Projective integration methods for kinetic equations - Giovanni Samaey

Abstract

Projective integration methods for kinetic equations

 

We study a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path.  We present the basic result and an extension to higher-order methods that are fully explicit. We also show how this principle can be used to create arbitrary-order, general, explicit schemes for systems of nonlinear conservation laws in multiple dimensions. The analysis is illustrated with numerical results.

 

This talk is based on joint work with Pauline Lafitte, Annelies Lejon and Ward Melis.