2.1. Kinetic description of a plasma

The kinetic description of a collisionless relativistic plasma with particles of species ${s}$ with elementary mass, $m_{s}$, and elementary charge, $q_{s}$, is given by the Vlasov equation

(2.1)\begin{equation} {\frac{\partial{{f_{s}}}}{\partial{t}}} + \frac{\boldsymbol{u}}{\gamma} \boldsymbol{\cdot} \boldsymbol{\nabla} f_{s} + \boldsymbol{a}_{s} \boldsymbol{\cdot} \boldsymbol{\nabla}_{u} f_{s} = 0, \end{equation}

where $f_{s}=f_{s}(\boldsymbol {x}, {\boldsymbol{v}},t)$ is the distribution function, $\boldsymbol {u} = \gamma {\boldsymbol{v}}$ is the spatial component of the four-velocity with the Lorentz factor $\gamma = [1 + ({\boldsymbol{v}}/c)^2]^{-1/2}$ and $c$ is the light speed. The acceleration due to the Lorentz force is given by

(2.2)\begin{equation} \boldsymbol{a}_{s} = \frac{q_{s}}{m_{s}} [ \boldsymbol{E}(\boldsymbol{x}, t) + {\boldsymbol{v}} \boldsymbol{\times} \boldsymbol{B}(\boldsymbol{x}, t)], \end{equation}

where $\boldsymbol {E}(\boldsymbol {x}, t)$ and $\boldsymbol {B}(\boldsymbol {x}, t)$ are the electric and magnetic fields, respectively. The evolution of electric and magnetic fields is governed by Maxwell's equations

(2.3)\begin{gather} {\frac{\partial{\boldsymbol{B}}}{\partial{t}}} =- \boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{E}, \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B} = 0, \end{gather}

(2.4)\begin{gather}{\frac{\partial{\boldsymbol{E}}}{\partial{t}}} = c^2 \boldsymbol{\nabla}\boldsymbol{\times}\boldsymbol{B} - \frac{\boldsymbol{J}}{\varepsilon_0} , \quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}= \frac{\rho}{\varepsilon_0}, \end{gather}

where $c=1/\sqrt {\varepsilon _0 \mu _0}$ is the vacuum speed of light, and $\varepsilon _0$ and $\mu _0$ are the permittivity and the permeability of free space, respectively. The evolution of the electromagnetic fields is influenced by the charge density, $\rho$, and current density, $\boldsymbol {J}$. They are given by the charge-weighted sum over all species of the number densities $n_{s}$ and bulk velocities ${\boldsymbol{w}}_{s}$, respectively,

(2.5)\begin{gather} \rho(\boldsymbol{x}, t) = \sum_s q_s n_s (\boldsymbol{x}, t)= \sum_s q_s \int f_s(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v}, \end{gather}

(2.6)\begin{gather}\boldsymbol{J}(\boldsymbol{x}, t) = \sum_s q_s n_s (\boldsymbol{x}, t) {\boldsymbol{w}}_s (\boldsymbol{x}, t) = \sum_s q_s \int {\boldsymbol{v}} f_s (\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} . \end{gather}

2.2. The PIC method

We use the PIC method to solve for the evolution of plasma species that are modelled with the kinetic description. The PIC method initializes a number of computational macroparticles to approximate the distribution function in a Lagrangian fashion. Each macroparticle represents multiple physical particles and, as such, each macroparticle has a shape in position space which can be represented by a spline function. By depositing the particle motions and positions onto the numerical grid (or computational cells), the electromagnetic fields can be computed. This step is followed by a back interpolation of these fields to the particle positions so that the Lorentz forces on the particles can be computed. In our implementation, these equations are solved using one spatial dimension and three velocity dimensions (1D3V), i.e. $\boldsymbol {\nabla } = (\partial /\partial x, 0, 0)^\mathrm {T}$.

The code quantities are defined as multiples of the fiducial units given for time, fields (electric and magnetic), charge, current density and length

(2.7)\begin{equation} \left.\begin{array}{c@{}} t_0 = \sqrt{ m_0 \epsilon_0/( q_0^2 n_0)}, \quad E_0 = \sqrt{n_0 m_0 c^2/\epsilon_0},\\ \rho_0 = q_0 n_0, \quad J_0 = \rho_0 c,\quad x_0 = c t_0. \end{array} \right\} \end{equation}

This enables us to select a fixed time step of

(2.8)\begin{equation} \Delta t = C_\mathrm{cfl} c \Delta x,\end{equation}

where $C_\mathrm {cfl}< 0.5$ to satisfy the Courant–Friedrichs–Lewy (CFL) condition. The value of the reference density $n_0$ is chosen such that the code time scale, $t_0$, obeys $\omega _{p}^{-2}= t_0^2$. The total plasma frequency is $\omega _{p}=(\sum _s \omega _s^2)^{1/2}$, and is related to the plasma frequencies of the individual species, $\omega _{s}^2 = q_{s}^2 n_{s}/(m_{s} \epsilon _0)$. We define the discretized time $t^{k} = k \Delta t$, position $x_i = i \Delta x$ and quantities at discrete position and times as $\boldsymbol {E}^{k}_i = \boldsymbol {E}(t^{k}, x_i)$. For details on the PIC code SHARP, the reader is referred to Shalaby et al. (Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b, Reference Shalaby, Thomas and Pfrommer2021). Here, we focus on describing how SHARP is extended to include fluid treatment of some plasma species.

2.3. Fluid description of plasma

A straightforward way of coarse graining the Vlasov equation (2.1) is to reduce its dimensionality. By taking the $j$th moment over velocity space, i.e. $\int {\boldsymbol{v}}^j f \,\mathrm {d}^3 {v}$, we retrieve the fluid quantities and reduce the dimensionality of the 1D3V kinetic description to one dimensions. The number density $n_{s}$ and the bulk velocity ${\boldsymbol{w}}_{s}$ are defined through the zeroth and first moments of the distribution function, respectively, while the total energy density per unit mass $\epsilon _s$ and the scalar pressure per unit mass $p_s$ are related to the second moment (Wang et al. Reference Wang, Hakim, Bhattacharjee and Germaschewski2015)

(2.9)\begin{gather} n_{s} (\boldsymbol{x}, t) = \int f_{s}(\boldsymbol{x}, {\boldsymbol{v}}, t) \, \mathrm{d}^3 {v} , \end{gather}

(2.10)\begin{gather}{\boldsymbol{w}}_{s} (\boldsymbol{x},t)= \int\frac{1}{n_{s}(\boldsymbol{x}, t) } {\boldsymbol{v}} f_{s}(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} , \end{gather}

(2.11)\begin{gather}\epsilon_{s} (\boldsymbol{x}, t) =\int \frac{1}{2} {\boldsymbol{v}}^2 f_{s}(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} , \end{gather}

(2.12)\begin{gather}p_{s}(\boldsymbol{x}, t) = \int ({v}_x - {w}_{{s},x})^2 f_{s}(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} = \frac{{\Gamma } - 1}{2} \int ({\boldsymbol{v}} - {\boldsymbol{w}}_{s})^2 f_{s}(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} . \end{gather}

Here, the pressure tensor is under the adiabatic assumption and the degrees of freedom are encoded in the adiabatic index ${\Gamma }$. The following relation is found from the definitions:

(2.13)\begin{equation} \epsilon_{s} = \frac{p_{s}}{{\Gamma } - 1} + \frac{1}{2} n_{s} {\boldsymbol{w}}_{s} \boldsymbol{\cdot} {\boldsymbol{w}}_{s}. \end{equation}

The first three moments of the Vlasov equation are called the continuity, momentum and energy conservation equations. A set of these equations is found for each fluid species, but the subscript ${s}$ is neglected here for simplicity

(2.14)\begin{gather} \frac{\partial n}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} (n {\boldsymbol{w}} ) = 0, \end{gather}

(2.15)\begin{gather}\frac{\partial n{\boldsymbol{w}} }{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} [\,p \boldsymbol{1} + n {\boldsymbol{w}} {\boldsymbol{w}} ]= \frac{q}{m} \boldsymbol{S}_{w} (n, {\boldsymbol{w}}, \boldsymbol{B}, \boldsymbol{E}), \end{gather}

(2.16)\begin{gather}\frac{\partial \epsilon}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} [ (\,p + \epsilon ) {\boldsymbol{w}} ] + \frac{1}{{\Gamma } - 1} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{Q}= \frac{q}{m} {\boldsymbol{w}}\boldsymbol{\cdot}\boldsymbol{S}_{w} (n, {\boldsymbol{w}}, \boldsymbol{B}, \boldsymbol{E}). \end{gather}

We assumed the non-relativistic limit and an isotropic pressure tensor with vanishing non-diagonal components, i.e. the inviscid limit. The notation ${\boldsymbol{w}}{\boldsymbol{w}}$ indicates the dyadic product of the two vectors and $\boldsymbol {1}$ is the unit matrix. Similar to the definition of the scalar pressure in (2.12) we use a definition of the heat flux vector, which is normalized to the degrees of freedom as well

(2.17)\begin{equation} \boldsymbol{Q} (\boldsymbol{x}, t) = \frac{{\Gamma } - 1}{2} \int ({\boldsymbol{v}}-{\boldsymbol{w}})^2 ({\boldsymbol{v}} - {\boldsymbol{w}}) f(\boldsymbol{x}, {\boldsymbol{v}}, t)\, \mathrm{d}^3 {v} .\end{equation}

The electromagnetic source term is given by

(2.18)\begin{equation} \boldsymbol{S}_{w} (n, {\boldsymbol{w}}, \boldsymbol{B}, \boldsymbol{E}) = n (\boldsymbol{E} + {\boldsymbol{w}} {\boldsymbol{\times}} \boldsymbol{B}). \end{equation}

The general form of the fluid equations can be written as

(2.19)\begin{equation} {\frac{\partial{\tilde{\boldsymbol{U}}}}{\partial{t}}} + \boldsymbol{\nabla} \boldsymbol{\cdot}{\boldsymbol{\mathsf{F}}}(\tilde{\boldsymbol{U}}) = \boldsymbol{S}(\tilde{\boldsymbol{U}}),\end{equation}

where $\tilde {\boldsymbol {U}} = \tilde {\boldsymbol {U}}(\boldsymbol {x}, t) = (n, n{\boldsymbol{w}}, \epsilon )^\mathrm {T}$ is the fluid state vector at position $(\boldsymbol {x}, t)$, ${\boldsymbol{\mathsf{F}}}$ is the flux matrix and $\boldsymbol {S}$ is the source vector.

Numerically, the complexity of solving (2.19) can be reduced by splitting the operator into less complex sub-operators using Strang operator splitting (Strang Reference Strang1968; Hakim et al. Reference Hakim, Loverich and Shumlak2006). This enables us to use the most appropriate solver for each subsystem sequentially. We split the fluid update into three parts: the flux ${\boldsymbol{\mathsf{F}}}$ excluding the heat flux (see § 2.4), the electromagnetic source $\boldsymbol {S}_\textrm {em}=\boldsymbol {S}_{w} q/m$ (see § 2.5.1) and the heat flux $\boldsymbol {Q}$ (see § 2.6). For commuting operators $\exp (\Delta t \boldsymbol {Q})$ and $\exp (\Delta t \boldsymbol {S}_\textrm {em})$ a second-order accurate Strang splitting is obtained as

(2.20)\begin{equation} \boldsymbol{U}^{n+ 1/2} = \mathrm{e}^{({\Delta t}/{2}) {\boldsymbol{\mathsf{F}}}} \,\mathrm{e}^{\Delta t \boldsymbol{Q}} \,\mathrm{e}^{\Delta t \boldsymbol{S}_{\rm em} } \,\mathrm{e}^{({\Delta t}/{2}) {\boldsymbol{\mathsf{F}}}} \boldsymbol{U}^{n- 1/2} + O(\Delta t^3) .\end{equation}

If $\boldsymbol {Q}$ and $\boldsymbol {S}_\textrm {em}$ act independently on the entries $p$ and ${\boldsymbol{w}}$, respectively, then the order of applying them can be varied and they need to be evaluated only once. In practice, the formulation of $\boldsymbol {Q}$ might partially depend on ${\boldsymbol{w}}$. In this case, Strang splitting is performed on this part of the operator $\boldsymbol {Q}$ as well, see (2.47). In order to apply $\boldsymbol {S}_\textrm {em}$ (and $\boldsymbol {Q}$, which depends on the direction of $\boldsymbol {B}$) one needs to find the electromagnetic quantities at time $t^n$ first. The components of $\boldsymbol {E}$ along the simulated spatial direction can only be updated from time $t^{n-1}$ to $t^n$ after applying $\exp ({\Delta t/2\times {\boldsymbol{\mathsf{F}}}})$ for the first time (see § 2.5.2). Therefore, electromagnetic quantities need to be calculated between these updates. This is unproblematic as ${\boldsymbol{\mathsf{F}}}$ is independent of the electromagnetic field, and we can defer updating $\boldsymbol {E}$ without reducing accuracy.

2.4. Finite volume scheme

The 1D3V fluid equations are solved using a finite volume method, where the fluid equations are averaged over the cell volume, which is an interval of length $\Delta x$ in one dimension,

(2.21)\begin{equation} \boldsymbol{U}_i(t) = \frac{1}{\Delta x}\int_{x_{i- 1/2}}^{x_{i+ 1/2}} \tilde{\boldsymbol{U}}(x, t)\,\mathrm{d} x. \end{equation}

This enables us to correctly conserve the overall fluid mass, fluid momentum and fluid energy, even in the presence of large gradients, by utilizing Gauss’ theorem

(2.22)\begin{equation} \frac{1}{\Delta x} \int_{x_{i- 1/2}}^{x_{i+ 1/2}} {\frac{\partial{\boldsymbol{F}(\tilde{\boldsymbol{U}})}}{\partial{x}}}\, \mathrm{d} x = \frac{1}{\Delta x} [ \boldsymbol{F}_{i+ 1/2} - \boldsymbol{F}_{i- 1/2}], \end{equation}

where the flux through an interface at $x_i$ is $\boldsymbol {F}_{i}(t) = \boldsymbol {F}[\tilde {\boldsymbol {U}}(x_{i}, t)]$, leading to the update equation

(2.23)\begin{equation} {\frac{\partial{\boldsymbol{U}_i(t)}}{\partial{t}}} = \frac{1}{\Delta x} \left[ - \boldsymbol{F}_{i+ 1/2} + \boldsymbol{F}_{i- 1/2} + \int \boldsymbol{S}(\tilde{\boldsymbol{U}}(x, t))\, \mathrm{d} x \right]. \end{equation}

Integrating equation (2.23) in time is achieved by using at least second-order Runge–Kutta methods (Butcher Reference Butcher2016), which is the limit set by the operator splitting scheme. We could not find examples yet, where higher-order Runge–Kutta methods have performed noticeably different from second-order methods in the fluid-PIC code. In contrast to the finite difference scheme used for electromagnetic fields and particles, where electromagnetic quantities are point values, fluid quantities discretized with the finite volume method are cell averages. This is useful, because the finite difference method does not guarantee the conservation of the conservation equations (2.14) through (2.16), which are governing the fluid; while on the other hand using the finite volume method for the electromagnetic fields needs additional steps to satisfy the constraint $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {B}=0$. Hybridization of both schemes to combine the advantages of each has been used before in other contexts, i.e. Soares Frazao & Zech (Reference Soares Frazao and Zech2002).

The maximum time step in the 1D3V Euler equations, which allows for stable simulations, is $\Delta t< C_\mathrm {cfl}\Delta x / (|{w}| + c_{s})$, with the speed of sound $c_{s} = ({\Gamma } p/n)^{1/2}$. For all realistic set-ups these velocities are limited naturally by the speed of light, $|{w}| < c$ and $c_{s} < c$, and this condition is automatically fulfilled by the time step criterion in (2.8). In practice, only (2.8) together with a suitable Courant number of $C_\mathrm {cfl}\leq 0.5$ is used to determine the time step of the simulation.

2.4.1. Reconstruction

To approximate the flux at interfaces, we need to reconstruct the fluid state at cell interfaces. The accuracy of the reconstruction has a crucial influence on the diffusivity. A lower-order reconstruction can lead to excessive damping of waves, which might suppress relevant physical effects on longer time scales.

For reconstructing the point value $\tilde {\boldsymbol {U}}(x_{i+{1/2}}, t)$, which is needed to compute $\boldsymbol {F}_{i+{1/2}}$, we employ a central-weighted essentially non-oscillatory reconstruction (C-WENO) scheme of spatial order five. The reconstruction computes two point values at each interface $x_{i+{1/2}}$, an interpolation from the left- and right-hand sides. We reconstruct the primitive variables $n$, ${\boldsymbol{w}}$ and $p$ individually.

Our implementation of the C-WENO method is based on the fifth-order scheme presented in Capdeville (Reference Capdeville2008). An introduction to the topic can be found in Cravero et al. (Reference Cravero, Puppo, Semplice and Visconti2018a). The C-WENO reconstruction uses a convex combination of multiple low-order reconstruction polynomials to achieve high-order interpolations of the interface values while it employs a nonlinear limiter to degrade this high-order interpolation to a lower order if the reconstructed quantity contains discontinuities. The fifth-order C-WENO uses three third-order polynomials $P_{L}(x), P_{C}(x), P_{R}(x)$ for each cell $i$ to interpolate the four adjacent cells in the following way:

while the optimal fifth-order polynomial interpolates all of them:

$P_\mathrm {opt}(x)\quad$ interpolates values at $\quad i-2\quad i-1\quad i\quad i+1\quad i+2$.

We define an additional polynomial

(2.24)\begin{equation} P_0(x) = \frac{1}{d_0} \left[P_\mathrm{opt}(x) - \sum_{q \in [{L, C, R}]} d_q P_q(x) \right], \end{equation}

where $d_0 + d_{L} + d_{C} + d_{R} = 1$. The polynomials $P_0$, $P_{L}$, $P_{C}$ and $P_{R}$ are a convex representation of the $P_\mathrm {opt}$ polynomial. We use $d_0 = 3 / 4$, $d_{C} = 2 / 16$ and $d_{L} = d_{R} = 1 /16$.

In general, we would like to use the reconstruction provided by the $P_\mathrm {opt}$ polynomial as frequently as possible because of its high-order nature. But this high-order reconstruction can cause oscillations similar to the Gibbs phenomenon at discontinuities. Therefore, we need to employ a limiting strategy to avoid such behaviour. In order to accomplish this, we re-weight all of our $d$-coefficients by taking the smoothness of the associated polynomial into account (Jiang & Shu Reference Jiang and Shu1996). We define

(2.25)\begin{equation} \alpha_q = d_q \left[1 + \left(\frac{\tau}{\text{IS}[P_q] + 10^{-9} \Delta x }\right)^2 \right] \quad \text{for } q \in [{0, L, C, R}], \end{equation}

where $\tau$ is a measure for the overall smoothness of the reconstructed variables, and $\mathrm {IS}[P_q]$ defines a smoothness indicator of the low-order polynomials. Because the formulae for these smoothness indicators are quite cumbersome, we list them in Appendix A. These coefficients define a new set of normalized weights given by

(2.26)\begin{equation} w_q = \frac{\alpha_q}{\alpha_0 + \alpha_{L} + \alpha_{C} + \alpha_{R}} \quad \text{for } q \in [{0, L, C, R}]. \end{equation}

The final reconstructed polynomial is then given by the convex combination of the low-order polynomials using this set of normalized weights

(2.27)\begin{equation} P_\mathrm{rec}(x) = w_0 P_0(x) + w_{L} P_{L}(x) + w_{C} P_{C}(x) + w_{R} P_{R}(x), \end{equation}

which we evaluate at the cell interfaces to calculate the required left- and right-handed interface values for the Riemann solver. We detail how these polynomials are evaluated in Appendix A.

The smoothness indicators $\text {IS}[P_q]$ vanish if the underlying polynomials are smooth. In this case, the re-weighted coefficients reduce to their original value $\alpha _q \rightarrow d_q$ and the reconstructed polynomial reduces to the optimal polynomial $P_\mathrm {rec}(x) \rightarrow P_\mathrm {opt}(x)$.

2.4.2. Riemann solver

The previous reconstruction step determines two, potentially different, values $\tilde {\boldsymbol {U}}_{L}$ and $\tilde {\boldsymbol {U}}_{R}$ for each quantity to the left and right of every interface, thereby providing the initial conditions for the Riemann problem

(2.28)\begin{gather} {\frac{\partial{\boldsymbol{U}}}{\partial{t}}} =- \boldsymbol{\nabla} \boldsymbol{\cdot}{\boldsymbol{\mathsf{F}}}(\tilde{\boldsymbol{U}}) , \end{gather}

(2.29)\begin{gather}\tilde{\boldsymbol{U}}(x, 0) = \begin{cases} \tilde{\boldsymbol{U}}_{L} & x<0 \\ \tilde{\boldsymbol{U}}_{R} & x>0 \end{cases}. \end{gather}

An (approximate) Riemann solver is employed to compute the numerical flux ${\boldsymbol{\mathsf{F}}}(\tilde {\boldsymbol {U}})$. While a number of different families of Riemann solvers have been developed with individual strengths and weaknesses, we have decided to implement multiple solvers which can be changed on demand. Implemented solvers in fluid-SHARP include a Roe solver with entropy fix (Roe Reference Roe1981; Harten & Hyman Reference Harten and Hyman1983) and an HLLC (Harten–Lax–van Leer contact) solver (Toro, Spruce & Speares Reference Toro, Spruce and Speares1994). While the Roe solver yields more accurate solutions and fewer overshoots in our tests in comparison with the HLLC solver, it becomes unstable in near vacuum flows and strong expansion shock waves. Even though differences between the solvers are easily visible in some shock set-ups and artificially extreme conditions, they are typically negligible in most applications common for thermal plasmas. We opt to employ the HLLC solver as our standard for stability purposes and use the Roe solver in cases where stronger shocks with overshoots are expected.

2.4.3. Importance of wave characteristics in approximating stable numerical fluxes

The characteristic curves of the Euler equations without sources correspond to the eigensystem of the flux Jacobian $\partial {\boldsymbol {F}}/\partial {\boldsymbol {U}}$. Approximate Riemann solvers use these characteristics for computing fluxes across small time steps and to introduce numerical dissipation to suppress spurious instabilities. For the hydrodynamic Euler equations in 1D3V (without a heat flux), five characteristics emerge with characteristic wave speeds $\lambda = {{w}_x-c_{s}, {w}_x+c_{s}, {w}_x}$, where the last eigenvalue ${w}_x$ has a multiplicity of 3. In particular, the Roe solver (without the entropy fix) computes the numerical flux at an interface by averaging the physical fluxes as follows (LeVeque Reference LeVeque2002):

(2.30)\begin{equation} \boldsymbol{F}_\mathrm{num} = \tfrac{1}{2} [\boldsymbol{F}(\tilde{\boldsymbol{U}}_{L}) + \boldsymbol{F}(\tilde{\boldsymbol{U}}_{R}) - \boldsymbol{D}_\lambda(\tilde{\boldsymbol{U}}_{R}, \tilde{\boldsymbol{U}}_{L})]. \end{equation}

The dissipation vector $\boldsymbol {D}_\lambda ={\boldsymbol{\mathsf{R}}}|{\boldsymbol { \varLambda }}|{\boldsymbol{\mathsf{R}}}^{-1} (\tilde {\boldsymbol U}_{R}-\tilde {\boldsymbol U}_{L})$ vanishes for $\tilde {\boldsymbol U}_{R}=\tilde {\boldsymbol U}_{L}$. The matrices ${\boldsymbol{\mathsf{R}}}(\tilde {\boldsymbol {U}}_{L}, \tilde {\boldsymbol {U}}_{R})$ and $|{\boldsymbol {\varLambda }}| (\tilde {\boldsymbol {U}}_{L}, \tilde {\boldsymbol {U}}_{R})$ are composed of eigenvectors and a diagonal of the absolute eigenvalues $|\lambda (\tilde {\boldsymbol {U}}_{L}, \tilde {\boldsymbol {U}}_{R})|$, respectively, where an appropriate averaging of left- and right-hand states at the interface is used to derive these eigenvectors and eigenvalues. That is, the dissipation is directly based on the jump at the interface of each wave multiplied by its characteristic wave velocity. The dissipation vector satisfies the subcharacteristic condition, i.e. in characteristic coordinates each eigenvalue is bounded by the dissipation $-D_{\lambda,i} \leq \lambda _i\leq D_{\lambda,i}$, and thus stabilizes the scheme without introducing excessive dissipation (Whitham Reference Whitham1974; Chen & Liu Reference Chen and Liu1993; Hsiao Reference Hsiao1997; LeVeque & Pelanti Reference LeVeque and Pelanti2001). In multi-dimensional scenarios, computing the projection of the difference $(\tilde {\boldsymbol U}_{R}-\tilde {\boldsymbol U}_{L})$ from a Cartesian grid onto the characteristics can result in violations of this condition or excessive dissipation. In these instances, it can be advantageous to artificially alter the wave speeds entering $\boldsymbol D_\lambda$. Reduced wave speeds can be used to successfully counter excessive dissipation leading to a wrong convergence for low Mach number flows (Dellacherie Reference Dellacherie2010), however, this leads to numerical instabilities when applied to high Mach number flows. On the other hand, increased wave speeds have been found to eliminate numerical instabilities at shocks (Peery & Imlay Reference Peery and Imlay1988). While the HLLC solver makes more sophisticated approximations to the wave speeds of the nonlinear system, the principle of artificially increasing selected wave speed estimates yields the same result (Simon & Mandal Reference Simon and Mandal2019).

