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We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.
In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.
Macroscale “continuum” level predictions are made by a new way to construct computationally efficient “wrappers” around fine-scale, microscopic, detailed descriptions of dynamical systems, such as molecular dynamics. It is often significantly easier to code a microscale simulator with periodicity: so the challenge addressed here is to develop a scheme that uses only a given periodic microscale simulator; specifically, one for atomistic dynamics. Numerical simulations show that applying a suitable proportional controller within “action regions” of a patch of atomistic simulation effectively predicts the macroscale transport of heat. Theoretical analysis establishes that such an approach will generally be effective and efficient, and also determines good values for the strength of the proportional controller. This work has the potential to empower systematic analysis and understanding at a macroscopic system level when only a given microscale simulator is available.
In this paper we investigate gradient estimation for a class of contracting stochastic systems on a continuous state space. We find conditions on the one-step transitions, namely differentiability and contraction in a Wasserstein distance, that guarantee differentiability of stationary costs. Then we show how to estimate the derivatives, deriving an estimator that can be seen as a generalization of the forward sensitivity analysis method used in deterministic systems. We apply the results to examples, including a neural network model.
This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier–Stokes flows based on Stokes preconditioning  which has been used successfully in studies of pattern formation in convection. The preconditioner takes the form of the Helmholtz operator I–ΔtL which maps the identity (no preconditioner) for Δt≪1 to Laplacian preconditioning for Δt≫1. It is built on a first order Euler time-discretization scheme and is part of the family of matrix-free methods. The preconditioner is tested on two fluid configurations: three-dimensional doubly diffusive convection and a two-dimensional projection of a shear flow. In the former case, it is found that Stokes preconditioning is more efficient for , away from the values used in the literature. In the latter case, the simple use of the preconditioner is not sufficient and it is necessary to split the system of equations into two subsystems which are solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications and the flexibility of the approach described, this preconditioner is expected to help in a wide range of applications.
We study a novel class of numerical integrators, the adapted nested force-gradient schemes, used within the molecular dynamics step of the Hybrid Monte Carlo (HMC) algorithm. We test these methods in the Schwinger model on the lattice, a well known benchmark problem. We derive the analytical basis of nested force-gradient type methods and demonstrate the advantage of the proposed approach, namely reduced computational costs compared with other numerical integration schemes in HMC.
In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.
The 2D Maxwell eigenproblems are studied from a new point of view. An electromagnetic problem is cast from the Lagrangian system with single variable into the Hamiltonian system with dual variables. The electric and magnetic components transverse to the wave propagation direction are treated as dual variables to each other. Higher order curl-conforming and divergence-conforming vector basis functions are used to construct dual vector spectral elements. Numerical examples demonstrate some unique advantages of the proposed method.
We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.
This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.
In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however they also have the significant difference, for example there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.
Many physical processes are described by elliptic or parabolic partial differential equations. For linear stability problems associated with such equations, the inverse Laplacian provides a very effective preconditioner. In addition, it is also readily available in most scientific calculations in the form of a Poisson solver or an implicit diffusive timestep. We incorporate Laplacian preconditioning into the inverse Arnoldi method, using BiCGSTAB to solve the large linear systems. Two successful implementations are described: spherical Couette flow described by the Navier-Stokes equations and Bose-Einstein condensation described by the nonlinear Schrödinger equation.
Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.
This paper analyses the steady-state operation of a generalized bioreactor model that encompasses a continuous-flow bioreactor and an idealized continuous-flow membrane bioreactor as limiting cases. A biodegradation of organic materials is modelled using Contois growth kinetics. The bioreactor performance is analysed by finding the steady-state solutions of the model and determining their stability as a function of the dimensionless residence time. We show that an effective recycle parameter improves the performance of the bioreactor at moderate values of the dimensionless residence time. However, at sufficiently large values of the dimensionless residence time, the performance of the bioreactor is independent of the recycle ratio.
Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.
Microbial competition for nutrients is a common phenomenon that occurs between species inhabiting the same environment. Bioreactors are often used for the study of microbial competition since the number and type of microbial species can be controlled, and the system can be isolated from other interactions that may occur between the competing species. A common type of competition is the so-called “pure and simple” competition, where the microbial populations interact in no other way except the competition for a single rate-limiting nutrient that affects their growth rates. The issue of whether pure and simple competition under time-invariant conditions can give rise to chaotic behaviour has been unresolved for decades. The third author recently showed, for the first time, that chaos can theoretically occur in these systems by analysing the dynamics of a model where both competing species grow following the biomass-dependent Contois model while the yield coefficients associated with the two species are substrate-dependent. In this paper we show that chaotic behaviour can occur in a much simpler model of pure and simple competition. We examine the case where only one species grows following the Contois model with variable yield coefficient while the other species is allowed to grow following the simple Monod model with constant yield. We show that while the static behaviour of the proposed model is quite simple, the dynamic behaviour is complex and involves period doubling culminating in chaos. The proposed model could serve as a basis to re-examine the importance of Contois kinetics in predicting complex behaviour in microbial competition.
It has been known for a long time that the equivariant
wave map into the
-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.
In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.
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