Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

TL;DR

This paper introduces a fifth-order finite difference Hermite WENO scheme for shallow water equations that is well-balanced, highly accurate, and computationally efficient, effectively preserving steady states and reducing oscillations.

Contribution

It develops a compact, fifth-order HWENO scheme that maintains the well-balance property and improves accuracy and efficiency over existing methods.

Findings

Achieves fifth-order accuracy in benchmarks.

Preserves steady-state solutions effectively.

Demonstrates superior resolution and efficiency.

Abstract

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balance property. Instead of using a five-point stencil in the same fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.