indeed, for designing a numerical scheme, we need to perform Taylor expansions, and these two maps perform them at the level of the algebra. The main difference with the tools used for singular SPDEs [Reference Bruned, Hairer and Zambotti13] is the length of the Taylor expansion which is now dictated by the order of the scheme.

The structure we propose in Section 2 is new and reveals the universality of deformed Butcher–Connes–Kreimer coproducts which appear in [Reference Bruned, Hairer and Zambotti13]. The nondeformed version of this map comes from the analysis of B-series in [Reference Butcher16Reference Chartier, Hairer and Vilmart26Reference Calaque, Ebrahimi-Fard and Manchon19], which is itself an extension of the Connes–Kreimer Hopf algebra of rooted trees [Reference Connes and Kreimer31Reference Connes and Kreimer32] arising in perturbative quantum field theory (QFT) and noncommutative geometry.

One can notice that our approximation $ \Pi ^{n,r} $ depends on $ n $ which has to be understood as the regularity we assume a priori on the solution. We design our framework such that for smooth solutions the numerical schemes are simplified, recovering in the limit classical linearised approximations as in Table 1.

Remark 1.1. The term $ \mathcal {L}^{r}_{\tiny {\text {low}}}(T,n) $ in the approximation (34) is obtained by performing several Taylor expansions. Depending on the value $ n $ , we get different numerical schemes (see also the applications in Section 5). In the sequel, we focus on two specific values of $ n $ associated to two particular schemes. We consider $ n^{r}_{\tiny {\text {low}}}(T) $ and $ n^{r}_{\tiny {\text {full}}}(T) $ given by

$$ \begin{align*} n^{r}_{\tiny{\text{low}}}(T) = \deg(\mathcal{L}^{r}_{\tiny{\text{low}}}(T)), \quad n^{r}_{\tiny{\text{full}}}(T) = \deg(\mathcal{L}^{r}_{\tiny{\text{full}}}(T)), \end{align*} $$

where $ \mathcal {L}^{r}_{\tiny {\text {low}}}(T) $ corresponds to the error obtained when we integrate exactly the dominant part $ \mathcal {L}_{\tiny {\text {dom}}}(T)$ and Taylor expand only the lower-order part $ \mathcal {L}_{\tiny {\text {low}}}(T) $ , while the term $\mathcal {L}^{r}_{\tiny {\text {full}}}(T)$ corresponds to the error one obtains when we Taylor expand the full operator $\mathcal {L}(T) = \mathcal {L}_{\tiny {\text {dom}}}(T) + \mathcal {L}_{\tiny {\text {low}}}(T) $ . One has

(41) $$ \begin{align} \deg(\mathcal{L}^{r}_{\tiny{\text{low}}}(T,n)) = \left\{ \begin{array}{lll} \displaystyle n^{r}_{\tiny{\text{low}}}(T), & \quad \text{if }\displaystyle \, n \leq n^{r}_{\tiny{\text{low}}}(T) , \\[2pt] n, & \quad \text{if }\displaystyle \, n^{r}_{\tiny{\text{low}}}(T) \leq n \leq n^{r}_{\tiny{\text{full}}}(T) , \\[2pt] \displaystyle n^{r}_{\tiny{\text{full}}}(T), & \quad \text{if }\displaystyle \, n \geq n^{r}_{\tiny{\text{full}}}(T). \\ \end{array} \right. \end{align} $$

At the level of the scheme, we get

(42) $$ \begin{align} \Pi^{n,r} T = \left\{ \begin{array}{lll} \displaystyle \Pi_{\tiny{\text{low}}}^{r} T, & \quad \text{if }\displaystyle \, n \leq n^{r}_{\tiny{\text{low}}}(T) , \\[2pt] \displaystyle \Pi^{n,r} T , & \quad \text{if }\displaystyle \, n^{r}_{\tiny{\text{low}}}(T) \leq n \leq n^{r}_{\tiny{\text{full}}}(T) , \\[2pt] \displaystyle \Pi_{\tiny{\text{full}}}^{r} T , & \quad \text{if } \displaystyle \, n \geq n^{r}_{\tiny{\text{full}}}(T) \\ \end{array} \right. \end{align} $$

where we call $ \Pi _{\tiny {\text {low}}}^{r} T $ the minimum regularity resonance-based scheme. This scheme corresponds to the minimisation of the local error and we can observe a plateau. Indeed, if $ n $ is too small, then by convention we get this scheme. This could be the case if one does not compute the minimum regularity needed a priori.

The other scheme $ \Pi _{\tiny {\text {full}}}^{r} T $ corresponds to a classical exponential type discretisation, where enough regularity is assumed such that the dominant components of the iterated integrals can also be expanded into a Taylor series as in (19). Then, we observe a second plateau: indeed, assuming more regularity will not change the scheme as we have already Taylor-expanded all the components.

Compared to $ \Pi _{\tiny {\text {low}}}^{r} T $ , the scheme $ \Pi _{\tiny {\text {full}}}^{r} T $ is in general much simpler as no nonlinear frequency interactions are taken into account. This comes at the cost that a smaller class of equations can be solved as much higher regularity assumptions are imposed.

Between these two schemes lies a large class of intermediate schemes $ \Pi ^{n,r} T $ which we call low regularity resonance-based schemes. They take advantage of Taylor expanding a bit more when more regularity is assumed. Therefore, the complexity of the schemes is decreasing as $n $ increases; see also Section 5. We can represent these different regimes through the diagram below.

Remark 1.2. Within our framework we propose a stabilisation technique. This will allow us to improve previous higher-order attempts breaking formerly imposed order barriers of higher-order resonance-based schemes, such as the order reduction down to $3/2$ suggested for Schrödinger equations in dimensions $d\geq 2$ in [Reference Knöller, Ostermann and Schratz60]. Details are given in Remark 3.2 as well as Section 5.

Remark 1.3. The aim is to choose the central approximation $\Pi ^{n,r} T$ as an interplay between optimising the local error in the sense of regularity while allowing for a practical implementation. We design our schemes in such a way that products of functions can always be mapped back to physical space. In practical computations, this will allow us to benefit from the fast Fourier transform (FFT) with computational effort of order $\mathcal {O}\left (\vert K\vert ^d \text {log}\vert K\vert ^d\right )$ in dimension d, where K denotes the highest frequency in the discretisation. However, it comes at the cost that the approximation error (34) involves lower-order derivatives. If, on the other hand, we would embed all nonlinear frequency interactions into the discretisation, the resulting schemes would need to be carried out fully in Fourier space, causing large memory and computational effort of order $\mathcal {O}\left (K^{d \cdot \text {deg}{p}}\right )$ , where $\text {deg}(p)$ denotes the degree of the nonlinearity p.

