We develop fifth-order A-WENO finite-difference schemes based on the path-conservative central-upwind method for nonconservative one- and two-dimensional hyperbolic systems of nonlinear PDEs. The main challenges in development of accurate and robust numerical methods for the studied systems come from the presence of nonconservative products. Semi-discrete second-order finite-volume path-conservative central-upwind (PCCU) schemes recently proposed in Castro Díaz et al. (2019) [8] provide one with a reliable Riemann-problem-solver-free numerical method for nonconservative hyperbolic system. In this paper, we extend the PCCU schemes to the fifth-order of accuracy in the framework of A-WENO finite-difference schemes.

We apply the developed schemes to the two-layer shallow water equations. We ensure that the developed schemes are well-balanced in the sense that they are capable of exactly preserving “lake-at-rest” steady states. We illustrate the performance of the new fifth-order schemes on a number of one- and two-dimensional examples, where one can clearly see that the proposed fifth-order schemes clearly outperform their second-order counterparts.

我们开发了基于路径保守中央迎风方法的五阶 A-WENO 有限差分格式，用于非线性 PDE 的非保守一维和二维双曲线系统。为所研究的系统开发准确和稳健的数值方法的主要挑战来自非保守产物的存在。Castro Díaz 等人最近提出的半离散二阶有限体积路径保守中央迎风 (PCCU) 方案。(2019) [8] 为非保守双曲系统提供了一种可靠的无黎曼问题求解器数值方法。在本文中，我们在 A-WENO 有限差分格式的框架内将 PCCU 格式扩展到五阶精度。

我们将开发的方案应用于两层浅水方程。我们确保开发的方案是平衡的，因为它们能够准确地保持“静止的湖泊”稳定状态。我们在许多一维和二维示例上说明了新五阶方案的性能，其中可以清楚地看到所提出的五阶方案明显优于其二阶对应方案。