To provide an understanding of how these characteristics influence the operator splitting, suppose the following decomposition of the total flux $\boldsymbol {F}_{{t}} = \boldsymbol {F}_A + \boldsymbol {F}_B$ into two fluxes. Hence, we need to compare the numerical estimate of the total flux with that of the individual subsystems, denoted by $\boldsymbol {F}_A$ and $\boldsymbol {F}_B$. The expansion of nonlinear systems provided by Strang (Reference Strang1968) yields $\boldsymbol {U}^{n+1}_i = \boldsymbol {U}^n_i + \Delta t [\boldsymbol {\Delta F}_{i,A}(\boldsymbol {U}^n) + \boldsymbol {\Delta F}_{i,B}(\boldsymbol {U}^n)] + {O}(\Delta t^2)$, where we only use terms up to first order for simplicity. The intracell flux $\Delta \boldsymbol {F}_i=-(\boldsymbol {F}_{\mathrm {num}, i+1/2} - \boldsymbol {F}_{\mathrm {num},i-1/2})/\Delta x$ is used for updating $\boldsymbol {U}$ (see (2.23)). The numerical flux at an interface for one time step is thus

(2.31)\begin{equation} \boldsymbol{F}_\mathrm{num} = \begin{cases} \tfrac{1}{2} [\boldsymbol{F}_{t}(\tilde{\boldsymbol{U}}_{L}) + \boldsymbol{F}_{t}(\tilde{\boldsymbol{U}}_{R}) - \boldsymbol{D}_{\lambda, {t}}(\tilde{\boldsymbol{U}}_{R}, \tilde{\boldsymbol{U}}_{L}) ] & \text{if unsplit} \\ \tfrac{1}{2} [\boldsymbol{F}_{t}(\tilde{\boldsymbol{U}}_{L}) +\boldsymbol{F}_{t}(\tilde{\boldsymbol{U}}_{R})- \boldsymbol{D}_{\lambda, A}(\tilde{\boldsymbol{U}}_{R}, \tilde{\boldsymbol{U}}_{L}) - \boldsymbol{D}_{\lambda, B}(\tilde{\boldsymbol{U}}_{R}, \tilde{\boldsymbol{U}}_{L})] & \text{if split} \end{cases}. \end{equation}

Both numerical fluxes converge to the total physical flux; for vanishing dissipation vectors, e.g. $\tilde {\boldsymbol {U}}_{R}=\tilde {\boldsymbol {U}}_{L}$, both formulations are equal. Intuitively, the total strength of the dissipation matrix in the unsplit scheme is smaller $\textrm {tr}|{\boldsymbol {\varLambda }}_t| \leq \textrm {tr}|{\boldsymbol {\varLambda }}_A| + \textrm {tr}|{\boldsymbol {\varLambda }}_B|$, while the split scheme is stable provided that the subsolvers are stable (Strang Reference Strang1968). As an important consequence, a split Riemann solver only needs to account for the characteristics in the subsystem. This conveniently allows us to use specific solvers for each subsystem without taking into consideration the other systems of equations. Another possibility is to convert the divergence of a flux into a source term, which eliminates the need for a Riemann solver but results in the loss of guaranteed conservation in the finite volume scheme. We provide two applications, for which this is useful.

First, the heat flux vector $\boldsymbol {Q}$, which is indeed a physical flux, results in a non-trivial change of the wave characteristics. Instead of including this complexity in the Riemann solver here, it is simpler to treat its divergence ($\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {Q}$) as a source term instead (see § 2.6).

Second, in the MHD limit, the evolution equation of the electromagnetic momentum $\epsilon _0 \partial _t \boldsymbol {E} {\boldsymbol {\times }} \boldsymbol {B} = -\rho \boldsymbol {E} - \boldsymbol {J} {\boldsymbol {\times }} \boldsymbol {B} + {\boldsymbol {\nabla }} {\boldsymbol {\cdot }} {\boldsymbol{\mathsf{T}}}_\mathrm {em}$ collapses to the constraint $\boldsymbol {J} {\boldsymbol {\times }} \boldsymbol {B} \sim {\boldsymbol {\nabla }} {\boldsymbol {\cdot }} (\boldsymbol {B} \boldsymbol {B} - \boldsymbol {1}B^2/2)/\mu _0$ (see, e.g. Braginskii Reference Braginskii1965), where the Maxwell stress tensor is given by ${\boldsymbol{\mathsf{T}}}_\mathrm {em} = \epsilon _0 \boldsymbol {E} \boldsymbol {E} + \boldsymbol {B} \boldsymbol {B}/\mu _0 - 0.5 (\epsilon _0 E^2+B^2/\mu _0 )\boldsymbol {1} \approx (\boldsymbol {B} \boldsymbol {B} - \boldsymbol {1}B^2/2)/\mu _0$. That is, the electromagnetic source term can be expressed as a divergence of an electromagnetic flux tensor. The MHD solvers make use of this constraint and, by including electromagnetic fluxes into their total flux, the MHD Riemann solvers add dissipation based on the full MHD wave characteristics. Failing to do so leads to numerical instabilities, especially for large magnetic field strengths, as in general the fast magnetosonic wave is faster than the characteristic waves treated in our scheme. However, because this source-flux equivalency is invalid without the MHD assumptions, we must include the Lorentz force as a source term (see the following § 2.5.1) and consequently do not use the MHD characteristic velocities in our Riemann solver. In § 3.2, we demonstrate the accuracy of our implemented algorithm for propagating MHD waves, demonstrating that the numerical dissipation is sufficient to suppress potential numerical instabilities.

2.5. Electromagnetic interaction with charged fluids

In this section, we first introduce the Lorentz force as a source term in (2.15). Furthermore, we describe how the fluid influences the electromagnetic fields. With these two additional parts, the description of an uncharged gas in § 2.4 is expanded here to include plasmas.

2.5.1. Treatment of electromagnetic source term

Instead of integrating the energy equation (2.16), which would require evaluating the source term on the right-hand side, we convert $\epsilon$ into $p$ before applying the source update $\exp (\Delta t \boldsymbol {S}_\textrm {em})$ (see (2.20)). Consequently, we compute the time evolution of the primitive pressure variable, for which the electromagnetic source term conveniently vanishes

(2.32)\begin{equation} \frac{\partial p}{\partial t} + {\Gamma } p \boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{w}} + {\boldsymbol{w}} \boldsymbol{\cdot} \boldsymbol{\nabla} p + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{Q} = 0. \end{equation}

Then, only the computation of the source term for the momentum equation (2.15) is left. Up until now we have only applied the C-WENO method for conservation laws, however, by adding the source term, we are left with a balance law. In C-WENO formulations for balance laws it is customary to approximate the integral of the source term ((2.23)) numerically to higher orders as well (Cravero et al. Reference Cravero, Puppo, Semplice and Visconti2018b). We use Simpson's formula for approximating (2.23)

(2.33)\begin{equation} \int_{x_{i-1/2}}^{x_{i+1/2}} \boldsymbol{S}(\tilde{\boldsymbol{U}})\, \mathrm{d} x = \tfrac{1}{6}(\boldsymbol{S}(\tilde{\boldsymbol{U}}_{i- 1/2}) + 4 \boldsymbol{S}(\tilde{\boldsymbol{U}}_{i}) + \boldsymbol{S}(\tilde{\boldsymbol{U}}_{i+ 1/2})) + O(\Delta x^5), \end{equation}

where the intracell values $\tilde {\boldsymbol {U}}_{i\pm 1/2}$ are interpolated by the same C-WENO scheme as used for solving the hydrodynamical equations, and the centre value is computed self-consistently with the numerical integration formula, i.e. $\tilde {\boldsymbol {U}}_{i} = (6 \boldsymbol {U}_{i} - \tilde {\boldsymbol {U}}_{i+1/2} - \tilde {\boldsymbol {U}}_{i-1/2})/4$. We also need to interpolate the electromagnetic field values to a comparable spatial order. This is achieved by performing finite difference interpolations for each component from the Yee mesh discretized fields, that is

(2.34)\begin{equation} \boldsymbol E_{i+ 1/2} = \frac{150 (\boldsymbol E_{i}+\boldsymbol E_{i+1}) - 25 (\boldsymbol E_{i-1}+\boldsymbol E_{i+2}) + 3 (\boldsymbol E_{i-2}+\boldsymbol E_{i+3})}{256} + O (\Delta x^6), \end{equation}

and temporal order, $\boldsymbol B^n=(\boldsymbol B^{n+1/2}+ \boldsymbol B^{n-1/2})/2$, again, for each component necessary. Lower-order approximations produce, in our tests, similar results, but converge to slightly lower wave frequencies when compared with the analytical solution of the dispersion relation. We apply $\boldsymbol {S}(\tilde {\boldsymbol {U}}_{i})$ by using an implicit velocity update

(2.35)\begin{equation} \frac{{\boldsymbol{w}}^{n+ 1/2}_i - {\boldsymbol{w}}^{n- 1/2}_i}{\Delta t} = \frac{q}{m}\left[\boldsymbol E^n_i + \frac{1}{2}({\boldsymbol{w}}^{n+ 1/2}_i + {\boldsymbol{w}}^{n- 1/2}_i) \boldsymbol{\times} \boldsymbol B^n_i\right], \end{equation}

which is solved using the Boris velocity integration (Boris et al. Reference Boris1970). The splitting of fluid flux update and Lorentz force ((2.20)) is reminiscent of pushing a particle with the PIC-method, where the Lorentz force for a full-time step is calculated in between half-time step updates of the position vector.

2.5.2. Deposition of charges

Equations (2.4) govern the electric field evolution, where Faraday's or Gauss’ law might be used to compute $\boldsymbol {E}$. In this section we focus on the one-dimensional set-up without particle contributions, which are explained in § 2.7. The perpendicular components’ update, $E_y$ and $E_z$, is received straightforwardly by discretizing Faraday's law

(2.36)\begin{gather} (E_{y})^{n+1}_{i+ 1/2} = (E_{y})^n_{i+ 1/2} -\sum_s \frac{\Delta t}{\epsilon_0} q_s (n {w}_y)_{i+1/2, s}^{n+ 1/2} - \frac{c^2 \Delta t}{\Delta x} [ (B_z)_{i+1}^{n+ 1/2} - (B_z)_{i}^{n+ 1/2} ], \end{gather}

(2.37)\begin{gather}(E_{z})^{n+1}_{i+ 1/2} = (E_{z})^n_{i+ 1/2} -\sum_s \frac{\Delta t}{\epsilon_0} q_s (n {w}_z)_{i+1/2, s}^{n+ 1/2} + \frac{c^2\Delta t}{\Delta x} [ (B_y)_{i+1}^{n+ 1/2} - (B_y)_{i}^{n+ 1/2} ], \end{gather}

where the sum is taken over all fluid species ${s}$ and $n {\boldsymbol{w}}$ are components of the fluid vector $\boldsymbol {U}$.