Remark 1.4. For notational simplicity, we focus on equations with polynomial nonlinearities (cf. (1)). Nevertheless, our scheme (33) allows for a generalisation to nonpolynomial nonlinearities of type

$$ \begin{align*}f(u) g(\overline{u})\end{align*} $$

for smooth functions f and g. In the latter case, the iteration of Duhamel’s formula boils down to a two-step algorithm. More precisely, imagine that we got a first expansion of the form $ e^{i s\mathcal {L}} v + A(v,s) $ where $ A(v,s) $ is a linear combination of iterated integrals. Then, when iterating Duhamel’s formula we need to plug this expansion into the nonlinearity and perform a Taylor expansion around the point $ e^{is \mathcal {L}}v $ :

$$ \begin{align*} f(e^{i s\mathcal{L}} v + A(v,s)) = \sum_{m \leq r} \frac{A(v,s)^m}{m!} f^{(m)}(e^{i s\mathcal{L}} v ) + \mathcal{O}(A(v,s)^{r+1}). \end{align*} $$

Carrying out the same manipulation for $ g(\overline {e^{i s\mathcal {L}} v + A(v,s)}), $ we end up with terms of type

$$ \begin{align*}\frac{A(v,s)^m}{m!} f^{(m)}(e^{i s\mathcal{L}} v ) \frac{\overline{A(v,s)}^n}{n!} g^{(n)}({e^{- i s\mathcal{L}}\overline{v}}). \end{align*} $$

At this point we cannot directly write down our resonance-based scheme due to the fact that the oscillations are still encapsulated inside $ f $ and g. In order to control these oscillations and their nonlinear interactions, we need to pull the oscillatory phases $ e^{\pm is\mathcal {L}} $ out of f and g. This is achieved via expansions of the form

$$ \begin{align*} f(e^{is \mathcal{L}}v) = \sum_{\ell \leq r} \frac{s^{\ell}}{\ell!} e^{is \mathcal{L}} \mathcal{C}^{\ell}[f,\mathcal{L}](v) + \mathcal{O}(s^{r+1} \mathcal{C}^{r+1}[f,\mathcal{L}](v)) \end{align*} $$

where $\mathcal {C}^{\ell }[f,\mathcal {L}]$ denote nested commutators which in general require (much) less regularity than powers of the full operator $\mathcal {L}^\ell $ . After these two linearisation steps, we are able to use the same machinery that leads to the construction of our scheme (33).

Such commutators were also recently exploited in [Reference Rousset and Schratz74] for second-order methods.

1.3 Outline of the article

Let us give a short review of the content of this article. In Section 2, we introduce the general algebraic framework by first defining a suitable vector space of decorated forests $ \hat {\mathcal {H}} $ . Next, we define the dominant frequencies of a decorated forest (see Definition 2.6) and show that one can map them back into physical space (see Corollary 2.9), which will be important for the efficiency of the numerical schemes (cf. Remark 1.3). Then, we introduced two spaces of decorated forests $ {\mathcal {H}}_+ $ and ${\mathcal {H}}$ . The latter $ {\mathcal {H}} $ is used for describing approximated iterated integrals. The main difference with the previous space is that now we project along the order $ r $ of the method. We define the maps for the coaction $ \Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+ $ and the coproduct $ {\Delta ^{\!+}} : {\mathcal {H}}_+ \rightarrow {\mathcal {H}}_+ \otimes {\mathcal {H}}_+ $ in (65) and (66). In addition, we provide a recursive definition for them in (69). We prove in Proposition 2.15 that these maps give a right-comodule structure for $ {\mathcal {H}} $ over the Hopf algebra $ {\mathcal {H}}_+ $ . Moreover, we get a simple expression for the antipode $ {\mathcal {A}} $ in Proposition 80.

In Section 3, we construct the approximation of the iterated integrals given by the character $ \Pi : \hat {\mathcal {H}} \rightarrow {{\cal C}} $ (see (82)) through the character $ \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}} $ (see (83)). The main operator used for the recursive construction is $ {{\cal K}} $ given in Definition 3.1. We introduce a new character $ \hat \Pi ^n : {\mathcal {H}} \rightarrow {{\cal C}} $ through a Birkhoff type factorisation obtained from the character $ \Pi ^n $ (see Proposition 3.8). Thanks to $ \hat \Pi ^n $ , we are able to conduct the local error analysis and show one of the main results of the article: the error estimate on the difference between $ \Pi $ and its approximation $ \Pi ^n $ (see Theorem 3.17). In Section 4, we introduce decorated trees stemming from Duhamel’s formula via the rules formalism (see Definition 4.2). Then, we are able to introduce the general scheme (see Definition 115) and conclude on its local error structure (see Theorem 4.8).

In Section 5 we illustrate the general framework on various applications and conclude in Subsection 5.4 with numerical experiments underlying the favourable error behaviour of the new resonance-based schemes for nonsmooth and, in certain cases, even for smooth solutions.

2 General framework

In this section, we present the main algebraic framework which will allow us to develop and analyse our general numerical scheme. We start by introducing decorated trees that encode the oscillatory integrals. Decorations on the edges represent integrals in time and some operators stemming from Duhamel’s formula. In addition, we impose decorations on the nodes for the frequencies and potential monomials. We will compute the corresponding dominant and lower-order frequency interactions associated to these trees via the recursive maps $\mathscr {F}_{\tiny {\text {dom}}} $ and $ \mathscr {F}_{\tiny {\text {low}}} $ given in Definition 2.6. These maps are chosen such that the solution is approximated at low regularity in Fourier space with the additional property that the approximation can be mapped back to physical space. The latter will allow for an efficient practical implementation of the new scheme, see Remark 1.3.

The second part of this section focuses on a different space of decorated trees that we name approximated decorated trees. The main difference from the trees previously introduced is the additional root decoration by some integer $ r $ . The approximated trees have to be understood as an abstract version of an approximation of order $ r $ of the corresponding oscillatory integral. In order to construct our low regularity scheme we want to carry out an abstract Taylor expansions of the time integrals at the level of these approximated trees in the spirit of (12): We will Taylor expand only the lower parts of the frequency interactions while integrating the dominant part exactly. For these operations, we need a deformed Butcher–Connes–Kreimer coproduct in the spirit of the one which has been introduced for singular SPDEs. We consider a coproduct ${\Delta ^{\!+}} $ and a coaction $ \Delta $ with a nonrecursive (see (65) and (66)) and a recursive definition (see (69)). We show the usual coassociativity/compatibility properties in Proposition 2.14. In the end we get a Hopf algebra and a comodule structures on these new spaces of approximated decorated trees. The antipode comes for free in this context since the Hopf algebra is connected, see Proposition 2.15. The main novelty of this general algebraic framework is the merging of two different structures that appear in dispersive PDEs (frequency interactions) and singular SPDEs (abstract Taylor expansion). They form the central part of the scheme – controlling the underlying oscillations and performing Taylor approximations. Within this construction we need to introduce new objects that were not considered before in such generality.

2.1 Decorated trees and frequency interactions

We consider a set of decorated trees following the formalism developed in [Reference Bruned, Hairer and Zambotti13]. These trees will encode the Fourier coefficients of the numerical scheme.

