TIME MULTISCALE MODELING OF SORPTION KINETICS I: UNIFORMLY ACCURATE SCHEMES FOR HIGHLY OSCILLATORY ADVECTION-DIFFUSION EQUATION

In this paper we propose a numerical method to solve a two-dimensional (2D) advection-diffusion equation in the highly oscillatory regime. We use an efficient and robust integrator which leads to an accurate approximation of the solution without any time step-size restriction. Uniform first and second order numerical approximations in time are obtained with errors, and at a cost, that are independent of the oscillation frequency. This work is part of a long time project, and the final goal is the resolution of a Stokes-advection-diffusion system, in which the expression for the velocity in the advection term, is the solution of the Stokes equations. This paper focuses on the time multiscale challenge, coming from the velocity that is an ε−periodic function, whose expression is explicitly known. We also introduce a two-scale formulation, as a first step to the numerical resolution of the complete oscillatory Stokes-advection-diffusion system, that is currently under investigation. This two-scale formulation is also useful to understand the asymptotic behavior of the solution.