Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

Abstract

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form Δt≤Ch (where h,Δt are mesh size and time step, respectively, and C is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.