We assume a finite set $ {\mathfrak {L}}$ and frequencies $ k_1,\ldots ,k_n \in {\mathbf {Z}}^{d} $ . The set $ {\mathfrak {L}}$ parametrises a set of differential operators with constant coefficients, whose symbols are given by the polynomials $ (P_{{\mathfrak {t}}})_{{\mathfrak {t}} \in {\mathfrak {L}}} $ . We define the set of decorated trees $ \hat {\mathcal {T}} $ as elements of the form $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}}) $ where

  • $ T $ is a nonplanar rooted tree with root $ \varrho _T $ , node set $N_T$ and edge set $E_T$ . We denote the leaves of $ T $ by $ L_T $ . $ T $ must also be a planted tree, which means that there is only one edge outgoing the root.

  • The map $ {\mathfrak {e}} : E_T \rightarrow {\mathfrak {L}} \times \lbrace 0,1\rbrace $ are edge decorations.

  • The map $ {\mathfrak {n}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow \mathbf {N} $ are node decorations. For every inner node $ v$ , this map encodes a monomial of the form $ \xi ^{{\mathfrak {n}}(v)} $ where $ \xi $ is a time variable.

  • The map $ {\mathfrak {f}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow {\mathbf {Z}}^{d}$ are node decorations. These decorations are frequencies that satisfy for every inner node $ u $ :

    (43) $$ \begin{align} (-1)^{\mathfrak{p}(e_u)}{\mathfrak{f}}(u) = \sum_{e=(u,v) \in E_T} (-1)^{\mathfrak{p}(e)} {\mathfrak{f}}(v) \end{align} $$
    where $ {\mathfrak {e}}(e) = ({\mathfrak {t}}(e),\mathfrak {p}(e)) $ and $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ . From this definition, one can see that the node decorations $ ({\mathfrak {f}}(u))_{u \in L_T} $ determine the decoration of the inner nodes. We assume that the node decorations at the leaves are linear combinations of the $ k_i $ with coefficients in $ \lbrace -1,0,1 \rbrace $ .
  • We assume that the root of $ T $ has no decoration.

When the node decoration $ {\mathfrak {n}} $ is zero, we will denote the decorated trees $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ as $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} = (T,{\mathfrak {f}},{\mathfrak {e}}) $ . The set of decorated trees satisfying such a condition is denoted by $ \hat {\mathcal {T}}_0 $ . We say that $ \bar T_{\bar {\mathfrak {e}}}^{\bar {\mathfrak {f}}} $ is a decorated subtree of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} \in \hat {\mathcal {T}}_0 $ if $ \bar T $ is a subtree of $ T $ and the restriction of the decorations $ {\mathfrak {f}}, {\mathfrak {e}} $ of $ T $ to $ \bar T $ are given by $ \bar {\mathfrak {f}} $ and $ \bar {\mathfrak {e}} $ . Notice that because trees considered in this framework are always planted, we look only at subtrees that are planted. A planted subtree of $ T $ is of the form $ T_e $ where $ e \in E_T $ and $ T_e $ corresponds to the tree above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ .

Example 5. Below, we give an example of a decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ where the edges are labelled with numbers from $ 1 $ to $ 7 $ and the set $ N_T \setminus \lbrace \varrho _T\rbrace $ is labelled by $\lbrace a,b,c,d,e,f,g \rbrace $ :

Remark 2.1. The structure imposed on the node decorations (43) is close to the one used in [Reference Christ27Reference Gubinelli43Reference Guo, Kwon and Oh44]. But in these works, the trees were designed only for one particular equation. In our framework, we cover a general class of dispersive equations by having more decorations on the edges given by $ {\mathfrak {L}} \times \lbrace 0,1 \rbrace $ . The set $ {\mathfrak {L}} $ keeps track of the differential operators in Duhamel’s formulation. The second edge decoration allows us to compute an abstract conjugate on the trees given in (111).

We denote by $\hat H $ (respectively $ \hat H_0 $ ) the (unordered) forests composed of trees in $ \hat {\mathcal {T}} $ (respectively $ \hat {\mathcal {T}}_0 $ ; including the empty forest denoted by ${\mathbf {1}}$ ). Their linear spans are denoted by $\hat {\mathcal {H}} $ and $ \hat {\mathcal {H}}_0 $ . We extend the definition of decorated subtrees to forests by saying that $ T $ is a decorated subtree of the decorated forest $ F $ if their exists a decorated tree $ \bar T$ in $ F $ such that $ T $ is a decorated subtree of $ \bar T $ . The forest product is denoted by $ \cdot $ and the counit is $ {\mathbf {1}}^{\star } $ which is nonzero only on the empty forest.

In order to represent these decorated trees, we introduce a symbolic notation. An edge decorated by $ o = ({\mathfrak {t}},{\mathfrak {p}}) $ is denoted by $ {\cal I}_{o} $ . The symbol $ {\cal I}_{o}(\lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow \hat {\mathcal {H}} $ is viewed as the operation that merges all of the roots of the trees composing the forest into one node decorated by $(\ell ,k) \in \mathbf {N} \times {\mathbf {Z}}^{d} $ . We obtain a decorated tree which is then grafted onto a new root with no decoration. If the condition (43) is not satisfied on the argument, then ${\cal I}_{o}( \lambda _{k}^{\ell } \cdot )$ gives zero. If $ \ell = 0 $ , then the term $ \lambda _{k}^{\ell } $ is denoted by $ \lambda _{k} $ as a shorthand notation for $ \lambda _{k}^{0} $ . When $ \ell = 1 $ , it will be denoted by $ \lambda _{k}^{1} $ . The forest product between $ {\cal I}_{o_1}( \lambda ^{\ell _1}_{k_1}F_1) $ and $ {\cal I}_{o_2}( \lambda ^{\ell _2}_{k_2}F_2) $ is given by

$$ \begin{align*} {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2) := {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) \cdot {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2). \end{align*} $$

Example 6. The following symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}(1),{\mathfrak{p}}(1))}( \lambda^{{\mathfrak{n}}(a)}_{{\mathfrak{f}}(a)}{\cal I}_{({\mathfrak{t}}(2),{\mathfrak{p}}(2))}( \lambda^{{\mathfrak{n}}(b)}_{{\mathfrak{f}}(b)}){\cal I}_{({\mathfrak{t}}(3),{\mathfrak{p}}(3))}( \lambda^{{\mathfrak{n}}(c)}_{{\mathfrak{f}}(c)})) \end{align*} $$

encodes the tree

(44)

We will see later (in Example 7) that the above tree with suitably chosen decorations describes the first iterated integral of the Kortweg–de Vries equation (3).

We are interested in the following quantity which represents the frequencies associated to this tree:

(45) $$ \begin{align} \mathscr{F}( T_{{\mathfrak{e}}}^{{\mathfrak{f}}} ) = \sum_{u \in N_T} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)) \end{align} $$

where $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ and

(46) $$ \begin{align} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)){ \, := \,} (-1)^{{\mathfrak{p}}(e_u)}P_{{\mathfrak{t}}(e_u)}((-1)^{{\mathfrak{p}}(e_u)}{\mathfrak{f}}(u)). \end{align} $$

The term $ \mathscr {F}( T_{{\mathfrak {e}}}^{{\mathfrak {f}}}) $ has to be understood as a polynomial in multiple variables given by the $ k_i $ .

