Finite difference alternative WENO schemes with Riemann invariant-based local characteristic decompositions for compressible Euler equations

TL;DR

This paper introduces a computationally efficient finite difference WENO scheme based on Riemann invariants for the Euler equations, reducing cost while maintaining accuracy and non-oscillatory properties.

Contribution

It proposes a novel finite difference WENO method using Riemann invariants to lower computational costs for Euler equations.

Findings

Reduced computational cost due to sparse eigenmatrix.

Maintains high-order accuracy in smooth regions.

Effectively resolves discontinuities without oscillations.

Abstract

The weighted essentially non-oscillatory (WENO) schemes are widely used for hyperbolic conservation laws due to the ability to resolve discontinuities and maintain high-order accuracy in smooth regions at the same time. For hyperbolic systems, the WENO procedure is usually performed on local characteristic variables that are obtained by local characteristic decompositions to avoid oscillation near shocks. However, such decompositions are often computationally expensive. In this paper, we study a Riemann invariant-based local characteristic decomposition for the compressible Euler equations that reduces the cost. We apply the WENO procedure to the local characteristic fields of the Riemann invariants, where the eigenmatrix is sparse and thus the computational cost can be reduced. It is difficult to obtain the cell averages of Riemann invariants from those of the conserved variables due to the nonlinear relation between them, so we only focus on the finite difference alternative WENO versions. The efficiency and non-oscillatory property of the proposed schemes are well demonstrated by our numerical results.