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

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 (Z. Zhao et al. (2020) [27]), 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 has better resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and is more efficient than the same order finite difference WENO scheme of Jiang and Shu (1996) [11] and WENO scheme with artificial linear weights of Zhu and Qiu (2016) [29]. Various benchmark numerical examples are presented to show the fifth-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.