In the numerical scheme, what matters are the terms with maximal degree of frequency, which are here the monomials of higher degree; cf. $\mathcal {L}_{\tiny {\text {dom}}}$ . We compute them using the symbolic notation in the next section. We assume fixed $ {\mathfrak {L}}_{+} \subset {\mathfrak {L}} $ . This subset encodes integrals in time that are of the form $ \int _0^{\tau } e^{s P_{(\mathfrak {t},\mathfrak {p})}(\cdot )} \cdots ds $ for $ (\mathfrak {t},\mathfrak {p}) \in {\mathfrak {L}}_+ \times \lbrace 0,1 \rbrace $ ; see also its interpretation given in (82).

2.2 Dominant parts of trees and physical space maps

Definition 2.2. Let $ P(k_1,\ldots ,k_n) $ a polynomial in the $ k_i $ . If the higher monomials of $ P $ are of the form

$$ \begin{align*} a \sum_{i=1}^{n} (a_i k_i)^{m}, \quad a_i \in { \lbrace 0,1 \rbrace}, \, a \in {\mathbf{Z}}, \end{align*} $$

then we define $ {\mathcal {P}}_{\tiny {\text {dom}}}(P) $ as

(47) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}(P) = a \left(\sum_{i=1}^{n} a_i k_i\right)^{m}. \end{align} $$

Otherwise, it is zero.

Remark 2.3. Given a polynomial $ P $ , one can compute its dominant part $ \mathcal {L}_{\text {dom}} $ and its lower part $\mathcal {L}_{\text {low}}$

$$ \begin{align*} \mathcal{L}_{\tiny{\text{dom}}} = {\mathcal{P}}_{\tiny{\text{dom}}}(P), \quad \mathcal{L}_{\tiny{\text{low}}} = \left({\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right)(P). \end{align*} $$

In our discretisation we will treat the dominant parts of the frequency interactions $\mathcal {L}_{\tiny {\text {dom}}}$ exactly, while approximating the lower-order parts $\mathcal {L}_{\tiny {\text {low}}} $ by Taylor series expansions (cf. also (12)). This will be achieved by applying recursively the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ introduced in Definition 2.6.

Note that in the special case that ${\mathcal {P}}_{\tiny {\text {dom}}}(P) = 0$ , we have to expand all frequency interactions into a Taylor series. The latter, for instance, arises in the context of quadratic Schrödinger equations

(48) $$ \begin{align} i \partial_t u = -\Delta u + u^2, \quad u(0,x) = v(x) \end{align} $$

for which we face oscillations of type (cf. (10))

$$ \begin{align*} \int_0^\tau e^{i \xi \Delta } (e^{-i \xi \ \Delta} v)^2 d\xi & = \sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -i \big(k_1+k_2\big)^2 \xi} e^{ i \big(k_1^2 + k_2^2 \big)\xi} d\xi \\ &=\sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -2 i k_1 k_2 \xi} d\xi. \end{align*} $$

Here we recall the notation $ k \ell = k_1 \ell _1 + \ldots + k_d \ell _d $ for $k, \ell \in {\mathbf {Z}}^d$ . In contrast to the cubic NLS (21) where we have that (cf. (26))

$$ \begin{align*} { \mathcal{L}_{\text{dom}}(k_1) = 2k_1^2 \quad \text{and}\quad \mathcal{L}_{\text{low}}(k_1,k_2,k_3) = - 2 k_1 (k_2+k_3) + 2 k_2 k_3}, \end{align*} $$

we observe for the quadratic NLS (48) with the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ given in (47) that

$$ \begin{align*} P(k_1,k_2) & = - (k_1 + k_2)^2 + (k_1^2 + k_2^2) = - 2k_1 k_2, \quad {\mathcal{P}}_{\tiny{\text{dom}}}(P) = 0, \\ \mathcal{L}_{\tiny{\text{dom}}} & = {\mathcal{P}}_{\tiny{\text{dom}}}(P) =0, \quad \mathcal{L}_{\tiny{\text{low}}} = P - {\mathcal{P}}_{\tiny{\text{dom}}}(P) = - 2k_1 k_2. \end{align*} $$

Hence, although $\mathcal {L}= -\Delta $ and $\mathcal {L}_{\text {dom}}= 0 $ (which means that no oscillations are integrated exactly), we ‘only’ lose one derivative in the local approximation error (cf. (16)) as

$$ \begin{align*} \mathcal{O}\left(\tau^2 \mathcal{L}_{\text{low}}v\right)= \mathcal{O}\left(\tau^2 \vert \nabla \vert v\right). \end{align*} $$

Remark 2.4. Terms of type (47) will naturally arise when filtering out the dominant nonlinear frequency interactions in the PDE. We have to embed integrals over their exponentials into our discretisation. For their practical implementation it will therefore be essential to map fractions of (47) back to physical space.

Indeed, if we apply the inverse Fourier transform $ \mathcal {F}^{-1} $ , we get

(49) $$ \begin{align} \mathcal{F}^{-1} & \left( \sum_{\substack{0 \neq k=k_1 +\ldots +k_n\\ k_\ell \neq 0} } \frac{1}{(k_1 +\ldots +k_n)^m} \frac{1}{k_1^{m_1}} \ldots \frac{1}{k_n^{m_n}}v^1_{k_1}\ldots v^n_{k_n} e^{i kx} \right) \end{align} $$
$$\begin{align*}& = (-\Delta)^{- m/2} \prod_{\ell = 1}^n \left( (-\Delta)^{-m_\ell/2} v^\ell(x)\right) \end{align*}$$

where by abuse of notation we define the operator $(-\Delta )^{-1}$ in Fourier space as $(-\Delta )^{-1} f(x) = \sum _{ k \neq 0} \frac { \hat {f}_k }{k^2}e^{i k x}.$

In the next proposition, we elaborate on (49) and give a nice class of functions depending on the $k_i$ that we can map back to the physical space.

Proposition 2.5. Assume that we have polynomials $ Q $ in $ k_1, \ldots , k_n $ and $ k $ is a linear combination of the $k_i$ such that

$$ \begin{align*} Q & = \prod_j (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad V_j \subset \lbrace 1,\ldots ,n \rbrace, \quad a_{u,V_j} \in \lbrace-1,1\rbrace, \\ k & = \sum_{u=1}^{n} a_u k_u, \quad a_u \in \lbrace -1,1 \rbrace, \end{align*} $$

where the $ V_i $ are either disjoint or if $ V_j \subset V_i $ we assume that there exist $ p_{i,j} $ such that

$$ \begin{align*} a_{u,V_i} = (-1)^{p_{i,j}} a_{u,V_j}, \quad u \in V_j. \end{align*} $$

We also suppose that the $ V_i $ are included in $ k $ in the sense that there exist $ p_{V_i} $ such that

$$ \begin{align*} a_u = (-1)^{p_{V_i}} a_{u,V_i}, \quad u \in V_i. \end{align*} $$

Then, one gets

$$ \begin{align*} \mathcal{F}^{-1} & \left( \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} \right)\\ & = \left(\prod_{V_i \subset V_j} (-1)^{p_{V_i}} (-\Delta)^{- m_i/2}_{V_i} \right) v^{1,a_1}\ldots v^{n,a_n} \end{align*} $$

where $ v^{i,1} = v^{i} $ and $ v^{i,-1} = \overline {v^{i}} $ . The operator $ (-\Delta )^{- m_i/2}_{V_i} $ acts only on the functions $\prod _{u \in V_i} v^{u,a_u} $ and the product starts by the smaller elements for the inclusion order.

