Fifth-Order Well-Balanced Path-Conservative A-WENO Scheme for the Ripa Model

Abstract

In this work, we introduce a fifth-order well-balanced (WB) path-conservative A-WENO scheme with the central-upwind numerical fluxes (PCCU-5) for the Ripa model. The proposed scheme is capable of exactly preserving a variety of steady states, including still-water, moving-water, isobaric, and constant water height ones. This goal is achieved with the help of a flux globalization technique: The source terms are incorporated into the fluxes, resulting in a quasi-conservative system, for which central-upwind numerical fluxes are computed using the path-conservative integration. The proposed A-WENO scheme utilizes a WENO interpolation of the equilibrium variables rather than the conservative ones to ensure the WB property. In addition, we perform the WENO interpolation of the local characteristic equilibrium variables to mitigate numerical oscillations near discontinuities. We perform a series of numerical experiments, which demonstrate that the proposed fifth-order WB PCCU-5 scheme achieves high resolution and clearly outperforms its second-order counterpart. Our numerical results also demonstrate the importance of the local characteristic projection for significantly reducing (eliminating) numerical oscillations near discontinuities.