Numerical Methods for Computing an Averaged Matrix Field. Application to the Asymptotic Analysis of a Parabolic Problem with Stiff Transport Terms
Thomas Blanc · Multiscale Modeling and Simulation · 2019
The numerical resolution of parabolic problems with terms is a challenge. When the stiff terms become dominant, multiple-scale effects occur and the classical numerical methods do not catch the microscopic effects. The aim of this paper is to provide a numerical method to study the behavior of parabolic problems with stiff transport terms based on recent results on the asymptotic analysis for such problems. Precisely, the behavior of the solutions can be described in terms of a composition product between a certain profile and the fast flow associated to the dominant transport operator, where the asymptotic profile solves an effective diffusion equation. A numerical method for determining the effective diffusion matrix is given, the computation of the limit profile is carried out, and the error with respect to the solution of the stiff problem is studied.