Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

TL;DR

This paper introduces a kernel-free boundary integral method combined with a modified MAC scheme to accurately solve two-phase Stokes equations with discontinuous viscosity on staggered grids, achieving second-order accuracy.

Contribution

The paper develops a novel kernel-free boundary integral approach integrated with a modified MAC scheme for two-phase Stokes problems with discontinuous viscosity, enabling accurate solutions on Cartesian grids.

Findings

Achieves second-order accuracy in velocity, pressure, and velocity gradient.

Effectively handles high contrast viscosity coefficients.

Numerical tests confirm the method's robustness and accuracy.

Abstract

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests.