Efficient and Long-Time Accurate Second-Order Methods for Stokes-Darcy System

TL;DR

This paper introduces two efficient second-order IMEX methods for the coupled Stokes-Darcy system, offering long-time stability and accuracy for modeling flows in karst aquifers, with proven stability and error estimates.

Contribution

The paper develops two novel second-order IMEX schemes for the Stokes-Darcy system that are unconditionally stable and suitable for long-term simulations, with comprehensive error analysis.

Findings

Both schemes are unconditionally stable and uniformly in time stable.

Numerical results confirm high accuracy and efficiency over long simulation periods.

Methods require solving only two decoupled problems per time step.

Abstract

We propose and study two second-order in time implicit-explicit (IMEX) methods for the coupled Stokes-Darcy system that governs flows in karst aquifers. The first is a combination of a second-order backward differentiation formula and the second-order Gear's extrapolation approach. The second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully-discretized schemes using finite element spatial discretizations are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes.