A dimensionally split Cartesian cut cell method for hyperbolic conservation laws

TL;DR

This paper introduces a new dimensionally split Cartesian cut cell method for hyperbolic conservation laws that improves stability and accuracy near stagnation points, with proven convergence in 1D and promising multi-dimensional results.

Contribution

It presents a novel stabilised cut cell flux within a dimensionally split framework, enhancing stability and accuracy over previous methods.

Findings

Reduces oscillations at higher Courant numbers

Achieves more accurate solutions near stagnation points

Easily extends to multiple dimensions

Abstract

We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a full description of its three-dimensional implementation in the dimensionally split framework of Klein et al. [1]. The convergence and stability of the method are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. When compared to the cut cell flux of Klein et al., it was found that the new flux alleviates the problem of oscillatory boundary solutions produced by the former at higher Courant numbers, and also enables the computation of more accurate solutions near stagnation points. Being dimensionally split, the method is simple to implement and extends readily to multiple dimensions.