Particle simulation methods for the Landau-Fokker-Planck equation with uncertain data

Abstract

The design of particle simulation methods for collisional plasma physics has always represented a challenge due to the unbounded total collisional cross section, which prevents a natural extension of the classical Direct Simulation Monte Carlo (DSMC) method devised for the Boltzmann equation. One way to overcome this problem is to consider the design of Monte Carlo algorithms that are robust in the so-called grazing collision limit. In the first part of this manuscript, we will focus on the construction of collision algorithms for the Landau-Fokker-Planck equation based on the grazing collision asymptotics and which avoids the use of iterative solvers. Subsequently, we discuss problems involving uncertainties and show how to develop a stochastic Galerkin projection of the particle dynamics which permits to recover spectral accuracy for smooth solutions in the random space. Several classical numerical tests are reported to validate the present approach.

Figures (19)

• Figure 1: Test 1. Time evolution of the relative $L^2$ errors with respect to the BKW solution of the distribution $f(v,t)$ (left) and of the fourth order moment $\textrm{M}4(t)$ (right), for the kernels $D^{(1)}_*$, $D^{(2)}_*$, and $D^{(3)}_*$, and different values of $\varepsilon=\rho\Delta t$. In all the tests, we use $5\times10^7$ particles and the Nanbu-Babovsky scheme. Initial conditions given by \ref{['eq:BKW_det']} with $t=0$ and $T=1$.
• Figure 2: Test 1. Comparison between the time evolution of the first and second order moments (left), and of the fourth order moment (right), obtained with the DSMC method and from the exact BKW solution. In all the tests, we use $5\times10^6$ particles, $\Delta t=\varepsilon/\rho=0.1$, the kernel $D^{(3)}_*$, and the Nanbu-Babovsky scheme. Initial conditions given by \ref{['eq:BKW_det']} with $t=0$ and $T=1$.
• Figure 3: Test 2. Time evolution of the relative temperature difference $\Delta T(t)/\Delta t(0)$ for the Maxwellian (left) and the Coulomb (right) cases. The solid black lines are the Trubnikov solutions with rate given, respectively, by \ref{['eq:trub_max']} and \ref{['eq:trub_coul']}. The particles are $N=5\times10^6$, the time step $\Delta t=0.1$, and the temperature $T=0.07$. We choose the Nanbu-Babovsky scheme with the kernel $D^{(3)}_*$. Initial conditions given by \ref{['eq:init_trub_det']}, with $z$-temperature $T^0_z=0.04$ for the Maxwellian case, and $T^0_z=0.04,\,0.001$ for the Coulomb case.
• Figure 4: Test 3. Left: $L^2$-Error in the evaluation of the temperature $T(\mathbf{z})$ at fixed time $t=1$ for increasing $M$, with respect to a reference solution, for different values of $\kappa$. Right: time evolution of the same error in the time span $[0,5]$ for the case $\kappa=0.95$. We consider in both cases $N=10^6$ particles, $\Delta t=\varepsilon/\rho=0.1$, and initial conditions given by \ref{['eq:initBKW']}. The kernel is $D^{(3)}_*$, the scheme is Nanbu-Babovsky. Reference solution computed with $M=30$.
• Figure 5: Test 3. Evolution at fixed times $t=0,1,5$ of the marginal $\mathbb{E}_{\mathbf{z}}[f(v,t,\mathbf{z})]$ and $\mathrm{Var}_{\mathbf{z}}[f(v,t,\mathbf{z})]$ of the BKW exact solution \ref{['eq:BKW']} and of the DSMC-sG approximation of the model for Maxwell molecules, with uncertainty in the initial temperature with $\kappa=0.95$ in \ref{['eq:initBKW']}. We consider $N=5\times10^7$ particles, $\Delta t=\varepsilon/\rho=0.1$ and $M=5$.

Theorems & Definitions (12)

• Lemma 1
• proof
• Proposition 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Lemma 2
• Lemma 3
• Theorem 1