Proof

We proceed by induction on the number of $ V_i $ . Let $ V_{\max } $ be an element among the $ V_i $ maximum for the inclusion order. Then, we get

$$ \begin{align*} \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} & \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} = \sum_{\substack{0 \neq k= r + \ell \\ \ell \neq 0}} \frac{(-1)^{p_{V_{\max}}}}{\ell^{m_{\max}}}\sum_{ \substack{0 \neq r= \sum_{u \notin V_{\max}} a_u k_u \\ R \neq 0}} \frac{1}{R} \\ & \left( \prod_{j \notin V_{\max}} v^{j,a_j}_{k_j} \right) e^{i r x} \times \sum_{ \substack{0 \neq \ell= \sum_{u \in V_{\max}} a_u k_u \\ S \neq 0}} \frac{1}{S} \left( \prod_{j \in V_{\max}} v^{j,a_j}_{k_j} \right) e^{i \ell x} \end{align*} $$

where

$$ \begin{align*} S = \prod_{V_j \varsubsetneq V_{\max}} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad R = \prod_{V_j \cap V_{\max} = \emptyset} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad Q = R S \ell^{m_{max}}. \end{align*} $$

Thus, by applying the inverse Fourier transform, we get the term $ (-1)^{p_{V_{\max }}} (-\Delta )^{- m_{\max }/2}_{V_{\max }} $ from $ \frac {(-1)^{p_{V_{\max }}}}{\ell ^{m_{\max }}} $ . We conclude from the induction hypothesis on the two remaining sums.

The next definition will allow us to compute the dominant part of the frequency interactions of a given decorated forest in $ \hat H_0 $ . The idea is to filter out the dominant part using the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ which selects the frequencies of highest order. The operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ only appears if we face an edge in $ {\mathfrak {L}}_+ $ which corresponds to an integral in time that we have to approximate.

Definition 2.6. We recursively define $\mathscr {F}_{\tiny {\text {dom}}}, \mathscr {F}_{\tiny {\text {low}}} : \hat H_{0} \rightarrow \mathbb {R}[{\mathbf {Z}}^d]$ as

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}({\mathbf{1}}) = 0 \quad \mathscr{F}_{\tiny{\text{dom}}}(F \cdot \bar F) & =\mathscr{F}_{\tiny{\text{dom}}}(F) + \mathscr{F}_{\tiny{\text{dom}}}(\bar F) \\ \mathscr{F}_{\tiny{\text{dom}}}\left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left\{ \begin{array}{l} \displaystyle {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right), \, \text{if } {\mathfrak{t}} \in {\mathfrak{L}}_+ , \\ P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F), \quad \text{otherwise} \\ \end{array} \right. \\ \mathscr{F}_{\tiny{\text{low}}} \left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left( {\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right) \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right). \end{align*} $$

We extend these two maps to $ \hat H $ by ignoring the node decorations $ {\mathfrak {n}} $ .

Remark 2.7. The definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ can be adapted depending on what is considered to be the dominant part. For example, if for $ {\mathfrak {t}}_2 \in {\mathfrak {L}}_+$ , one has (cf. (9))

$$ \begin{align*} P_{({\mathfrak{t}}_2,p)}( \lambda) = \frac{1}{\varepsilon^{\sigma}} + F_{({\mathfrak{t}}_2,p)}( \lambda). \end{align*} $$

Then we can define the dominant part only depending on $ \varepsilon $ (see Example 5.3):

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}}_2,p)}(k) \right)= \frac{1}{\varepsilon^{\sigma}}. \end{align*} $$

The definition considers only the case where one does not have terms of the form $ 1/\varepsilon ^{\sigma } $ . It is sufficient for covering many interesting examples.

In the following we compute the dominant part $ \mathscr {F}_{\text {dom}}$ for the underlying trees of the cubic Schrödinger (2) and KdV (3) equation.

Example 7 (KdV). We consider the decorated tree $ T $ given in Example 44, where we fix the following decorations:

$$ \begin{align*} {\mathfrak{p}}(1) & = {\mathfrak{p}}(2) = {\mathfrak{p}}(3) = 0, \quad {\mathfrak{t}}(2) = {\mathfrak{t}}(3) = {\mathfrak{t}}_1, \quad {\mathfrak{t}}(1) = {\mathfrak{t}}_2, \\[-3pt] {\mathfrak{f}}(b) & = k_1, \quad {\mathfrak{f}}(c) = k_2, \quad {\mathfrak{f}}(a) = k_1 + k_2, \quad P_{{\mathfrak{t}}_2}( \lambda) = \lambda^3 , \quad P_{{\mathfrak{t}}_1}( \lambda) = - \lambda^3. \end{align*} $$

Now, we suppose $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ . Then the tree (44) takes the form

(50)

This tree corresponds to the first iterated integral for the KdV equation (3). In more formal notation, we denote this tree by

(51)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ and a brown edge is used for $(\mathfrak {t}_1,0)$ . The frequencies are given on the leaves. The ones on the inner nodes are determined by those on the leaves. On the left-hand side, we have given the symbolic notation. Together with Definition 2.6, one gets

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (k_1+k_2)^{3} - k_1^{3} - k_2^{3} \right) = 0\\ \mathscr{F}_{\tiny{\text{low}}} (T) & = (k_1+k_2)^{3} - k_1^{3} - k_2^{3} = 3k_1 k_2 (k_1+k_2). \end{align*} $$

The fact that $ \mathscr {F}_{\tiny {\text {dom}}}(T) $ is zero comes from the fundamental choice of definition of the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ in Definition 2.2.

Example 8 (Cubic Schrödinger). Next we consider the symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{-k_1+k_2+k_3}{\cal I}_{({\mathfrak{t}}_2,1)}( \lambda_{k_1}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_2}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_3})) \end{align*} $$

with $P_{{\mathfrak {t}}_2}( \lambda )= \lambda ^2$ , $P_{{\mathfrak {t}}_1}( \lambda ) = - \lambda ^2$ , $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ which encodes the tree

(52)

This tree corresponds to the frequency interaction of the first iterated integral for the cubic Schrödinger equation (2). In a more formal notation, we denote this tree by

(53)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ , a brown edge is used for $ (\mathfrak {t}_1,0) $ and a dashed brown edge is for $ (\mathfrak {t}_2,1) $ . With Definition 2.6, we get

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 \right) \\& = {\mathcal{P}}_{\tiny{\text{dom}}} \left( 2k_1^2 - 2k_1(k_2+k_3) + 2 k_2 k_3\right) = 2 k_1^2\\ \mathscr{F}_{\tiny{\text{low}}} (T)& = (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 - 2k_1^2\\ & = - 2k_1(k_2+k_3) + 2 k_2 k_3. \end{align*} $$

The map $ \mathscr {F}_{\tiny {\text {dom}}}$ has a nice property regarding the tree inclusions given in the next proposition. This inclusive property will be important in practical computations; see also Remark 1.3 and the examples in Section 5.

