High-order stochastic integration schemes for the Rosenbluth-Trubnikov collision operator in particle simulations

Abstract

In this study, we consider a numerical implementation of the nonlinear Rosenbluth-Trubnikov collision operator for particle simulations in plasma physics in the framework of the finite element method (FEM). The relevant particle evolution equations are formulated as stochastic differential equations, both in the Stratonovich and Itô forms, and are then solved with advanced high-order stochastic numerical schemes. Due to its formulation as a stochastic differential equation, both the drift and diffusion components of the collision operator are treated on an equal footing. Our investigation focuses on assessing the accuracy of these schemes. Previous studies on this subject have used the Euler-Maruyama scheme, which, although popular, is of low order, and requires small time steps to achieve satisfactory accuracy. In this work, we compare the performance of the Euler-Maruyama method to other high-order stochastic methods known in the stochastic differential equations literature. Our study reveals advantageous features of these high-order schemes, such as better accuracy and improved conservation properties of the numerical solution. The main test case used in the numerical experiments is the thermalization of isotropic and anisotropic particle distributions.

Figures (8)

• Figure 1: Top left: The strong errors of the numerical solution at the last step for the Euler-Maruyama scheme, the 2-stage PL scheme, and the 4-stage CL scheme using 25 sample paths are depicted. Top right: The standard deviation of the mean of the errors. Bottom: The same cases are run using 50000 sample paths. The strong (Left) and weak (Right) errors are depicted.
• Figure 2: The time evolution of the Kubo oscillator for $\gamma=0.25$ (left) and $1$ (right).
• Figure 3: The strong error at the end of the simulation (top left), the relative error of the total energy (top right) for the Kubo oscillator with 100 particles. The same cases of the Kubo oscillator are run using 50000 particles as shown at the bottom, with another four schemes included for comparison.
• Figure 4: The structure of the drag and diffusion coefficients in Eq. \ref{['eq:langevin_dfdt']}.
• Figure 5: The error of the Rosenbluth-Trubnikov potentials $h$ and $g$ from the mixed-MC-FEM solver. The bottom $x$ and top $x$ labels are the marker number per degree of freedom of the finite element solver and the total marker number, respectively.