Genuinely multi-dimensional stationarity preserving Finite Volume formulation for nonlinear hyperbolic PDEs

TL;DR

This paper introduces a new multi-dimensional finite volume method that preserves stationary states in nonlinear hyperbolic PDEs, outperforming existing methods in accuracy and stability.

Contribution

It proposes a novel multi-dimensional quadrature strategy for stationarity preservation in finite volume methods, enhancing accuracy for nonlinear hyperbolic PDEs.

Findings

Significantly outperforms existing methods on stationary and non-stationary solutions.

Preserves multi-dimensional stationary states effectively.

Applicable to nonlinear conservation and balance laws.

Abstract

Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g.\ via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a possibly existing balance of contributions coming from different directions, such as the one characterizing multi-dimensional stationary states. Instead of being preserved, they are usually diffused away by such methods. Stationarity preserving methods use a better suited stabilization term that vanishes at the stationary state, allowing the method to preserve it. This work presents a general approach to stationarity preserving Finite Volume methods for nonlinear conservation/balance laws. It is based on a multi-dimensional stationarity preserving quadrature strategy that allows to naturally introduce genuinely multi-dimensional numerical fluxes. The new methods are shown to significantly outperform existing ones even if the latter are of higher order of accuracy and even on non-stationary solutions.