For Proposition 2.8, we need an additional assumption.

Assumption 1. We consider decorated forests whose decorations at the leaves form a partition of the $ k_1,\ldots ,k_n $ , in the sense that for two leaves $ u $ and $ v $ , $ {\mathfrak {f}}(u) $ (respectively $ {\mathfrak {f}}(v) $ ) is a linear combination of $ (k_i)_{i \in I} $ (respectively $ (k_i)_{i \in J} $ ) with $ I,J \subset \lbrace 1,\ldots ,n \rbrace $ and $ I \cap J = \emptyset $ . This will be the case in the examples given in Section 5.

With this assumption, for a decorated forest $ F = \prod _i T_i $ such that $ {\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) \neq 0 $ , one has the nice identity

(54) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = {\mathcal{P}}_{\tiny{\text{dom}}}\left(\sum_i {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align} $$

We will illustrate this property and give a counterexample in an example below.

Example 9. We give a counterexample in the setting of the Schrödinger equation to (54) when ${\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) = 0 $ . We consider the following forest $ F = \prod _{i=1}^3 T_i$ where the decorated trees $ T_i $ are given by

Then, we can check Assumption 1 for $ F $ . One has

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T_1) & = - k_4^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_2) = - k_5^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_3) = - (-k_1 + k_2 + k_3)^2 + 2 k_1^2, \\ \mathscr{F}_{\tiny{\text{dom}}}(F) & = - k_4^2 - k_5^2 - (-k_1 + k_2 + k_3)^2 + 2 k_1^2 \end{align*} $$

and

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right) = - k_4^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_2) \right) = - k_5^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_3) \right) = 0. \end{align*} $$

Therefore, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^3 {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right) = - (k_4 + k_5)^2. \end{align*} $$

But, on the other hand,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = 0. \end{align*} $$

Proposition 2.8. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ be a decorated tree in $ \hat H_0 $ and $ e \in E_T $ . We recall that $ T_e $ corresponds to the subtree of $ T $ above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ . Under Assumption 1 one has

(55) $$ \begin{align} \begin{split} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad m \in \mathbf{N}, a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\ {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}((T_e)_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = b \left( \sum_{u \in \bar V } b_u k_u \right)^{m_e}, \quad m_e \in \mathbf{N}, b \in {\mathbf{Z}}, \, \bar V \subset L_T, \, b_u \in \lbrace -1,1 \rbrace \end{split} \end{align} $$

and $\bar V \subset V \text { or } \bar V \cap V = \emptyset $ . If $ \bar V \subset V $ , then there exists $ \bar p \in \lbrace 0,1 \rbrace $ such that $ a_u = (-1)^{\bar p} b_u $ for every $ u \in \bar V $ .

Proof

We proceed by induction over the size of the decorated trees. We consider $T= {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k} F) $ where $ F = \prod _{i=1}^m T_i $ .

(i) If $F ={\mathbf {1}}$ , then one has

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

We conclude from the definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ .

(ii) If $ m> 2 $ , then one gets

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + \sum_{i=1}^m \mathscr{F}_{\tiny{\text{dom}}}(T_i) \right). \end{align*} $$

Using Assumption 1, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m (P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i)) \right) \end{align*} $$

where $ k^{(i)} $ corresponds to the frequency attached to the node connected to the root of $ T_i $ . If $ {\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(T) \right ) \neq 0 $ , then from (54) we have

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align*} $$

We apply the induction hypothesis to each decorated tree $ \tilde T_i = {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k^{(i)}} T_i) $ and we recombine the various terms in order to conclude.

(iii) If $ m=1 $ , then $ F = {\cal I}_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}( \lambda _{ \bar k} T_1) $ where $ \bar k $ is equal to $ k $ up to a minus sign. We can assume without loss of generality that

(56) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = \mathscr{F}_{\tiny{\text{dom}}}(F). \end{align} $$

Indeed, otherwise,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T)\right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + P_{({\mathfrak{t}}_1,{\mathfrak{p}}_1)}(\bar k) + \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right). \end{align*} $$

We can see $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k) + P_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}(\bar k) $ as a polynomial and then apply the induction hypothesis. We are down to the case (56) and we consider

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) & = {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) \right) \end{align*} $$

where now $ F = \bar T$ is just a decorated tree. We apply the induction hypothesis on $ \bar T $ and we get

(57) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\k & = \sum_{u \in L_T } c_u k_u, \quad c_{ u} = (-1)^{ p} a_{u} , \quad u \in V. \notag \end{align} $$

If the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is higher than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ , we obtain that

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

On the other hand, if the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is lower than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ ,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right). \end{align*} $$

If $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ and $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ have the same degree $ m $ , we get using the definition of $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ in (46) as well as the induction hypothesis on $\bar T$ given in (57) that

$$ \begin{align*} P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = \sum_{u \in V } \left( a (-1)^{p + m + {\mathfrak{p}}} + (- 1)^{{\mathfrak{p}}} \right)( (-1)^{{\mathfrak{p}}}c_u k_u)^{m} \\ & \quad{}+ \sum_{u \in L_T \setminus V } (- 1)^{{\mathfrak{p}}} ((-1)^{{\mathfrak{p}}} c_u k_u)^{m} + R \end{align*} $$

where $ R $ terms of lower orders.

By applying the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ defined in (47), we thus obtain an expression of the form

(58) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = b \left( \sum_{u \in \tilde{V} } (-1)^{{\mathfrak{p}}} c_u k_u \right)^{m} \,\text{for some } b \in {\mathbf{Z}} \end{align} $$

where $ \tilde V $ could be either $ L_T $ or $ L_T \setminus V $ .

Let $ e \in E_T $ , $ T_e \neq T $ , then $ T_e $ is a subtree of $ \bar T $ . By the induction hypothesis, one obtains (55), meaning that if we denote by $ V $ (respectively $ \bar V $ ) the set associated to $ \bar T $ (respectively $ T_e $ ), we get $ \bar V \subset V $ or $ \bar V \cap V = \emptyset $ .

In the first case, $\bar V \subset V $ , the assertion follows as $ V \subset \tilde {V} $ or $ V \cap \tilde {V} = \emptyset $ such that necessarily $\bar V \subset \tilde V$ or $\bar V \cap \tilde V = \emptyset $ .

