A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

Abstract

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step ∆t in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [5] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit.

Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

“…Two recent review papers [35,19] contain an almost up to date bibliography on numerical methods for collisional kinetic equations of type (1.1) and AP schemes. For even more recent works on AP schemes, one can consult these two papers [20,15] on particle methods and the references therein.…”

Asymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling

“…Thus, in the situation in which the scaling parameter is small, the problem becomes stiff and in particular this causes the characteristic speeds to grow to infinity. Among the possible solutions, the class of asymptotic preserving method represents certainly a good choice to tackle the method [26,8,17,15], they permit to choose the time step independently of the stiffness of the underlying equation remaining consistent and stable. Here, we propose an alternative which enjoys the same consistency and stability properties but, in addition, it also permits to reduce the numerical complexity (and then the computational cost) of the problem.…”

“…Here, we propose an alternative which enjoys the same consistency and stability properties but, in addition, it also permits to reduce the numerical complexity (and then the computational cost) of the problem. Our first step is presented in the following paragraph and follows the strategy of [15]. Indeed, we propose a reformulation of the micro-macro system (2.3) which permits to surround the stiffness of the transport term.…”

“…In this work, we propose a numerical method which tries to take the best of the above two approaches: domain decomposition and asymptotic preservation. The method is based on a hybrid Monte Carlo strategy [12,46] for the underlying kinetic equation which follows the ideas introduced in [24,23,20] and more recently in [16,15,14] in the context of gas dynamics. The problem addressed is the solution of a kinetic type equation under a diffusive scaling which in the limit gives rise to a parabolic type equation.…”

“…To achieve our goal in this context, we proceed as follows: we couple a Monte Carlo method for the solution of the kinetic model with a finite volume method for the solution of the macroscopic one [24,23,20]. However, instead of solving directly the kinetic model, we reformulate the original problem by using a micro-macro decomposition strategy [47,44,3,17,14,15]. This permits to derive an equivalent set of equations which expresses the time evolution of the macroscopic part ρ(t, x)M (v) plus the time evolution of the perturbation g(t, x, v).…”