A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws

TL;DR

This paper introduces a fifth order Hermite WENO reconstruction method for hyperbolic conservation laws, leveraging interface values from the GRP solver to achieve high accuracy with a compact scheme, improving nonlinear wave resolution.

Contribution

The paper presents a novel HWENO scheme that directly uses interface values from the GRP solver, reducing computational steps and enhancing nonlinear wave resolution.

Findings

Achieves fourth order temporal accuracy with only two HWENO reconstructions.

The scheme is more compact and efficient compared to conventional HWENO or DG methods.

Numerical results demonstrate improved resolution of nonlinear waves.

Abstract

This paper develops a new fifth order accurate Hermite WENO (HWENO) reconstruction method for hyperbolic conservation schemes in the framework of the two-stage fourth order accurate temporal discretization in [{\em J. Li and Z. Du, A two-stage fourth order time-accurate discretization {L}ax--{W}endroff type flow solvers, {I}. {H}yperbolic conservation laws, SIAM, J. Sci. Comput., 38 (2016), pp.~A3046--A3069}]. Instead of computing the first moment of the solution additionally in the conventional HWENO or DG approach, we can directly take the {\em interface values}, which are already available in the numerical flux construction using the generalized Riemann problem (GRP) solver, to approximate the first moment. The resulting scheme is fourth order temporal accurate by only invoking the HWENO reconstruction twice so that it becomes more compact. Numerical experiments show that such compactness makes significant impact on the resolution of nonlinear waves.