In the second case, $ \bar V \cap V = \emptyset $ , we apply the induction hypothesis on $ T_e $ . Then, for $ v $ of $ T_e $ , there exists $ p $ such that the decoration $ {\mathfrak {f}}(v) $ is given by

$$ \begin{align*} {\mathfrak{f}}(v) = \sum_{u \in L_{T_v} } d_u k_u, \quad d_{ u} = (-1)^{ p} b_{u} , \quad u \in \bar V. \end{align*} $$

As $ {\mathfrak {f}}(v) $ appears as a subfactor in $ k $ , one has $ \bar V \subset L_T $ . Then, $ \bar {V} \cap V = \emptyset $ also gives that $ \bar V \subset L_T \setminus V $ . Therefore, we have $ \bar V \subset \tilde {V} $ , which concludes the proof.

Corollary 2.9. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ a decorated tree in $ \hat {\mathcal {T}}_0 $ . We assume that Assumption 1 holds true for a set $ A $ of decorated subtrees of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ such that ${\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \right ) =\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \neq 0$ for $ \bar T \in A $ . Moreover, we assume that the $ \bar T $ are of the form $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ where $ e \in E_T $ . Then, the following product

$$ \begin{align*} \prod_{\bar T\kern-1pt \in A} \frac{1}{\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right)^{m_T}} \end{align*} $$

can be mapped back to physical space using operators of the form $(- \Delta )^{-m/2}_V $ as defined in Proposition 2.5.

Proof

Proposition 2.8 gives us the structure needed for applying Proposition 2.5 which allows us to conclude.

Example 10. We illustrate Corollary 2.9 via an example extracted from the cubic Schrödinger equation (2). We consider the following decorated trees:

(59)

We observe that these trees satisfy Assumption 1 and that $ T_1 $ is a subtree of $ T_2 $ . One has that

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}( T_1) = 2 k_1^2 , \quad\mathscr{F}_{\tiny{\text{dom}}}( T_2) = 2 (k_1 + k_4)^2 \end{align*} $$

and the following quantity can be mapped back into physical space:

$$ \begin{align*} &\sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5 \\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_1)} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_2)} \overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x}\\ &\quad= \sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5\\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{4 (k_1^2) (k_1 + k_4)^2}\overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x} \\ &\quad = \frac14 v(x)^3(-\Delta)^{-1}\left(\overline{v}(x) (-\Delta)^{-1} \overline{v}(x)\right). \end{align*} $$

2.3 Approximated decorated trees

We denote by $ {\mathcal {T}} $ the set of decorated trees $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},r) $ where

  • $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} \in \hat {\mathcal {T}} $ .

  • The decoration of the root is given by $ r \in {\mathbf {Z}} $ , $ r \geq -1 $ such that

    (60) $$ \begin{align} r +1 \geq \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) \end{align} $$
    where $ \deg $ is defined recursively by
    $$ \begin{align*} \deg({\mathbf{1}}) & = 0, \quad \deg(T_1 \cdot T_2 ) = \max(\deg(T_1),\deg(T_2)), \\ \deg({\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda^{\ell}_{k}T_1) ) & = \ell + {\mathbf{1}}_{\lbrace{\mathfrak{t}} \in {\cal L}_+\rbrace} + \deg(T_1) \end{align*} $$
    where $ {\mathbf {1}} $ is the empty forest and $ T_1, T_2 $ are forests composed of trees in $ {\mathcal {T}} $ . The quantity $\deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}})$ is  the maximum number of edges with type in $ {\mathfrak {L}}_+ $ and node decorations $ {\mathfrak {n}} $ lying on the same path from one leaf to the root.

We call decorated trees in $ {\mathcal {T}} $ approximated decorated trees. The main difference with decorated trees introduced before is the adjunction of the decoration $ r $ at the root. The idea is that these trees correspond to different analytical objects. We summarise this below:

$$ \begin{align*} T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Iterated integral} \\ T_{{\mathfrak{e}},r}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Approximation of an iterated integral of order } r. \end{align*} $$

The interpretation (83) of approximated decorated trees gives the numerical scheme; see Definition 4.4.

Remark 2.10. The condition (60) encodes the fact that the order of the scheme must be higher than the maximum number of iterated integrals and monomials lying on the same path from one leaf to the root. Moreover, we can only have monomials of degree less than $ r+1 $ at order $ r+2 $ .

Example 11. We continue Example 6 with the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},\mathfrak {f}} $ given in (44). We suppose that $ {\mathfrak {t}}(1) $ is in $ \mathfrak {L}_+ $ but not $ {\mathfrak {t}}(2) $ and ${\mathfrak {t}}(3) $ . We are now in the context of Example 7. Then, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) = {\mathfrak{n}}(a) + 1+ \max({\mathfrak{n}}(b),{\mathfrak{n}}(c)). \end{align*} $$

We denote by $ {\mathcal {H}} $ the vector space spanned by forests composed of trees in $ {\mathcal {T}}$ and $ \lambda ^n $ , $ n \in \mathbf {N} $ where $ \lambda ^n $ is the tree with one node decorated by $ n $ . When the decoration $ n $ is equal to zero, we identify this tree with the empty forest: $ \lambda ^{0} ={\mathbf {1}} $ . Using the symbolic notation, one has

$$ \begin{align*} {\mathcal{H}} = \langle \lbrace \prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i), \, {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i) \in \hat {\mathcal{T}} \rbrace \rangle \end{align*} $$

where the product used is the forest product. We call decorated forests in $ {\mathcal {H}} $ approximated decorated forests. The map $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ is defined the same as for $ {\cal I}_{o}( \lambda _{k}^{\ell } \cdot ) $ except now the root is decorated by $ r $ and it could be zero if the inequality (60) is not satisfied. We extend this map to $ {\mathcal {H}} $ by

$$ \begin{align*} {\cal I}^{r}_{o}( \lambda_{k}^{\ell} (\prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))) { \, := }{\cal I}^{r}_{o}( \lambda_{k}^{\ell+\sum_j m_j} (\prod_i {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))). \end{align*} $$

In the extension, we remove the decorations $ r_i $ and we add up the decorations $ m_j $ with $ \ell $ . In the sequel, we will use a recursive formulation and move from $ \hat {\mathcal {H}} $ to $ {\mathcal {H}} $ . Therefore, we define the map $ {\cal D}^{r} : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ which replaces the root decoration of a decorated tree by $ r $ and performs the projection along the identity (60). It is given by

(61) $$ \begin{align} {\cal D}^{r}({\mathbf{1}})= {\mathbf{1}}_{\lbrace 0 \leq r+1\rbrace} , \quad {\cal D}^r\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{r}_{o}( \lambda_{k}^{\ell} F) \end{align} $$

and we extend it multiplicatively to any forest in $ \hat {\mathcal {H}} $ . The map $ {\cal D}^r $ projects according to the order of the scheme $ r $ . We disregard decorated trees having a degree bigger than $ r $ .

We denote by $ {\mathcal {T}}_{+} $ the set of decorated forests composed of trees of the form $ (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},(r,m)) $ where

  • $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} \in {\mathcal {T}} $ .

  • The edge connecting the root has a decoration of the form $ ({\mathfrak {t}},{\mathfrak {p}}) $ where $ {\mathfrak {t}} \in {\mathfrak {L}}_+ $ .

