A second-order pressure-accurate finite-difference scheme for the Stokes problem with rigid non-conforming boundaries

TL;DR

This paper introduces a second-order accurate finite-difference scheme for the Stokes problem with non-conforming interfaces, ensuring precise pressure and velocity solutions by leveraging Helmholtz decomposition and ghost methods.

Contribution

The novel scheme achieves second-order accuracy on pressure and velocity in complex geometries using Helmholtz decomposition and ghost methods, reducing boundary layer errors.

Findings

Achieves second-order accuracy for pressure and velocity.

Ensures strict mass conservation in complex geometries.

Reduces numerical boundary layer errors.

Abstract

We present a finite-difference scheme which solves the Stokes problem in the presence of curvilinear non-conforming interfaces and provides second-order accuracy on physical field (velocity, vorticity) and especially on pressure. The gist of our method is to rely on the Helmholtz decomposition of the Stokes equation: the pressure problem is then written in an integral form devoid of the spurious sources known to be the cause of numerical boundary layer error in most implementations, leading to a discretization which guarantees a strict enforcement of mass conservation. The ghost method is furthermore used to implement the boundary values of pressure and vorticity near curved interfaces.