For the $E_x$ component in spatial direction however, in order to enforce charge conservation, Gauss’ law in discretized form needs to be enforced for all $i\ge 1$ as well

(2.38)\begin{equation} (E_{x})^{n}_i = (E_{x})^n_0 + \sum_s \frac{q_s}{\epsilon_0} \sum_{j=0}^{i-1}n^n_{j+1/2, s} \Delta x =(E_{x})^n_0 + \sum_s \frac{q_s}{\epsilon_0} \int_{x_{0}}^{x_{i}} \tilde{n}_s^n\, \mathrm{d} x, \end{equation}

where the second equality uses the definition of cell averages in the finite volume scheme (see (2.21)) and shows, that this numerical formula is exact. Another formula for updating $(E_x)_0$ to the time step $n$ is still needed. In the analytical case Gauss’ law in combination with the density conservation equation (2.14) for the analytical flux (or cell values) $J_x\propto q n{w}_x$ can be shown to be equivalent to Faraday's law; in the numerical case this equivalency is shown using the discretized conservation equation and corresponding numerical flux $J_x\propto q F_n(\tilde {\boldsymbol {U}}) \simeq q n {w}_x$ for the current density $J_x$. Taking the time derivative of (2.38) in conjunction with the discretized density update (2.23) leads to the expression

(2.39)\begin{align} & \frac{(E_{x})^{n+1}_i - (E_{x})^n_i}{\Delta t}+\frac{ (E_{x})^{n+1}_0- (E_{x})^n_0}{\Delta t} \nonumber\\ & \quad =\sum_s \frac{q_s}{\epsilon_0\Delta t}\int_{t_n}^{t_{n+1}}[-( F_{n, s})_{i} + (F_{n, s})_{0}]\, \mathrm{d} t. \end{align}

The integration in time using Runge–Kutta methods is the same as used to solve (2.23). Faraday's law using fluxes in one spatial dimension is then given by

(2.40)\begin{equation} (E_{x})^{n+1}_i = (E_{x})^n_i - \sum_s \frac{q_s}{\epsilon_0} \int_{t_n}^{t_{n+1}} [F_n(\tilde{\boldsymbol{U}})]_{i, s} \,\mathrm{d} t, \end{equation}

and enables us to identify $J_x$ by comparison with the charge conservation equation ((2.14) multiplied by $q_{s}$)

(2.41)\begin{equation} (J_{x})^{n+{1/2}}_{i} =\sum_s \frac{q_s}{\Delta t} \int_{t_n}^{t_{n+1}} [ F_n(\tilde{\boldsymbol{U}})]_{i, s} \,\mathrm{d} t. \end{equation}

Note that the numerical flux also includes numerical diffusion and is directly related to changes in $\rho$. Due to this, other formulations for $J_x$ violate the charge conservation equation and can lead to numerical instabilities.

2.5.3. Magnetic field evolution

Because the fluid evolution influences the magnetic field only indirectly, the finite difference time-domain (FDTD) update for the magnetic field is unchanged from the previous SHARP code. For completeness we reproduce the formulae here (Shalaby et al. Reference Shalaby, Thomas and Pfrommer2021)

(2.42)\begin{gather} (B_y)_i^{n+ 1/2} = (B_y)_i^{n- 1/2} + \frac{\Delta t}{\Delta x}( (E_z)_{i+ 1/2}^{n} - (E_z)_{i- 1/2}^{n} ), \end{gather}

(2.43)\begin{gather}(B_z)_i^{n+ 1/2} = (B_z)_i^{n- 1/2} - \frac{\Delta t}{\Delta x}( (E_z)_{i+ 1/2}^{n} - (E_y)_{i- 1/2}^{n} ). \end{gather}

Here, $B_x$ is constant in the 1D3V model because of the requirement $\boldsymbol {\nabla } \boldsymbol {\cdot }\boldsymbol {B}=0$.

2.6. Landau closure for fluid species

The highest retained fluid moment, which is in our case the specific heat flux $\boldsymbol {Q}$, is not evolved in our set of equations. Instead, we need to estimate its value dynamically using an appropriate closure. The simple ideal gas closure sets $\boldsymbol {Q}=\boldsymbol {0}$, which, however, prevents the energy dissipation of plasma waves. One important mechanism of such a dissipation is the collisionless damping of electrostatic waves achieved through Landau damping. Landau damping is a microphysical kinetic wave–particle interaction, where particles resonate with the wave to exchange energy as a function of time. In essence, the resonant particles accelerate or decelerate to approach the wave's phase velocity, thereby picking up energy or releasing it, respectively. For Maxwellian phase-space distributions, there are more particles at velocities smaller than the phase velocity, which yields a net damping, i.e. energy loss of the wave (Boyd & Sanderson Reference Boyd and Sanderson2003).

Various attempts, e.g. by Hammett & Perkins (Reference Hammett and Perkins1990), were carried out to approximate the heat flux $\boldsymbol {Q}$ of an almost Maxwellian distributed plasma, such that the kinetic phenomenon of Landau damping is mimicked in the linearized fluid equations. Landau damping is a non-isotropic effect, which can be reflected in the fluid descriptions. Accounting for the gyrotropy of the system around the magnetic field, the double-adiabatic law with two adiabatic coefficients parallel and perpendicular to the magnetic field can be adopted, which might be extended using an appropriate heat flux vector (Snyder, Hammett & Dorland Reference Snyder, Hammett and Dorland1997; Hunana et al. Reference Hunana, Tenerani, Zank, Goldstein, Webb, Khomenko, Collados, Cally, Adhikari and Velli2019a,Reference Hunana, Tenerani, Zank, Khomenko, Goldstein, Webb, Cally, Collados, Velli and Adhikarib). This is achieved by first decomposing the pressure tensor in a coordinate system that is aligned with the direction of the magnetic field $\boldsymbol {b}=\boldsymbol {B}/|\boldsymbol {B}|$, yielding ${\boldsymbol{\mathsf{p}}}=p_\parallel \boldsymbol {b}\boldsymbol {b}+p_\perp (\boldsymbol {1}-\boldsymbol {b}\boldsymbol {b})$. The transport of parallel and perpendicular heat along the magnetic field then corresponds to the terms $Q_\parallel \boldsymbol {b}$ and $Q_\perp \boldsymbol {b}$, respectively. In this publication, our algorithm is restricted to pressures with only one common adiabatic coefficient for parallel and perpendicular pressures. We leave the possibility of implementing anisotropic double-adiabatic systems open for future extensions of our algorithm. Hence, we model only the scalar heat flux $Q=\boldsymbol {b}\boldsymbol {\cdot }\boldsymbol {Q}$, which is a simplification of the double-adiabatic modelling from the aforementioned literature, i.e. $2Q/({\Gamma } -1)=Q_\parallel + 2 Q_\perp$. In our model, we can assume an isotropized pressure tensor ${\boldsymbol{\mathsf{p}}}=p \boldsymbol {1}$ using the adiabatic coefficient ${\Gamma } = 5/3$. To do this, we set $p = p_\parallel =p_\perp$ everywhere without explicitly modelling $Q_\perp$ and the corresponding perpendicular pressure equation. Instead, we choose $Q$ such that the linear Landau-damping rate of the isotropic system is comparable to that of an anisotropic electrostatic system with the same $p_\parallel$ (for details refer to comments below (2.46)). This is a simplifying assumption, which does not follow from the kinetic equations, because a rigorous treatment necessitates solving the perpendicular pressure equation. Furthermore, isotropization mainly results from collisions, while collisionless systems are rarely isotropic and Maxwellian as we assume. Nevertheless, this is a convenient choice for damping waves with an isotropic background plasma when particle heating is negligible compared with their thermal energy. A more physically motivated anisotropic pressure tensor ${\boldsymbol{\mathsf{p}}}=p\boldsymbol {e}_x\boldsymbol {e}_x$ is attained for ${\Gamma } = 3$, where $\boldsymbol {e}_x$ is the unit vector in the $x$-direction. This approximates the kinetic equations well when $p\simeq p_\parallel$. Since we model only components of $\boldsymbol {Q}$ parallel to $\boldsymbol {b}$, the heat flux enters into the pressure evolution equation (2.32) as

(2.44)\begin{equation} {\frac{\partial{p}}{\partial{t}}} \propto-\boldsymbol{\nabla} \boldsymbol{\cdot} {(\boldsymbol{b} Q)}. \end{equation}

In practice, we make the assumption of locally constant (or slowly varying) magnetic fields on top of the 1D3V geometry, that is $\boldsymbol {\nabla }\boldsymbol {\cdot } (\boldsymbol {b} Q) \simeq \cos (\varTheta ) \partial _x Q$, where the angle $\varTheta$ is measured between the background magnetic field $\boldsymbol {B}_0$ and the $x$-axis.

Here, we will introduce two different formulae for heat flux closures. The first and most popular collisionless electrostatic closure was proposed by Hammett & Perkins (Reference Hammett and Perkins1990). We refer to it as the $R_{32}$ closure throughout this paper, and it approximates the heat flux at a fixed ${\Gamma } = 3$, in Fourier space, by

(2.45)\begin{equation} \hat{Q} =- {\mathrm{i}}\,\mathrm{sign}(k) \frac{2}{\sqrt{{\rm \pi} }} \sqrt{2 \theta_0} c n_0 {k_{\rm B}} \frac{\hat{T}}{m}\equiv\hat{Q}_T.\end{equation}

Here, hats are used to denote quantities in Fourier space along the magnetic field line, i.e. $\hat {Q}=\mathcal {F}_\parallel (Q)$, and the subscript $0$ refers to simulation box averages, that is $n_0= \sum _{i=0}^{N_{c}} n_i/ N_{c}$ is an average over all $N_{c}$ cells. Furthermore ${k_{\rm B}}$ is the Boltzmann constant, and ${k_{\rm B}} \hat {T} = ( m \hat {p} - {k_{\rm B}} T_0\hat {n})/n_0$. Since the plasma average or equilibrium temperature evolves slowly as a function of time, we adjust the background temperature $T_0$ after every time step to synchronize it with the mean pressure, ${k_{\rm B}} T_0(t)/m = p_0 (t) / n_0$, while the density conservation ensures that $n_0$ stays constant. Note also that $Q_0=0$. The dimensionless mass-normalized temperature is $\theta _0 = {k_{\rm B}} T_0/(m c^2)$.

A more recent approximation was proposed by Hunana et al. (Reference Hunana, Zank, Laurenza, Tenerani, Webb, Goldstein, Velli and Adhikari2018), who restricts this closure to ${\Gamma } =3$ only, for reasons mentioned already. We use an ad hoc formulation of their closure with a variable ${\Gamma }$, thereby allowing our simplified model to be used. They also introduce the nomenclature $R_{mn}$ adopted here, which is used to denote that the kinetic plasma response function $R$ is mimicked for this closure by a Padé approximant with polynomials $P_m/Q_n=R_{mn}$ of orders $m$ and $n$. We refer to their closure as $R_{31}$ and it approximates the heat flux, in Fourier space, by

(2.46)\begin{equation} \hat{Q} = \underbrace{\left(\frac{4}{4-{\rm \pi}} - {\Gamma }\right) p_0 \hat{w}}_{\hat{Q}_{w}} + \underbrace{\left(-{\mathrm{i}}\,\mathrm{sign}(k) \frac{\sqrt{2{\rm \pi}\theta_0}}{4-{\rm \pi}} c n_0 \frac{{k_{\rm B}} \hat{T}}{m}\right)}_{\hat{Q}_T}.\end{equation}

The $R_{31}$ closure may be seen as a generalization of the $R_{32}$ closure with an additional degree of freedom in $\hat {w}$, which can mimic the kinetic damping more accurately over a broader spectrum. We choose this closure as our fiducial model because it requires only a relatively inexpensive local derivative to compute the additional term dependent on $\hat {w}$. This additional term effectively increases the speed of sound obtained from the non-electromagnetic fluid equations and allows us to retrieve the correct damping rate with our ad hoc assumption of a specific value of ${\Gamma }$, see Appendix C. This means, that adopting a value of ${\Gamma }$, e.g. ${\Gamma } =5/3$ in our isotropic model, does not change the linear dispersion relation associated with this closure. For ${\Gamma } = 3$, we retrieve the coefficient for $\hat {w}$ from the aforementioned literature ${4}/({4-{\rm \pi} }) - 3 = ({3{\rm \pi} - 8})/({4-{\rm \pi} })$.

In only one spatial dimension, as assumed in our code, the global integration along a magnetic field line is approximated to be along the spatial direction, i.e. $\mathcal {F}_\parallel =\mathcal {F}_x$. An extension to multiple spatial dimensions with an anisotropic pressure tensor is not straightforward because, in this case, this approach can lead to spurious instabilities (Passot et al. Reference Passot, Henri, Laveder and Sulem2014) and the integration would need to be carried out along magnetic field lines.

A kinetic code does not need global communication to accurately reproduce Landau damping, since each particle (or particle bin) tracks its own interaction with each wave mode as a function of time and accumulates this information in the particle velocity. However, after integrating out the individual particle velocities when building the evolution equations for the phase-space distribution function, i.e. (2.14)–(2.16), information about the individual particle–wave interaction is no longer collected. Because some information about this interaction is also contained in the wave, such non-local information can be used to approximate the gradient of the physical heat flux, i.e. a closure of the fluid moments that incorporates such missing information. This non-local information is approximated in (2.45) and (2.46), and is manifested by the term ${\mathrm {i}}\, \mathrm {sign}(k)$ in Fourier space, which is also referred to as the Hilbert transform.

Numerically, we do not include the heat flux in the Riemann solver used to compute the fluid fluxes. Instead, we compute the spatial derivative of the heat flux $\boldsymbol {\nabla }_\parallel \boldsymbol {\cdot } \boldsymbol Q$ separately. We use Strang splitting for the ${\boldsymbol{w}}$-dependent part $\boldsymbol {Q}_{w}$ and the temperature dependent part $\boldsymbol {Q}_T$ to expand (2.20) into

(2.47)\begin{equation} \boldsymbol{U}^{n+ 1/2} = \mathrm{e}^{({\Delta t}/{2}) {\boldsymbol{\mathsf{F}}}} \,\mathrm{e}^{({\Delta t}/{2}) \boldsymbol{Q}_{w}} \, \mathrm{e}^{\Delta t \boldsymbol{Q}_T} \,\mathrm{e}^{\Delta t \boldsymbol{S}_{\rm em} } \,\mathrm{e}^{({\Delta t}/{2}) \boldsymbol{Q}_{w}} \, \mathrm{e}^{({\Delta t}/{2}) {\boldsymbol{\mathsf{F}}}} \boldsymbol{U}^{n- 1/2} + O(\Delta t^3), \end{equation}

such that only one non-global evaluation of $\boldsymbol {Q}_T$ is needed. Using Heun's method together with the fast Fourier transform (FFT) the update formulae for the pressure with respect to operators $\boldsymbol {Q}_{w}$ and $\boldsymbol {Q}_T$ are, respectively,

(2.48)\begin{gather} p^{n+1}|_{Q_{w} } = \mathrm{e}^{\Delta t Q_{w} } p^n = p^n + \Delta t a_{w} p_0 \boldsymbol{\nabla}_\parallel \boldsymbol{\cdot} {\boldsymbol{w}}, \end{gather}

(2.49)\begin{gather}p^{n+1}|_{Q_T} = \mathrm{e}^{\Delta t Q_T} p^n = p^n + \Delta t \mathcal{F}^{-1}_\parallel \left[|k| a_T \left(1+\frac{\Delta t}{2} |k| a_T\right) \hat{T}^n \right], \end{gather}

where the derivative in Fourier space was obtained by multiplying with ${\mathrm {i}} k$ and the inverse FFT is denoted by $\mathcal {F}^{-1}$. For the $R_{31}$ closure the coefficients are given by $a_{w} =4/(4-{\rm \pi} )$ and $a_T= (4-{\rm \pi} )^{-1} (2{\rm \pi} \theta _0)^{1/2} c n_0 {k_{\rm B}}/m$, while for the $R_{32}$ closure these are given by $a_{w} = 0$ and $a_T=2 (2 \theta _0/{\rm \pi} )^{1/2} c n_0 {k_{\rm B}} /m$. Both closures compute a term proportional to $\hat {T}$ (cf. (2.49))

(2.50)\begin{equation} {\mathrm{i}} k \hat{Q} \propto- {\mathrm{i}}\, \mathrm{sign}(k) {\mathrm{i}} k a_T \hat{T} = |k| a_T \hat{T}. \end{equation}

Computing this term naively using the FFT is expensive. This is why, in the following, we present local, semi-local and efficient global (Fourier transform-based) numerical approximations of the Landau closures, which we have implemented in the fluid-SHARP code.

2.6.1. Local approximations of the Hilbert transform

The phase shift between the wanted derivative ${\mathrm {i}} k \hat {Q}$ and the input of $\hat {T}$ in (2.50) is exactly $0$, while the amplitude is proportional to $|k|$. This is therefore a special case ($a=1$) of the fractional Riesz derivative $\partial ^{a}/\partial |x|^a$ with Fourier representation

(2.51)\begin{equation} \mathcal{F}\left( \frac{\partial^a f(x)}{\partial|x|^a} \right) =-|k|^a \hat{f}(k), \end{equation}

where $a \in \mathbb {R}$. Note that all approximations mentioned here only introduce errors in the amplitude of $|k|$, but not in its phase. This makes them easier to integrate into simulations in comparison with approximations which are not designed to prevent phase errors, because large phase errors (between ${\rm \pi} /2$ and $3{\rm \pi} /2$) in any wave mode transform the damping term into an exponentially growing numerical instability. The local approximations make use of the fact, that the fractional Riesz derivative is local and cheap to evaluate for the special case $a = 2m$ with $m \in \mathbb {N}^0$, where it reproduces the usual derivative $\partial ^{2m}/\partial {|x|}^{2m} = (-1)^{m+1}\partial ^{2m}/\partial x^{2m}$. Wang et al. (Reference Wang, Hakim, Bhattacharjee and Germaschewski2015) use $a=0$, while Allmann-Rahn et al. (Reference Allmann-Rahn, Trost and Grauer2018) and Ng et al. (Reference Ng, Hakim, Wang and Bhattacharjee2020) approximate the non-isotropic pressure tensors with $a=2$. These approximations are scaled to a characteristic wavenumber $k_0$ at which the damping is expected to occur.

The choice of $a=0$ means that the approximation is a scalar

(2.52)\begin{equation} {\mathrm{i}} k \hat{Q} \propto |k_0| \hat{T}, \end{equation}

while the gradient-driven closures with $a=2$ use

(2.53)\begin{equation} {\mathrm{i}} k \hat{Q} \propto \frac{k^2}{|k_0|} \hat{T} . \end{equation}

The gradient-driven closures are equal to the FFT solution at two wavelengths, $0$ and $k_0$, while the scalar closure is only exact at $k_0$, see figure 1. Since ${\mathrm {i}} k\hat {Q}$ is not computed alongside the conserved fluxes in the Riemann solver, energy conservation is only preserved if the mean energy does not increase. To achieve this, the approximation for the derivative of the heat flux needs to vanish at wavenumber $0$, which the scalar approximation does not fulfil.

Figure 1. The magnitude of the frequency response, which is a quantification of how much the amplitude at a specific frequency is amplified or suppressed, of different approximations of the derivative of the Hilbert transform. Here, $\hat {k}$ is given in normalized frequencies (with regards to the Nyquist frequency), while the negative frequencies in the interval $[-{\rm \pi}, 0]$ are not shown here due to the symmetric dependence of all plotted values on $|k|$. The FFT-based approach reproduces the correct, linear response. The scalar- and gradient-driven closures are given by (2.52) and (2.53), respectively, with the parameter $k_0$ marked as a grey vertical line. The FIR filter is described by (2.55).

Because fluid closures are only approximately mimicking kinetic Landau damping anyway, these local approximations to the fluid closures are useful to save computational cost. Furthermore they are easier to implement, especially when the full pressure tensor is computed. However, they may lead to misleading results in multiscale simulations, where multiple characteristic damping lengths are present and depend on the estimate of $k_0$. For example, Allmann-Rahn, Lautenbach & Grauer (Reference Allmann-Rahn, Lautenbach and Grauer2022) show a case where ion and electron heating intensities are switched qualitatively.

2.6.2. Semi-local approximations of the Hilbert transform

While the less accurate local approximations use an arbitrary value of $k_0$, the FFT is expensive and depends on periodic boundary conditions. Here, we aim to have a fallback algorithm as a compromise between both approaches.

A digital FIR filter can be designed to approximate the non-local effects by convolving the simulation data with adjacent auxiliary data points, where the filter length determines the maximum distance. For example, an asymmetric filter with an even number of entries is applied on an input $x$ using filter coefficients $b_j$, producing the output $y$

(2.54)\begin{equation} y_{i+0.5} = \sum_{j=-(N_f/2-0.5)}^{N_f/2-0.5} b_j x_{i+j+0.5} . \end{equation}

A numerical derivative is then an asymmetrical filter with $N_f=2$ and coefficients $b_{\pm 0.5} = \pm 1/\Delta x$, such that $y_{i+0.5} = (x_{i+1} - x_i)/\Delta x$. Figure 1 shows the magnitude of the frequency response. The gradient-driven case shows a quadratic $k^2$ dependence, which is suppressed for larger $k$. This is due to the relatively small uneven filter length of $7$ used here; the filter length is an important parameter, since it influences the accuracy of the approximation. With a filter length corresponding to the simulation box size the results can converge to the FFT-based algorithm (i.e. the $k^2$ dependence is not suppressed at higher $k$), if the filter is designed appropriately. As noted previously, the local closures do not converge to $\partial /\partial |x|$. A correct convergence for approximating $\partial /\partial |x|$ is obtained through the high-order formulation by Ding, Li & Chen (Reference Ding, Li and Chen2015). However, this filter violates energy conservation for smaller filter length and is thus not suitable for our case. Instead, we construct the filter by adopting a convolution of two sub-filters, each of which has an odd amount of asymmetric entries (termed a type IV filter) similar to the numerical derivative mentioned already. By design, their output has a vanishing mean, thereby guaranteeing energy conservation. A symmetric splitting into the sub-filters $\partial /\partial |x| = (\partial ^{1/2}/\partial |x|^{1/2} )^2$ is possible, however, its frequency response is not monotonic (and has visible ripples) for small filter lengths. This leads to the non-physical case that some waves at a particular wavenumber $k$ are damped less than their slightly larger scale waves at $k-\delta k$.

Instead, we opt to use the intuitive splitting of $\partial /\partial |x|=\partial /\partial {x} \mathcal {H}$ where the Hilbert-transform filter $\mathcal {H}$ is equivalent to $-{\mathrm {i}}\, \mathrm {sign}(k)$ in Fourier space. The filter $\mathcal {H}$ has coefficients $b_{j}=1/({\rm \pi} j)$. We derive an equivalent formulation to (2.49), which is first order in time, by applying the derivative and Hilbert-transform filters successively, i.e.

(2.55)\begin{equation} p^{n+1} = p^n + \Delta t a_T {\frac{\partial{}}{\partial{x}}} \sum_{j=-(N_f/2-0.5)}^{N_f/2-0.5} \frac{1}{{\rm \pi} j} T^n_{i+j+0.5}. \end{equation}

Note that the derivative is also computed by convolution and has a separate filter length corresponding to its spatial order. We opt to use the same spatial order as in the C-WENO reconstruction for the finite volume scheme.

Even for small Hilbert-transform filter lengths in comparison with the number of cells, e.g. $N_f/N_{c} = 0.04$ as shown in figure 1, this formulation dramatically improves the accuracy of multiscale problems in comparison with local approximations. Here, $N_f$ is critical for the accuracy at small wavenumbers $k$, while the spatial order of the derivative is critical for the accuracy at large $k$. Most importantly, this semi-local approach does not require setting of an arbitrary damping scale $k_0$ such as the local approximations mentioned before. The only parameter of this approach is the filter length, which should be chosen to be sufficiently large.

2.6.3. Efficient FFT-based computation of the Hilbert transform

Provided the plasma background is uniform and periodic, the most accurate while computationally most expensive results are achieved by computing the heat flux of the fluid in Fourier space. While the FFT is easy to compute on a single computer using standard numerical libraries, our code is parallelized using MPI and an efficient one-dimensional FFT is needed. The computation of the Fourier transform is expensive for two reasons:

1. (i) globally, each Fourier component needs to be informed about data from every other computational cell (which may be stored on a different processor); and

2. (ii) the Fourier transform is not easily parallelizable in one dimension, which precludes an efficient scalable Fourier algorithm.

This naturally limits the overall computational scalability of the fluid part of the code. Communication over multiple MPI processes is time consuming because of latency and finite bandwidth. For this reason, parallel FFT algorithms are prone to become a computational bottleneck. However, using non-blocking MPI routines to perform communication in the background can be used while the high computational load of the particles is carried out. Thus, in our case of a combined fluid and PIC algorithm, the communication required for an accurate FFT-based heat flux computation is comparatively computationally cheaper, even with relatively small numbers of PIC particles. Hence, in our case the FFT algorithm does not necessarily become a bottleneck for larger problems.

