Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit (Q2904804)

This paper is devoted to considerations related to the projective integration scheme for a kinetic equation in the limit of a vanishing mean free path in which the kinetic description approaches a diffusion phenomenon. One can see that, in many applications, the underlying physical system consists of a large number of moving and colliding particles. Such systems can be accurately modeled using a kinetic mesoscopic description that governs the evolution of the particle distribution in position-velocity phase space. When the mean free path of the particles is small with respect to the length scale, the usually used macroscopic description involves only a few low-order moments of the particle distribution, and, as a result, it can give a rough idea of the behavior. On the other hand, the description is difficult because if the physical model becomes much simpler in the diffusion limit, a direct numerical simulation of the kinetic model tends to be expensive due to the additional dimensions in the velocity space and stability restrictions. Considering the equation NEWLINE\[NEWLINE\partial_t f+v\cdot\nabla_x f=Q(f),NEWLINE\]NEWLINE the authors attempt to derive a scheme, based on methods developed for large multiscale systems of ODEs, that allows computations where no linear system needs to be solved. This is based on projective integration methods that can offer a number of important advantages for simulations. They do not need any splitting in time or microscopic and macroscopic values.NEWLINENEWLINEThis paper consists of 7 sections. After an introduction, in Section 2, all necessary preliminaries, used in Section 3 for the development of the numerical scheme, are given. Section 4 is devoted to considerations about stability conditions for the proposed method, and Section 5 deals with the consistency of the method. The next section presents numerical illustrations; after this final conclusions are given.