  • The decoration $ (r,m) $ is at the root of $ T $ and $ m \in \mathbf {N} $ is such that $ m \leq r +1 $ .

The linear span of $ {\mathcal {T}}_+ $ is denoted by $ {\mathcal {H}}_+ $ . One can observe that the main difference with ${\mathcal {H}}$ is that $ \lambda ^m \notin {\mathcal {T}}_+ $ . We can define the same grafting operator as before $ {\cal I}^{(r,m)}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}} $ but now we add the decoration $ (r,m) $ at the root where $ m \leq r+1 $ . We also define $ \hat {\cal D}^{(r,m)}: \hat {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal D}^{r} $ . It is given by

(62) $$ \begin{align} \hat {\cal D}^{(r,m)}({\mathbf{1}})= {\mathbf{1}}, \quad \hat {\cal D}^{(r,m)}\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{(r,m)}_{o}( \lambda_{k}^{\ell} F). \end{align} $$

Example 12. For the tree (44) we obtain when applying $ {\cal D}^r $ and $ \hat {\cal D}^{(r,m)} $

(63)

For the decorated tree in (53), one obtains

(64)

2.4 Operators on approximated decorated forests

In the subsection, we introduce two maps $ \Delta $ and $ \Delta ^{\!+} $ that will act on approximated decorated forests, splitting them into two parts by the use of the tensor product. They act at two levels. First, on the shapes of the trees, they extract a subtree at the root. Then, they induce subtle changes in the decorations that could be interpreted as abstract Taylor expansions.

We define a map $\Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+$ for a given $T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} \in {{\mathcal T}}$ by

(65) $$ \begin{align} \Delta T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},r} & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} (A,{\mathfrak{n}} + \pi{\mathfrak{e}}_A, {\mathfrak{f}}, {\mathfrak{e}},r)\\ & \otimes \prod_{e \in \partial(A,T)}( T_{e}, {\mathfrak{n}} , {\mathfrak{f}}, {\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))), \notag \\ & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},r} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))} \notag\end{align} $$

where we use the following notation:

  • We write $T_e $ as the planted tree above the edge $ e $ in $ T $ . For $g : E_T \rightarrow \mathbf {N}$ , we define for every $x \in N_T$ , $(\pi g)(x) = \sum _{e=(x,y) \in E_T} g(e)$ .

  • In $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} $ , the maps $ {\mathfrak {n}}, {\mathfrak {f}} $ and $ {\mathfrak {e}} $ are restricted to $ N_A $ and $ E_A $ . The same is valid for $ (T_{e})^{{\mathfrak {n}} , {\mathfrak {f}}}_{{\mathfrak {e}}, (r-\deg (e),{\mathfrak {e}}_A(e))} $ where the restriction is on $ N_{T_e} \setminus \lbrace \varrho _{T_e}\rbrace $ and $ E_{T_e} $ , $ \varrho _{T_e} $ is the root of $ T_e $ . When $ A $ is reduced to a single node, we set $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} = \lambda ^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A} $ .

  • The first sum runs over ${\mathfrak {A}}(T)$ , the set of all subtrees A of T containing the root $ \varrho $ of $ T $ . The second sum runs ${\mathfrak {e}}_{A} : \partial (A,T) \rightarrow \mathbf {N}$ where $\partial (A,T)$ denotes the edges in $E_T \setminus E_A$ of type in $ {\mathfrak {L}}_+ $ that are adjacent to $N_A$ .

  • Factorial coefficients are understood in multi-index notation.

  • We define $ \deg (e) $ for $ e \in E_T $ as the number of edges having $ {\mathfrak {t}}(e) \in {\mathfrak {L}}_+ $ lying on the path from $ e $ to the root in the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ . We also add up the decoration $ {\mathfrak {n}} $ on this path.

Let us comment briefly that the play on decorations can be interpreted as asbtract Taylor expansions. We can use the following dictionary:

$$ \begin{align*} (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \int_{0}^{\tau} e^{i\xi P(k)} f(k_1,\ldots ,k_n,\xi) d\xi \end{align*} $$

where $ k_1,\ldots ,k_n $ are the frequencies appearing on the leaves of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}}$ , $ P $ is the polynomial associated to the decoration of the edge $ e $ and $ k $ is the frequency on the node connected to the root. This iterated integral appears inside the iterated integral associated to $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . For our numerical approximation, we need to approximate this integral by giving a scheme of the form

$$ \begin{align*} (T_e)_{{\mathfrak{e}}, \tilde{r}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \sum_{\ell \leq \tilde{r}} \frac{\tau^{\ell}}{\ell!} f_{\tilde{r},\ell}(k_1,\ldots ,k_n), \quad (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (\tilde{r},\ell)} \equiv f_{\tilde{r},\ell}(k_1,\ldots ,k_n) \end{align*} $$

where the order of the expansion is given by $ \tilde {r} = r - \deg (e) $ . Then, the $ \tau ^{\ell } $ are part of the original iterated integrals and cannot be detached as the $ f_{\tilde {r},\ell } $ . That is why we increase the polynomial decorations where the tree $ T_e $ was originally attached via the term $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A $ . The choice for $ \tilde {r} $ is motivated by the fact that we do not need to go too far for the approximation of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . We take into account all of the time integrals and polynomials which lie on the path connecting the root of $ T_e $ to the root of $ T $ .

The map $ \Delta $ is compatible with the projection induced by the decoration $ r $ . Indeed, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}}) = \max_{e \in E_T} \left( \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) \right). \end{align*} $$

Therefore, if $ \deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}}) < r $ , then for every $ e \in E_T $

$$ \begin{align*} \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) < r. \end{align*} $$

We deduce that if $ T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} $ is zero, then the $ (T_e)_{{\mathfrak {e}},(r-\deg (e),{\mathfrak {e}}_A(e))}^{{\mathfrak {n}}, {\mathfrak {f}}} $ are zero, too.

We define a map ${\Delta ^{\!+}} : {\mathcal {H}}_+ \rightarrow {\mathcal {H}}_+ \otimes {\mathcal {H}}_+$ given for $T^{{\mathfrak {n}}, {\mathfrak {f}}}_{{\mathfrak {e}},(r,m)} \in {{\mathcal T}}_+$ by

(66) $$ \begin{align} {\Delta^{\!+}} T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},(r,m)} = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},(r,m)} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))}. \end{align} $$

We require that $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},(r,m)} \in {\mathcal {H}}_+ $ , so we implicitly have a projection on zero when $ A $ happens to be a single node with $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A \neq 0 $ . We illustrate this coproduct on a well-chosen example.

Example 13. We continue with the tree in Example 5. We suppose that $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}(2), {\mathfrak {t}}(3), {\mathfrak {t}}(4), {\mathfrak {t}}(5) \rbrace $ . Below, the subtree $ A \in {\mathfrak {A}}(T)$ is coloured in blue. We have $ N_A = \lbrace \varrho , a,b\rbrace $ , $ E_A = \lbrace 1,2 \rbrace $ and $ \partial (A,T) = \lbrace 3, 4,5 \rbrace $ .

We also get