A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws

TL;DR

This paper introduces a simplified fifth-order HWENO scheme with limiter for hyperbolic conservation laws, achieving higher accuracy and efficiency with fewer oscillations compared to previous methods.

Contribution

A new HWENO scheme that applies derivatives only in time discretization, simplifying implementation and improving accuracy and resolution over existing HWENO and WENO schemes.

Findings

Achieves fifth-order accuracy in numerical tests.

Demonstrates higher efficiency and resolution than classical WENO schemes.

Shows robustness and effectiveness through benchmark examples.

Abstract

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two- dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (J. Sci. Comput., 85:29, 2020), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient and higher resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and greater efficiency than the classical fifth-order finite difference WENO scheme of Jiang and Shu. Various benchmark numerical examples are presented to show the fifth-order accuracy, great efficiency, high resolution and robustness of the proposed HWENO scheme.