Time multiscale modeling of sorption kinetics I: uniformly accurate schemes for highly oscillatory advection-diffusion equation

TL;DR

This paper introduces a numerical method for 2D advection-diffusion equations in highly oscillatory regimes, achieving uniform accuracy regardless of oscillation frequency, and lays groundwork for solving coupled Stokes-advection-diffusion systems.

Contribution

It presents a robust, efficient integrator that provides uniform first and second order temporal accuracy for highly oscillatory advection-diffusion equations, independent of oscillation frequency.

Findings

Achieves uniform accuracy without time step restrictions

Develops a two-scale formulation for oscillatory systems

Lays foundation for solving coupled Stokes-advection-diffusion systems

Abstract

In this paper we propose a numerical method to solve a 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 $\varepsilon-$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 behaviour of the solution.