Jump splicing schemes for elliptic interface problems and the incompressible Navier-Stokes equations

TL;DR

This paper introduces a versatile, second-order accurate finite difference framework for elliptic interface problems and incompressible Navier-Stokes equations with discontinuities, using level set representations and jump condition extrapolations.

Contribution

It develops a general, robust method for accurately handling interface discontinuities in PDEs, applicable in 2D and 3D, with proven convergence and ease of implementation.

Findings

Achieves second-order accuracy for Navier-Stokes with surface tension.

Compatible with symmetric positive-definite solvers for elliptic problems.

Works effectively with non-smooth geometries and in three dimensions.

Abstract

We present a general framework for accurately evaluating finite difference operators in the presence of known discontinuities across an interface. Using these techniques, we develop simple-to-implement, second-order accurate methods for elliptic problems with interfacial discontinuities and for the incompressible Navier-Stokes equations with singular forces. To do this, we first establish an expression relating the derivatives being evaluated, the finite difference stencil, and a compact extrapolation of the jump conditions. By representing the interface with a level set function, we show that this extrapolation can be constructed using dimension- and coordinate-independent normal Taylor expansions with arbitrary order of accuracy. Our method is robust to non-smooth geometry, permits the use of symmetric positive-definite solvers for elliptic equations, and also works in 3D with only a change in finite difference stencil. We rigorously establish the convergence properties of the method and present extensive numerical results. In particular, we show that our method is second-order accurate for the incompressible Navier-Stokes equations with surface tension.