Second order convergence of a modified MAC scheme for Stokes interface problems

TL;DR

This paper provides a rigorous proof of second order convergence for a modified MAC scheme applied to Stokes interface problems with moving interfaces, validating its accuracy through numerical experiments in 2D and 3D.

Contribution

It introduces a novel error analysis framework for the MAC scheme that does not require problem reformulation, proving second order convergence of the velocity gradient.

Findings

Proves second order convergence of velocity, pressure, and velocity gradient.

Numerical experiments confirm theoretical convergence rates in 2D and 3D.

Method applies to problems with moving interfaces and constant viscosity.

Abstract

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $\ell^2$-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.