Weakly convergent stochastic simulation of electron collisions in plasmas

Collisions between charged particles can be described and solved using Monte Carlo methods in the framework of stochastic differential equations (SDEs). In this paper, we start from an SDE including the extended Lorentz collision operator, which can recover the collisions between a sampling electron and background ions and electrons. On this basis, we construct a second order weakly convergent algorithm (WCA2) to simulate collisional effects of electrons in plasmas. Superseding the Weiner process by a three-point distribution, WCA2 possesses high weakly convergent accuracy as well as low computational costs. The definition and properties of weak convergence are discussed in detail. The weakly convergent order of WCA2 is verified both theoretically and numerically. Through two trial moment functions, we carefully analyze the numerical solutions of the SDE using rigorous statistical tests in the sense of weak convergence. The criteria and practical operations of finding the benchmark solution of SDEs are introduced at length. In order to illustrate the power of WCA2, we apply it to simulate the backward runaways in plasmas, which is a dramatic physical phenomenon. By comparison with the Euler-Maruyama method and the Cadjan-Ivanov method, the advantage and efficiency of WCA2 is exhibited. The backward runaway probability and its dependence on initial conditions are accurately studied using WCA2.