Unconditional stability over long time intervals of a two-level coupled MacCormack/Crank–Nicolson method for evolutionary mixed Stokes-Darcy model

Abstract

This paper analyzes the stability of a two-level coupled explicit MacCormack/Crank–Nicolson method for solving the nonstationary mixed Stokes-Darcy model. The approach is systematically derived based on the monolithic weak formulations of the explicit MacCormack technique and the implicit Crank–Nicolson discretization. The stability of the explicit scheme does not require a time step restriction. The two-step MacCormack provides an approximate solution at the coarse grid level while the implicit Crank–Nicolson algorithm uses this approximation to compute the desired numerical solution at the fine grid stage. The proposed method is unconditionally stable over long time intervals and the computational cost is reduced. This combination is efficient than a wide set of numerical schemes applied to time dependent mixed Stokes-Darcy problem. A large range of numerical examples which confirm the theoretical study are presented to illustrate the performance of the proposed method.