A robust fifth order finite difference Hermite WENO scheme for compressible Euler equations

Abstract

In this paper, we develop a robust fifth order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations following the HWENO with limiter (HWENO-L) scheme (Zhang and Zhao, 2023). The HWENO-L scheme reduced storage and increased efficiency by using restricted derivatives only for time discretizations. However, it cannot control spurious oscillations well when facing strong shocks since the derivatives are directly used in spatial discretizations without any restrictions. To address such an issue, our proposed HWENO scheme performs flux reconstructions in the finite difference framework without using the derivative value of a target cell, which can result in a simpler and more robust scheme. The resulting scheme is simpler while achieving fifth order accuracy, making it more efficient. Besides, numerically we find it is very robust for some extreme problems even without positivity-preserving limiters. The proposed scheme also inherits advantages of previous HWENO schemes, including arbitrary positive linear weights in flux reconstructions, compact reconstructed stencils, and high resolution. Extensive numerical tests are performed to demonstrate the fifth order accuracy, efficiency, robustness, and high resolution of the proposed HWENO scheme.