A DSMC method for the space homogeneous multispecies Landau equation

TL;DR

The paper introduces a mesh-free Direct Simulation Monte Carlo method for the spatially homogeneous multispecies Landau equation, derived from a first-order grazing-collision approximation of the multispecies Boltzmann operator and equipped with a regularized angular kernel to avoid iterative solvers. The DSMC scheme uses a time-discretized, convex-combination update with intra- and inter-species collision handling, enabling accurate treatment of realistic mass ratios up to $m_p/m_e \,\approx\,1836$ and straightforward coupling to PIC solvers. Numerical tests against the BKW Maxwellian benchmark and Coulomb relaxation demonstrate conservation of mass, momentum, and energy and correct relaxation toward Maxwellians, with clear mass-ratio-dependent kinetics. The method's mesh-free, particle-based nature and compatibility with PIC frameworks make it a practical tool for fully kinetic, multispecies plasma simulations, including potential extensions to spatially inhomogeneous Vlasov–Maxwell–Landau dynamics.

Abstract

We present a Direct Simulation Monte Carlo (DSMC) method for the spatially homogeneous multispecies Landau-Fokker-Planck equation. The scheme is derived from a first-order approximation of the multispecies Boltzmann operator in the grazing collision limit and employs a regularized, easy-to-sample scattering kernel that removes the need for iterative solvers while preserving the fundamental invariants of the Landau dynamics. The method is fully mesh-free -- being a Monte Carlo particle algorithm -- which makes it naturally scalable to high-dimensional velocity spaces and straightforward to couple with particle-in-cell (PIC) solvers via operator splitting. A notable feature of our approach is its ability to treat realistic mass ratios: we show accurate simulations up to the physical proton-electron ($p$-$e$) mass ratio $m_p/m_e \approx 1836$. We validate the method against the multispecies BKW benchmark for Maxwellian interactions and study collisional relaxation for Coulomb interactions, showing conservation of mass, momentum, and energy, and the expected trend towards Maxwellian equilibria.

Figures (6)

• Figure 1: Test 1: L2 Error. Time evolution of the relative $L^2$ error with respect to the multispecies BKW solution of the functions $f_\alpha(v,\,t)$ and $f_\beta(v,\,t)$, for different values of the time step $\Delta t$. Upper row: multispecies system with $m_\alpha=1$ and $m_\beta=2$. Bottom row: multispecies system with $m_\alpha=1$ and $m_\beta=1836$. Notice how the problem becomes more stiff as the mass ratio increases. The number of particles is $N=5\cdot10^7$, the distribution is reconstructed with histograms with $N_v=100$ bins per dimension in the domain with $L_\alpha=5$ and with $L_\beta=L_\alpha\sqrt{m_\alpha/m_\beta}$.
• Figure 2: Test 1: Marginals with $m_\alpha=1$ and $m_\beta=2$. Evolution at fixed times $t=0.1,\,0.5,\,5$ of the marginals $f_\alpha(v_x,t)$ and $f_\beta(v_x,t)$ of the BKW solution (solid black lines) and the DSMC simulations at different time steps $\Delta t=0.1$ (red circles) and $\Delta t=0.01$ (blue stars). In this test, we consider $m_\alpha=1$ and $m_\beta=2$. The number of particles is $N=5\cdot10^7$, the distribution is reconstructed with histograms with $N_v=100$ bins per dimension in the domain with $L_\alpha=5$ and with $L_\beta=L_\alpha\sqrt{m_\alpha/m_\beta}$.
• Figure 3: BKW Test: Marginals with $m_\alpha=1$ and $m_\beta=1836$. Evolution at fixed times $t=0.1,\,0.5,\,5$ of the marginals $f_\alpha(v_x,t)$ and $f_\beta(v_x,t)$ of the BKW solution (solid black lines) and the DSMC simulations at different time steps $\Delta t=0.1$ (red circles) and $\Delta t=0.001$ (blue stars). In this test, we consider $m_\alpha=1$ and $m_\beta=1836$. The number of particles is $N=5\cdot10^7$, the distribution is reconstructed with histograms with $N_v=100$ bins per dimension in the domain with $L_\alpha=5$ and with $L_\beta=L_\alpha\sqrt{m_\alpha/m_\beta}$.
• Figure 4: Test 2: Relaxation to the equilibrium $m_\beta/m_\alpha=2$. Time evolution of the temperature (top-left panel), the $x$-components of the velocities (top-right panel), the $y$-component of the velocities (bottom-left panel), and the $z$-component of the velocities (bottom-right panel), for the system with mass ratio $m_\beta/m_\alpha=2$. In all the panels, the solid red line is the species $\alpha$, the solid blue line the species $\beta$, and the solid black line represents the total quantities, that are conserved. The number of particles is $N=10^7$.
• Figure 5: Test 2: Relaxation to the equilibrium $m_\beta/m_\alpha=1836$. Time evolution of the temperature (top-left panel), the $x$-components of the velocities (top-right panel), the $y$-component of the velocities (bottom-left panel), and the $z$-component of the velocities (bottom-right panel), for the system with mass ratio $m_\beta/m_\alpha=1836$. In all the panels, the red line (solid or star-solid) is the species $\alpha$, the blue line (solid or circle-solid) the species $\beta$, and the solid black line represents the total quantities, that are conserved. In the top-left panel, we display also the detail for small times of the electron temperature $T_\alpha(t)$ to highlight its fast evolution toward the maximum before converging towards the equilibrium. The number of particles is $N=10^7$.

Theorems & Definitions (2)

• Remark 1
• Remark 2