In order to distribute the computational load of the FFT, we employ a four-step algorithm in the first step of the computation (Bailey Reference Bailey1990; Takahashi & Kanada Reference Takahashi and Kanada2000), which extends the Cooley–Tukey algorithm (Cooley & Tukey Reference Cooley and Tukey1965) for multiple processors. We shortly describe the algorithm for complex input data as found in the literature and afterwards adapt the parallel FFT for real input data in our implementation. The four-step algorithm interprets the complex data vector $x_j$ of length $N$ as a two-dimensional vector $x_j= x_{j_1, j_2}$ with lengths $n_1$ and $n_2$, respectively, and volume $n_1 n_2 = N$. The mapping $j = j_1 + j_2n_1$ and $k = k_2 + k_1n_2$ is inserted into the definition of the discrete Fourier transform, where $\varPsi = \exp {(-2 {\rm \pi}{\mathrm {i}})}$

(2.56)\begin{gather} \hat{x}_k = \sum_{j=0}^{N-1} x_j \varPsi^{jk/N}, \end{gather}

(2.57)\begin{gather}\hat{x}_{k_2,k_1}= \sum_{j_1 = 0}^{n_1-1} \sum_{j_2 = 0}^{n_2-1} x_{j_1,j2} \varPsi^{ j_2k_2/n_2} \varPsi^{j_1k_2/N} \varPsi^{j_1k_1/n_1} . \end{gather}

This way, a complex-to-complex parallel FFT of length $N$ is distributed to $n_1$ local FFTs of length $n_2$, a multiplication by the twiddle factors $\varPsi ^{j_1k_2/N}$ and finally $n_2$ FFTs of length $n_1$, with a communication intensive transpose in between. All-to-all communication takes place two times, in the first step – cyclically distributing $j$ to $j_1$ and $j_2$ – and for the transpose. A third all-to-all communication would be needed to properly sort the values in Fourier space. However, a scrambled output suffices for computing the heat flux. Furthermore, since often two FFTs, i.e. electrons and ions, need to be computed simultaneously, they can be computed on different nodes. This has the advantage that the second all-to-all communication for the transpose is not completely global resulting in reduced communication times.

Adapting this algorithm to a real-to-complex FFT, where, due to Hermitian symmetry, only values of $k\leq \lfloor N/2\rfloor$ need to be computed, a large amount of computational and communicational savings can be realized. A real-to-complex parallel FFT of length $N$ is distributed to $n_1$ local real-to-complex FFTs of length $n_2$, a multiplication by the twiddle factors $\varPsi ^{j_1k_2/N}$ and, now only, $\lfloor n_2/2\rfloor + 1$ complex-to-complex FFTs of length $n_1$. Up to two of the latter FFTs can be replaced by real-to-complex FFTs, along the axis $k_2=0$ and, if $n_2$ is even, $k_2=n_2/2$. A scrambled output is received, which, due to Hermitian symmetry, needs to be partially complex conjugated.

A key point in ensuring the efficiency of the parallel four-step algorithm consists in choosing large $n_1$ and $n_2$; $n_1 \simeq n_2 \simeq \sqrt {N}$ is the optimal choice for the distributed complex-to-complex FFT, the real-to-complex FFT should prefer $n_1 \simeq \lfloor n_2/2\rfloor +1\simeq (\sqrt {2N+1} + 1)/2$. The computational scaling with $P$ processors and roughly optimally distributed $n_1$ and $n_2$ is akin to ${O}(N/P\log {N})$, but degrades if $N$ is a prime number or, more generally, if $n_1$ or $n_2/2$ is smaller than the number of processors. This easily avoidable because $N$ is a free parameter, and so are $n_1$ and $n_2$. While this does not scale favourably in comparison with the ${O}(N/P)$ scaling that dominates the rest of the fluid code, still, the FFT is trivially independent of the numbers of particles per cell $N_\mathrm {pc}$. The PIC module on the other hand scales as ${O}(N_\mathrm {pc} N/P)$ and typical applications have $N_\mathrm {pc}\gtrsim 100$. In many applications the cost of the Fourier transform is, even with worse scaling, subdominant in comparison with the cost of the PIC part. In the remaining cases, local approximations, discussed above, are favourable.

2.7. Current-coupled fluid-PIC algorithm

The coupling in our code between various fluid and kinetic (PIC) species is achieved through a current-coupling scheme. Namely, both fluid and kinetic species contribute to the charge and current densities. The electromagnetic fields then evolve in response to the total contributions. The fields are staggered on a Yee mesh and are updated with the FDTD scheme. Subsequently, both fluid and kinetic species evolve in response to the new electromagnetic fields. That is our current-coupling scheme does not make any assumption on the velocity distribution of the species modelled using the kinetic description (Park et al. Reference Park, Parker, Biglari, Chance, Chen, Cheng, Hahm, Lee, Kulsrud, Monticello, Sugiyama and White1992).

The PIC species, using fifth-order spline interpolation, are deposited to specific points on the Yee grid for which the charge density is defined at full-time steps while the current density is defined at half-time steps, as discussed by Shalaby et al. (Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b, Reference Shalaby, Thomas and Pfrommer2021). Table 1 gives an overview on the staggering of our implementation of the fluid-PIC method. We initialize the staggered quantities directly, with one exception: $(J_x)_{f}$, see (2.41), necessitates an integral over the flux from $t^0$ to $t^1$. We approximate the integral between $t^0$ and $t^{1/2}$ using an interpolation at $x_i$ of the cell centre values $J_x \simeq q n {w}|_{t^{1/2}}$, while the remaining part of the integral to $t^1$ is obtained through the fluxes again. If $E_x$ is updated using $J_x$, then $\rho$ does not need to be calculated and vice versa. Otherwise, another complication may be seen when obtaining $\rho$ from (2.5), which necessitates $\boldsymbol {U}$ to be defined at full-time steps. While $\boldsymbol {U}$ is formally defined only at half-time steps, we define $\boldsymbol {U}^{n*} = \exp (\Delta t/2\times {\boldsymbol{\mathsf{F}}})\boldsymbol {U}^{n-1/2}$ (i.e. the first part of the splitting in (2.20)), from which $\rho$ is obtained. Note that $\rho _{s}=q_s n_s$ stays constant when computing the Lorentz force and heat flux updates and therefore $\rho ^{n}=\rho ^{n*}$ is defined consistently, while bulk velocity and pressure are not well defined at full-time steps.

Table 1. The grid assignment and time staggering of the fluid-SHARP 1D3V code. Here, $\boldsymbol {U}_{f}$ refers to the fluid state vector. Note that it is sufficient to compute either $\rho$ or $J_x$, but not both.

Our algorithm does not apply any approximations to the electrical field components or to Ohm's law, requiring electron time scales and motions to be fully resolved. Consequently, we apply the same algorithm to fluid electrons and protons. This is accomplished using the modular design of the fluid SHARP code, where each fluid species is represented by initializing a fluid code class. Each instance of this code class is initialized using the values of the mass and the charges of their respective particle species. The algorithms which define the evolution of each particle species are implemented as functions of the fluid class. This allows us to set up simulations with multiple species, all of which are evolved with the same numerical algorithms, with little effort.

In figure 2 the main loop of the fluid-PIC algorithm is presented schematically. It can be seen that the usual PIC-algorithm loop of electromagnetic update, interpolation to particle position, particle push and field deposition, is retrieved when no fluid species is initialized. On the other hand, without PIC particles, we retrieve a multispecies fluid plasma code. While our fluid-PIC algorithm can simulate an arbitrary mixture of species, it is most efficient if fluids are used for background species and particles for non-thermal particle distributions. Possibilities for task parallelization are shown in figure 2 by dashed lines, which allows maximizing of computation–communication overlap. The full main loop of our algorithm can be schematically described as follows (referencing the corresponding equations):

Figure 2. Schematic representation of the interaction of the different modules in the fluid-SHARP code. Red boxes belong to the particle class, violet boxes to the electromagnetic class and blue boxes to the fluid class. Dashed lines show branches which are task parallelizable, i.e. where non-blocking MPI communication can be used for overlapping communication and computation. The particle and fluid modules might be instantiated arbitrarily often, where each instance represents a species.

Our fluid implementation is included within the SHARP code, which uses a fifth-order spline function for deposition and back-interpolation for PIC species (Shalaby et al. Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b, Reference Shalaby, Thomas and Pfrommer2021). The PIC part of the code does not make use of filtering grid quantities and results in comparatively small numerical heating per time step, which (if present) would affect the reliability of the simulation results on long time scales (see § 5 in Shalaby et al. Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b). This property is important because we are specifically interested in studying microphysical effects on long time scales with our fluid-PIC code. Noise generated by the PIC particles could influence the fluid through the electromagnetic coupling. The CFL condition keeps the propagation of this noise within a single cell during one time step, and the PIC noise at the next time step will be uncorrelated with this noise so that we do not expect a systematic numerical error emerging from this. Indeed, we have not yet observed a case where this leads to a numerical instability. It has also been observed, that the larger physical dissipation through Landau closures replaces the need for numerical dissipation completely (Passot et al. Reference Passot, Henri, Laveder and Sulem2014).

Due to the modularity of our code, each part can be tested individually. These tests, ranging from the uncharged fluid solver to full fluid-PIC simulations, are shown in the next section.

3.1. Shock tube

As the fluid approximation will be primarily used for background plasmas without excessive gradients, the accuracy of resolving sharp discontinuities is of secondary importance in practical applications. Still, we stress test our implementation of the fluid equations without electromagnetic coupling to ensure its numerical robustness and to compare the numerical dispersion for different Riemann solvers. For the shock tests a numerical grid of $100$ cells is used with a constant CFL number $C_{\mathrm {cfl}}=0.2$. The boundary conditions are transmissive and the initial conditions for the tests are given in table 2 with the adiabatic coefficient of ${\Gamma } = 1.4$. These test set-ups are the same as used by Toro (Reference Toro2009), where, unlike the tests performed here, a CFL number of $0.2\times 0.95$ is used only in the first five steps and $0.95$ afterwards. The units used for these non-electromagnetic tests are arbitrary units and do not coincide with the usual simulation units.

Table 2. Parameters adopted for the shock-tube tests described in § 3.1; $x_0$ divides the domain into two halves, where values to the left of $x_0$ ($x< x_0$) are initialized by the parameters with subscript ${l}$. Similarly, subscript ${r}$ indicates parameters to the right of $x_0$.

Test 1, as shown in figure 3(a), is a modified Sod shock-tube test. The sonic rarefaction wave on the left-hand side as well as the shock front on the right are well resolved without noticeable oscillations. The contact discontinuity in the middle introduces small oscillations in the density and is smeared out more than the shock front. While the Roe and HLLC solvers yield almost the same results, the HLLC solver is slightly better at resolving the sonic point at the head (to the left) of the sonic rarefaction wave, which the Roe solver can only resolve because an entropy fix is applied.

Figure 3. The 1D1V hydrodynamical shock-tube tests with initial conditions given in table 2. The simulations carried out with the HLLC and Roe Riemann solvers are compared with the exact solutions. Density, bulk velocity in the $x$-direction and pressure are plotted for each test.

Figure 3(b) shows a test of a stationary contact discontinuity with a shock front of a high Mach number travelling to the right and a rarefaction wave to the left. It can be seen that, while the HLLC method introduces more oscillations, it is also better at resolving the contact discontinuity.

In low-density flows the Roe solver is not suitable because it is not robust without further modifications (Einfeldt et al. Reference Einfeldt, Munz, Roe and Sjögreen1991), making the HLLC method slightly more robust while the Roe method is slightly less dispersive. We use the HLLC solver as our default, however, for most practical applications, both methods produce similar results.

A natural extension to the hydrodynamic shock tubes are the MHD shock tubes, which also test the evolution of shocks in the electromagnetic variables. A two-fluid model is not expected to exactly replicate the MHD shock tubes used to test MHD codes, because the characteristic waves are different for both system of equations. Finite volume two-fluid models have been used to replicate the MHD shock tubes with some success, even without informing the Riemann solver about the MHD characteristic wave velocity (Shumlak & Loverich Reference Shumlak and Loverich2003; Hakim et al. Reference Hakim, Loverich and Shumlak2006). Because the Maxwell solver in our implementation uses the finite difference scheme, the most common choice for PIC codes, it is unable to capture electromagnetic shock tubes properly. Their relevance for two-fluid codes rarely extends beyond testing purposes, as physical shocks stretch over a length scale larger than $c/\omega _{\rm i}\gg c/\omega _{\rm e}$, which appears smooth in simulations resolving the electron skin depth. However, shock acceleration is not properly captured using the fluid-PIC algorithm at the shock interface. This is because efficient shock acceleration mechanisms are only experienced by the computational particles, but not the fluids. Injection prescriptions for CRs might be used to mitigate this (e.g. Pfrommer et al. Reference Pfrommer, Pakmor, Schaal, Simpson and Springel2017). We focus on cases where the electromagnetic quantities are smoothly varying, i.e. wave transport. The choice of the Riemann solver and its characteristic waves are less important for smooth waves, especially when employing a high-order interpolation routine. This is because different Riemann solvers should converge to the same results when the interface state is unambiguous, for example if $\tilde {\boldsymbol {U}}_{L}=\tilde {\boldsymbol {U}}_{R}$ (cf. (2.29)).

