A DSMC-PIC coupling method for the Vlasov-Maxwell-Landau system

Abstract

We present a numerical framework for the simulation of collisional plasma dynamics, based on a coupling between Direct Simulation Monte Carlo (DSMC) and Particle-in-Cell (PIC) methods for the Vlasov-Maxwell-Landau system. The approach extends previously developed DSMC techniques for the homogeneous Landau equation to the fully inhomogeneous, electromagnetic regime. The Landau collision operator is treated through a stochastic particle formulation inspired by the grazing-collision limit of the Boltzmann equation, which enables an efficient and physically consistent representation of Coulomb interactions without relying on the full Boltzmann structure. The resulting collisional solver is combined, via operator splitting, with standard PIC schemes for the Vlasov-Maxwell dynamics, providing flexibility in the choice of field discretisation and time integration. The overall method preserves the main physical invariants of the system while maintaining computational efficiency and simplicity of implementation. Numerical experiments on benchmark problems demonstrate the accuracy, robustness, and effectiveness of the coupled DSMC-PIC approach across a wide range of collisional regimes.

Figures (16)

• Figure 1: Test 1 - Linear Landau Damping: kernel comparison. Logarithm of the electric energy $\mathcal{E}(t)$ as a function of the time. Top row: Maxwellian collisions with $\gamma=0$; bottom row: Coulomb collisions with $\gamma=-3$. We compare different collisional scenarios corresponding to the choices $\nu=1,10^{-1},10^{-2}$, in the left column, center column, and right column respectively. In every plot, we display the three kernels $D_*^{(i)}$, with $i=1,2,3$, and we note a good accordance. We other parameters are $N=5\cdot 10^7$, $k=0.5$, $\Delta t=\varepsilon=0.1$, and $\alpha=0.1$.
• Figure 2: Test 1 - Linear Landau Damping: weak Coulomb collisions. Logarithm of the electric energy $\mathcal{E}(t)$ as a function of the time (solid black line), for Coulomb collisions ($\gamma=-3$) with $D_*^{(3)}$. We compare the collisionless case with different collisional scenarios corresponding to the choices $\nu=10,20,40,60,80,100$. We choose $N=5\cdot 10^7$, $k=0.5$, $\Delta t=\varepsilon=0.1$, and $\alpha=0.1$. The theoretical damping rate in the collisionless case is $\gamma_L=-0.1514$ (blue dashed line); the collisional correction depends on $\nu$ and it is given by $\gamma_C$ in \ref{['eq:gammaC']} (dashed red line).
• Figure 3: Test 1 - Nonlinear Landau Damping: electric energy. Logarithm of the electric energy $\mathcal{E}(t)$ as a function of the time for Maxwellian ($\gamma=0$) and Coulomb ($\gamma=-3$) collisions with $D_*^{(3)}$. We compare different collisional regime corresponding to the choice $\nu=1$ (solid red line), $\nu=0.1$ (solid blue line), $\nu=0.01$ (solid green line), with respect to the collisionless scenario (solid black line). We choose $N=5\cdot 10^7$, $k=0.5$, $\Delta t=\varepsilon=0.1$, and $\alpha=0.5$.
• Figure 4: Test 1 - Nonlinear Landau Damping: Maxwellian collisions. Marginals of the particle distribution $f(x,v_x,t)$ for the nonlinear Landau damping test with Maxwellian collisions ($\gamma=0$). Each row corresponds to a fixed time (top: $t=25)$; bottom: $t=50$). Each column shows a different collisional regime: left collisionless scenario, centre intermediate collisions ($\nu=0.1$); right strong collisions approaching the hydrodynamic limit ($\nu=0.01$). The parameters are $N=5\cdot 10^7$, $k=0.5$, $\Delta t=\varepsilon=0.1$, and $\alpha=0.5$. The distribution is reconstructed with $N_\ell=100$ cells in the spatial domain, and $N_v=200$ cells in the velocity domain.
• Figure 5: Test 1 - Nonlinear Landau Damping: Coulomb collisions. Marginals of the particle distribution $f(x,v_x,t)$ for the nonlinear Landau damping test with Coulomb collisions ($\gamma=-3$). Each row corresponds to a fixed time (top: $t=25)$; bottom: $t=50$). Each column shows a different collisional regime: left collisionless scenario, centre intermediate collisions ($\nu=0.1$); right strong collisions approaching the hydrodynamic limit ($\nu=0.01$). The parameters are $N=5\cdot 10^7$, $k=0.5$, $\Delta t=\varepsilon=0.1$, and $\alpha=0.5$. The distribution is reconstructed with $N_\ell=100$ cells in the spatial domain, and $N_v=200$ cells in the velocity domain.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3