Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method

TL;DR

This paper demonstrates that a finite difference method with immersed interface corrections can achieve nearly second-order uniform accuracy for low Reynolds number Navier-Stokes flow with a moving boundary, despite interface-induced jumps.

Contribution

It provides a rigorous proof of uniform second-order accuracy for velocity and pressure in Navier-Stokes flow with moving boundaries using an immersed interface method.

Findings

Velocity error is $O(h^2| ext{log} h|^2)$ near the interface.

Pressure error is $O(h^2| ext{log} h|^3)$.

Method achieves almost second-order accuracy uniformly across the domain.

Abstract

We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Difference operators are corrected to $O(h)$ near the interface using the immersed interface method, adding terms related to the jumps, on a regular grid with spacing $h$ and periodic boundary conditions. The force at the interface is assumed known within an error tolerance; errors in the interface location are not taken into account. The error in velocity is shown to be uniformly $O(h^2|\log{h}|^2)$, even at grid points near the interface, and, up to a constant, the pressure has error $O(h^2|\log{h}|^3)$. The proof uses estimates for finite difference versions of Poisson and diffusion equations which exhibit a gain in regularity in maximum norm.