3.2. Two-fluid dispersion relation

To test the interplay of the fluid solver with the electromagnetic coupling, we perform a test where the linear waves of an ion–electron plasma are reproduced. For an ideal two-fluid plasma the dispersion relation can be solved for six different wave branches (Stix Reference Stix1992). We show the solutions to the dispersion relation of a two-fluid plasma in figure 4 for a realistic mass ratio of $m_{\rm i} = 1836 m_{\rm e}$, $\beta _{\rm i} = n {k_{\rm B}} T_{\rm i}/[B_0^2/(2\mu _0)] = 0.2$ in an isothermal plasma and ${\Gamma } =3$. Here, $B_0$ is oriented along the $x$-axis and the Alfvén velocity is ${v}_{\rm A} = B_0/(\mu _0 n_{\rm i} m_{\rm i})^{1/2}=5.83\times 10^{-3} c$. Multiple simulations at different wavenumbers have been initialized that have all six wave modes simultaneously present and were run for a total time of $14 \boldsymbol {\times } 2{\rm \pi} /\min (\omega ) (20\boldsymbol {\times } 2{\rm \pi} /\min (\omega )$ for the smallest scale), where $\omega$ denotes the wave frequencies, which are always completely real for an ideal fluid. Consequently, the waves should be undamped in the linear regime. Initial conditions for all of our fluid simulations of waves are obtained as eigenvectors (in $\boldsymbol {U},\boldsymbol {E},\boldsymbol {B}$) while theoretical predictions of $\omega$ are obtained as eigenvalues using an extended algorithm based on the dispersion solver by Xie (Reference Xie2014), which can take into account the effects of both heat flux closures. We calculate the initial conditions to double precision, the machine precision of the simulation code. We normalize the amplitude of the eigenvectors by setting the maximum amplitude in any quantity to $10^{-4}$ for each wave mode, to suppress nonlinear effects. The resolution is $\Delta x = 0.1 c/\omega _{\rm e}$ for all simulations. The box size for the intermediate scale is $L_x = 214.2 c/\omega _{\rm e}$, covering waves with $k=n_{w} k_0$ of $n_{w}={1,3,5,10,25}$, where $k_0=2{\rm \pi} /L_x$. The largest and smallest scales use box sizes of $L_x=2142 c/\omega _{\rm e}$ ($n_{w}=1$) and $L_x=21 c/\omega _{\rm e}$ ($n_{w}=3$), respectively. A Fourier analysis in time has been performed and the six largest local extrema are shown as encircled bars extending over a Fourier bin in figure 4. It can be seen, that the simulation results are in excellent agreement with the analytical results. In the Fourier analysis of the slowest two waves, the Fourier mode closest to the theoretical wave frequency is always observed. The largest error measured in this analysis occurs in the whistler branch for $k\sim 31c/\omega _{\rm i}$ with less than $0.5$ %.

Figure 4. The six branches of the two-fluid dispersion relation are shown as lines, with two electrostatic wave branches (Langmuir and ion acoustic) as well as four electromagnetic wave branches, of which two are left- and two are right-hand circularly polarized (LCP and RCP). The lower RCP is referred to as the whistler or electron cyclotron branch and the lower LCP as ion cyclotron branch; for parallel propagation their phase velocities approach the Alfvén speed at small $k$. The upper RCP and LCP are modified light waves. We mark the six local extrema of the discrete Fourier-transformed fluid simulation outputs at each wavenumber with circles encapsulating the error bars extending over the frequency bin. For comparison, we plot the dispersion relation of the three MHD waves at scales larger than the ion inertial length, $1/k > c/\omega _{\rm i}$.

Because our Riemann solver is not explicitly informed about MHD wave speeds, for which the coupling between fluids and electromagnetic fields is especially strong, one could naively expect large errors or numerical instabilities in the MHD limit. In order to test the fidelity of the coupling, we set up a wave with a wave speed well separated from the propagation speeds of the uncoupled fluid Riemann and electromagnetic solvers, i.e. $c_{s}\ll {v}_\phi =\omega /k \ll c$. We test a fast magnetosonic wave (${v}_\phi =0.03728c$ in two fluid vs. ${v}_\mathrm {fast}=0.03703c$ in MHD) at very low $\beta _{\parallel,{\rm i}}=\beta _{\perp,{\rm i}}=0.02$ (corresponding to an oblique propagation angle of $\theta =45^\circ$ in the $B_y-B_x$ plane). The parameters of $k$, $\Delta x$, etc. are the same as for the parallelly propagating waves in the MHD limit in the previous test. While parallelly propagating electrostatic (electromagnetic) waves can be described using 1D1V (1D2V) models, oblique propagation requires the 1D3V model. In figure 5 the time evolution of two representative quantities along and perpendicular to the box size direction are shown. The time evolution of the different quantities are taken from the first cell in the simulation box. The fast magnetosonic wave is captured well, short and long term, without introducing numerical instabilities at low dissipation. The simulated wave velocity is $0.027$ % slower than the theoretical prediction. In conclusion, the current-coupled fluid and electromagnetic solvers numerically approximate the analytical dispersion relation with high fidelity.

Figure 5. An oblique fast magnetosonic wave in a highly magnetized plasma. We plot the absolute of the perturbations of the magnetic field strengths in the top panel, as well as the ion bulk velocity and electric field along the domain in the bottom panel. The theoretical predictions are shown as black-dashed lines while the black-dotted lines indicate the amplitude. Electric and magnetic field strengths are expressed in code units, denoted by the subscript ${c}$.

3.3. Langmuir wave damping

The electrostatic wave modes are directly subject to linear Landau damping, and thus present a good test for the heat flux closures. To test this, we initialize standing Langmuir waves in an electron plasma with immobile ions. We use the same grid layouts as in table 1 of Shalaby et al. (Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b), supplemented with fluid simulations run at $k/k_{\rm D} \in \{0.1, 0.2, 0.3\}$ with a resolution of $\lambda /\Delta x = 68$ cells per wavelength and a domain size of length $L = 10\lambda$ wavelengths. The wavenumber associated with the Debye length is the ratio of plasma frequency to thermal velocity, i.e. $k_{\rm D}=\omega _{p}/ \theta ^{1/2}c$. The amplitude of the wave is chosen, such that the density fluctuation to background ratio is fixed to $\delta n/n_0=10^{-3}$.

In order to find the numerical dispersion relation we perform curve fitting with the Powell algorithm on the time series for times up to $80 \omega _{p}^{-1}$, while the simulations at $k/k_{\rm D}=0.01$ and $0.05$ with small damping are analysed up to $240 \omega _{p}^{-1}$. The computation of the heat fluxes for the $R_{31}$ and $R_{32}$ closures is performed using the FFT-based method. The results are shown in figure 6, where the ideal gas closure and the kinetic results are also depicted for reference (using ${\Gamma } =3$).

Figure 6. The linear dispersion relations of a Langmuir wave with immobile ions. Shown are, on the left-hand side, the real frequency components and, on the right-hand side, the negative imaginary frequency components (which are responsible for damping). The crosses present data points obtained from simulations with the respective closure while the theoretical result is shown with a solid line. The relative error between simulation and theoretical results $(\omega ^{\mathrm {sim}} - \omega ^{\mathrm {theor}}) / \omega ^{\mathrm {theor}}$ is shown in the lower panels. For reference, the red crosses display the data points as given in table 1 of Shalaby et al. (Reference Shalaby, Broderick, Chang, Pfrommer, Lamberts and Puchwein2017b).

Generally, it can be seen that, at small scales, the closures show larger deviations from each other, which is also where the fluid description starts breaking down naturally as the particle distribution is not in equilibrium. At larger scales, the various descriptions of Landau damping converge and approach zero. The numerical relative error of the fluid code is small and stays below $0.003$ % for real frequencies and below $0.02$ % for decay rates in this set-up. The simulation at $k/k_{\rm D}=0.05$ performs worse than the one at $k/k_{\rm D}=0.1$ due to the significantly lower resolution. The error in $\omega$ decreases at second order with increasing spatial resolution, as shown in Appendix B.

3.4. Interacting Alfvén waves

A single Alfvén wave is purely transversal and not directly affected by Landau damping. However, two or more Alfvén waves drive a longitudinal electrostatic wave, which is susceptible to Landau damping, see figure 7. This leads to particle heating as a result of the collisionless damping of the Alfvén wave, also known as nonlinear Landau damping (Lee & Völk Reference Lee and Völk1973).

Figure 7. Two different Alfvén waves, with magnetic and velocity vectors $\boldsymbol {B}_1, \boldsymbol {B}_2$ and ${\boldsymbol{w}}_1, {\boldsymbol{w}}_2$, propagate transversally along the $x$-axis, where the electromagnetic vectors rotate (counter-)clockwise around it. Because of their phase difference $\Delta k x$ the overall Lorentz force $({\boldsymbol{w}}_1 + {\boldsymbol{w}}_2) \boldsymbol{\times} (\boldsymbol B_1 + \boldsymbol B_2)$ in the $x$-direction is non-zero, thereby generating the longitudinal wave shown in dark yellow.

Restricting ourselves to a set-up of pairwise interacting waves, we can identify two distinct cases. In the first case counter-propagating waves are interacting. In consequence, both waves damp, lose energy to the longitudinal wave and subsequently heat the particles. In the second case the waves are co-propagating. Here, the wave with the smaller wavelength will not only transfer energy to the particles, but also to the other Alfvén wave. Lee & Völk (Reference Lee and Völk1973) describe this mechanism in detail and formulate the following coupled set of differential equations while adopting a measure for the magnetic energy of a wave, $I_j = |B_j|^2$, where $j\in \{1, 2\}$:

(3.1)\begin{equation} \frac{\rm{d}}{\rm{d}t} I_j = 2 {\Gamma }_j I_j. \end{equation}

The coupling between the differential equations is implicit because the damping coefficient has the dependency ${\Gamma } _1 \propto I_2$. For the counter-propagating case with an isothermal ion–electron plasma in the high beta limit $\beta _{\rm i} = 2\mu _0 n_{\rm i} {k_{\rm B}} T_{\rm i}/B_0^2 = 2 \gg 1$, where $B_0$ is the background magnetic field strength, the damping rate ${\Gamma } _j$ is approximately equal for both wave polarizations with similar frequencies $\omega _j$ and may be approximated by (Holcomb Reference Holcomb2019)

(3.2)\begin{equation} {\Gamma }_1 =- \frac{\sqrt{\rm \pi}}{16} \frac{I_2}{B_0^2} \sqrt{\beta _{\rm i}} \omega_1. \end{equation}

Note that ${\Gamma } _2$ is found by substituting the subscripts $1\rightarrow 2$ and $2\rightarrow 1$. This prediction is using kinetic physics and also includes damping effects due to modulation in $B_\perp$ (see figure 5), which can electromagnetically heat or even trap particles, analogous to Landau damping in the electrostatic case. This is not captured in the Landau fluid approximation. Therefore, we do not expect our analytical and simulated damping rates to exactly match. However, they provide a good insight into whether wave modes are qualitatively correctly captured. Another prediction by Hollweg (Reference Hollweg1971) uses the fluid picture to derive the amplitude of the secondary, electrostatic wave, which is then damped according to kinetic prescriptions. This prediction agrees with our model. However, this analysis does not differentiate different wave types and therefore does not make individual predictions about interacting waves. In the case considered in the following, the damping rates are coincidentally similar.

In figure 8 we show simulations of a linearly polarized Alfvén wave, which consists of two counter-propagating waves of equal amplitude. The pure fluid simulations are shown with a box size of $L=252 c/\omega _{\rm i}$ and wavelengths $\lambda =L/3$. Right- and left-hand circularly polarized waves are initialized with phase velocities $\omega _\textrm {RCP}/k = 0.0342$ and $\omega _\textrm {LCP}/k = 0.0318$ with a perpendicular magnetic field amplitude of $\delta B = 0.1 B_0$. A reduced mass ratio of $m_{\rm i}/m_{\rm e} =100$ is adopted here.

Figure 8. Time evolution of the magnetic energy of a linearly polarized Alfvén wave in our fluid simulations with Landau damping. Time is measured in units of the period of the mean wave frequencies $P_\omega = 4{\rm \pi} (\omega _1 + \omega _2)^{-1}$. Analytical predictions for the damping rate are taken from Lee & Völk (Reference Lee and Völk1973, labelled L&V) and Hollweg (Reference Hollweg1971). The fluid simulations are presented with the different heat flux closures $R_{31}$ and $R_{32}$. We compare the time evolution of the total magnetic wave energy (a) and the magnetic wave energy of the different polarization states (b). The RCP wave has a higher phase velocity and loses energy more quickly in comparison with the LCP wave.

Our simulations are carried out with the different heat flux closures $R_{32}$ and $R_{31}$, as shown in figure 8. Both closures reproduce the theoretical predictions quite well. A PIC simulation with similar parameters has been shown in figure 6.4 by Holcomb (Reference Holcomb2019), which reproduces half of the predicted damping rate until $t\sim 2P_\omega$ and shows a quenching of the damping rate afterwards. In comparison with kinetic simulations, there is no saturation of the Landau-damping effect in fluids. This is because the distribution of the fluid particles is always assumed to be roughly Maxwellian and resonant particles are not depleted as a function of time. Hence, Landau fluids are implicitly assumed to have small thermalization time scale in comparison with the damping time scale. On the other hand, PIC simulations are plagued by Poisson noise and an insufficient resolution of velocity space might lead to a reduced Landau damping rate.

3.5. Gyrotropic CR streaming instability

To test the entire code, we run CR streaming instability simulations, where electron and ion CRs are modelled with the PIC method and the background electron and ion plasmas are modelled as fluids. The initial CR momentum distribution for ions (electrons) is assumed to be a gyrotropic distribution with a non-vanishing (zero) pitch angle, while both CR electrons and ions are assumed to drift at the same velocity ${v}_\textrm {dr}$. Namely, the phase space distributions for the electron and ion CR species $s\in \{{\rm e, i}\}$ are given by (Shalaby et al. Reference Shalaby, Thomas and Pfrommer2021, Reference Shalaby, Thomas, Pfrommer, Lemmerz and Bresci2023)

(3.3)\begin{equation} f_{\mathrm{cr,} s}(\boldsymbol{x},\boldsymbol{u}) = \frac{ n_{\mathrm{cr,} s} }{2{\rm \pi} u_\perp} \delta(u_{{\parallel}} - \gamma_s {v}_{\rm dr}) \delta(u_{{\perp}}-\gamma_s {v}_{{\perp}, s}) , \end{equation}

where $\gamma _s = (1- {v}^2_\textrm {dr} /c^2- {v}^2_{\perp, s}/c^2)^{-1/2}$ is the Lorentz factor and ${v}_{\perp, s}$ is the perpendicular component of the CR velocity. We choose ${v}_{\perp,{\rm e}} = 0$ and ${v}_{\perp,{\rm i}}=13.1 {v}_{\rm A}$, where the ion Alfvén velocity is given by ${v}_{\rm A} = B_0/(\mu _0 n_{\rm i} m_{\rm i})^{1/2} = 0.01c$ with the background magnetic field pointing along the spatial direction, and ${v}_\textrm {dr}$ of $5 {v}_{\rm A}$ resulting in a pitch angle for the ions of $\tan ^{-1} ({v}_{\perp,{\rm i}}/{v}_\textrm {dr} )= 69.1^{\circ }$. The thermal background species are isothermal with the temperatures $k_{\rm B} T/(m c^2)=10^{-4}$ and a mass ratio $m_{\rm i}/m_{\rm e} = 1836$. We use a periodic box of length $L_x = 10\ 971.5 c/\omega _{p}$ and resolution $\Delta x = 0.1 c/\omega _{p}$. The CR to background number density ratio $\alpha =n_{\textrm {cr}, {\rm i}}/n_{\rm i}=0.01$.

We run two simulations where the background plasmas are modelled as fluids. The first one uses an ideal gas closure without accounting for Landau damping (FPIC ideal gas) while we include the heat flux source term in the second simulation to mimic the impact of linear Landau damping using the $R_{31}$ closure of (2.46) (FPIC Landau $R_{31}$). We compare these two fluid-PIC simulations against PIC simulations where both CRs and background plasmas are modelled as PIC species. The number of CR ions per cell is $N_{\mathrm {pc}} = 25(75)$ and we call this simulation ‘PIC normal (high) $N_{\mathrm {pc}}$’ (Shalaby et al. Reference Shalaby, Thomas and Pfrommer2021). Like the ‘PIC normal $N_{\mathrm {pc}}$’ simulation, the fluid-PIC simulations also use $25$ particles per cell for modelling CRs.

Growth rates of the instability in the linear regime can be computed from the linear cold background plasma dispersion relation (Holcomb & Spitkovsky Reference Holcomb and Spitkovsky2019; Shalaby et al. Reference Shalaby, Lemmerz, Thomas and Pfrommer2022)

(3.4) \begin{align} 0 & = 1 - \frac{k^2c^2}{\omega^2} + \frac{\omega _{\rm i}^2}{\omega (-\omega \pm \varOmega_{{\rm i},0})} + \frac{\omega _{\rm e}^2}{\omega (-\omega \pm \varOmega_{{\rm e},0} )} + \frac{\alpha \omega _{\rm e}^2 }{ \gamma_{\rm e} \omega^2 } \left(\frac{ \omega -k {v}_{\mathrm{dr}} } { k {v}_{\mathrm{dr}}-\omega \pm \varOmega_{{\rm e},0}} \right) \nonumber\\ & \quad + \frac{\alpha \omega _{\rm i}^2 }{ \gamma_{\rm i} \omega^2 } \left( \frac{ \omega -k {v}_{\mathrm{dr}} } { k {v}_{\mathrm{dr}}-\omega \pm\varOmega_{\rm i}} - \frac{ {v}_{{\perp}}^2/c^2 (k^2 c^2-\omega^2)} {2 (k {v}_{\mathrm{dr}}-\omega \pm\varOmega_{\rm i} )^2} \right). \end{align}

The non-relativistic and relativistic cyclotron frequencies of each species are given by $\varOmega _{{s},0}=q_s B_0/m_s$ and $\varOmega _{{s}}=\varOmega _{{s},0}/\gamma _{s}$, respectively. The wavelength of the most unstable wave mode at the gyroscale is $\lambda _{{g}} = 2{\rm \pi} ({v}_\textrm {dr}-{v}_{\rm A})/\varOmega _{\rm i}$, which is properly captured in our set-up using a box size of $L_x \sim 10.15 \lambda _{{g}}$.

We show the amplification of the perpendicular magnetic field components as a function of time for this unstable set-up in figure 9 for various simulations. It shows that the noise level of the fluid-PIC simulations is orders of magnitude lower in comparison with the ‘PIC normal $N_{\mathrm {pc}}$’ resolution, even though the number of CR particles per cell is the same. Especially up to the saturation point ($t\varOmega _{\rm i}\sim 10$) the fluid-PIC simulation compares more favourably with the PIC results with lower noise than with the PIC simulation with fewer $N_{\mathrm {pc}}$.

Figure 9. Growth of the perpendicular magnetic field as a function of time for a gyrotropic CR streaming set-up. The maximum growth rate expected from the linear dispersion relation at intermediate scales is ${\Gamma } _{\mathrm {inter}} = 2.299 \varOmega _{\rm i}$ and shown in dashed grey. Because of the different initial seed populations for the particle species, the onset of the instabilities is not expected to happen at the same simulation time. Hence, we choose an arbitrary $t=0$ so that the different simulated growth phases roughly coincide.

After saturation, i.e. when Alfvén waves at many scales have built up and their interaction has created an electrostatic field, these waves start to lose some energy to Landau damping of the electrostatic waves (see § 3.4). At that point, the Landau closure becomes relevant. Qualitatively the ideal gas closure has no efficient mechanism for dissipating such electrostatic waves, resulting in a prolonged growth period leading to saturation at higher values at the cascading and intermediate scales. Utilization of a Landau closure leads to some damping, albeit it is quantitatively smaller than in the PIC simulations. While figure 6 indicates faster damping for the Landau closures in comparison with the kinetic results in the electron electrostatic branches, damping in the ion-acoustic branch might be underestimated in the Landau closures. We have compared the expected damping between kinetic and Landau fluid in the ion-acoustic branch for multiple wavenumbers, which confirmed that this is a likely scenario. The accuracy of this approximation is not the same at all scales, which can be seen in figure 10, where the magnetic field amplifications at various ranges of scales are compared. Especially at the highly Landau-damped scales, differences between fluid-PIC and PIC emerge. At ion gyro-scales, where most of the magnetic energy is stored at saturation, there is a good agreement over the entire time period. Exponential growth at every scale is also in good agreement between PIC and fluid-PIC simulations at all scales. The initial exponential growth can also be compared with the expected growth rates from the linear dispersion relation. The growth rates of the two local maxima are plotted alongside the simulated data, one at the intermediate scales around $c k=4.91\omega _{\rm i}$ and one at the gyro-scale at $c k = 0.38 \omega _{\rm i}$. The intermediate scale starts an inverse cascade to larger scales almost immediately, which causes a reduced growth rate in comparison with the expectation from linear theory. By contrast, the gyro-scale instability follows linear expectations to very good approximation.

Figure 10. Growth of the perpendicular magnetic field as a function of time at different scales for a gyrotropic CR streaming set-up. We show mean values of the fields that are averaged over a range of wave vectors $k$, as indicated in the legends. The maximum growth rates at the gyro-scale and the intermediate scale are given by ${\Gamma } _{\mathrm {gyro}} = 0.498 \varOmega _{\rm i}$ and ${\Gamma } _{\mathrm {inter}} = 2.299 \varOmega _{\rm i}$, and indicated by the grey dotted and dashed lines, respectively. At wavenumbers corresponding to cascading scales, there is no instability expected according to the linear dispersion relation, and wave growth solely arises as a result of cascading from other (unstable) scales.

While our fluid-PIC and PIC results are promisingly similar, differences after the saturation level might be attributed to multiple reasons. First, the Landau closures do not exactly reproduce the correct damping, and therefore will deviate quantitatively. Second, due to the high electron temperature chosen, relativistic effects might occur in PIC, but not in the non-relativistic fluid that we assumed for the background plasma. Third, the PIC method might exhibit more numerical dissipation at the given $N_\textrm {pc}$ in comparison with the fluid method. However, figure 10 seems to indicate numerical convergence at the intermediate and gyro-scale.

Even though our simulations were run at unrealistically high $\alpha$, the background particles did not deviate significantly from the Maxwellian distribution at the end of the simulation time. The pressure anisotropy measured from the PIC thermal particles is below $2\,\%$. This indicates that an isotropic fluid description for background species is a valid approach for this set-up, especially for smaller, more realistic values of $\alpha$.

3.6. Computational scaling

We show the strong scaling properties of our fluid-PIC code in figure 11. The tests were run on Intel Cascade 9242 processors with 96 processors per node at the HLRN Emmy cluster. Simulations with 3000 processors or more typically cause severe bottlenecks due to the latency and/or the finite bandwidth of input/output operations. For this number of processors the Fourier-based closures are roughly 20 % more costly in comparison with the ideal gas closures. This is in stark contrast to pure PIC simulations, which scale with the inverse ratio of CR-to-background density $\alpha ^{-1}$, consequently the fluid-PIC algorithm leads to a speed up of a factor of $100$ for the simulation performed in § 3.5, which adopted unrealistically large $\alpha$.

Figure 11. Strong scaling of the fluid-PIC code, with and without Fourier-based Landau closures. Shown is the wall-clock time needed to simulate $1250$ time integration steps with $180\ 000$ cells at $1000$ particles per cell at a varying number of processors. We show the perfect strong scaling that is proportional to the inverse number of processors as the grey dashed line for reference. For the disabled fluid module no background plasma was initialized and only CRs are initialized, showing that the bulk of the computational work is performed by the PIC routines.

The bottleneck in the communication procedure of our implementation is currently the ‘Ialltoallv’ MPI routine, which is not optimized for hierarchical architecture networks as of now. Further optimizations to this might provide fruitful in increasing the code's scalability further if necessary.

The fluid-PIC simulations in § 3.5 used only $N_\textrm {pc}=25$ and seem to be sufficiently resolved. For such a low particle number, the FFT is the bottleneck for scalability because the overlap of communication and computation is small, i.e. we measure a 260 % increase in time with 2880 processors, while at 192 processors the increase is below 20 %. This indicates that scalability of fluid-only simulations is dominated quickly by the FFT, while the cost is almost negligible for fluid-PIC simulations. Still, simulations with only a few particles per cell are computationally inexpensive so that there is no reason for performing such a simulation on thousands of processors. Furthermore, the example of a mono-energetic cold CR beam is not very demanding regarding the phase-space resolution. More realistic scenarios include power-law distributions for the CR population as well as larger spatial density inhomogeneities, both resulting in an increased requirement for the number of particles per cell in order to accurately resolve the velocity phase-space distribution along the entire spatial domain.