求解多孔浅水波方程组的高精度和谐数值方法

High-Accuracy Numerical Methods for Solving the System of Porous Shallow Water Wave Equations

摘要: 本文提出了一种高阶平衡(Runge-Kutta Discontinuous Galerkin, RKDG)有限元方法，用于解决具有不连续孔隙度和复杂底部地形的多孔浅水波方程组。该方法能够在静水稳态解的条件下保持良好的平衡特性，即当数值通量梯度与源项在连续意义上完全平衡时，离散格式也能准确保持稳态解。与传统的恒定横截面浅水波方程相比，由于孔隙度和底部地形的非连续性，构造具有良好平衡性质的高阶数值格式更加复杂。为应对这一挑战，我们首先对源项进行了重构，并结合基于Lax-Friedrichs通量的静水重构方法，同时合理设计离散格式，确保该方法在处理不连续区域时仍能保持高阶精度和稳定性。通过一系列标准数值算例验证了所提方法的有效性，结果表明，该方法在保持静水平衡、实现高阶精度以及分辨不连续解方面具有优异的表现。

Abstract: In this paper, we propose a high-order balanced Runge-Kutta Discontinuous Galerkin (RKDG) finite element method for solving the porous shallow water wave equations with discontinuous porosity and complex bottom topography. The proposed method preserves the good balance properties of the steady-state solution, meaning that when the numerical flux gradient and the source terms are fully balanced in a continuous sense, the discretization scheme can also accurately maintain the steady-state solution. Compared to traditional shallow water wave equations with constant cross-sections, the discontinuities in porosity and bottom topography make it more challenging to construct a high-order numerical scheme with good balance properties. To overcome this challenge, we first reconstruct the source terms and apply a hydrostatic reconstruction method based on the Lax-Friedrichs flux, while also designing the discretization scheme to ensure high-order accuracy and stability when handling discontinuous regions. A series of standard numerical test cases validate the effectiveness of the proposed method. The results demonstrate that the method performs exceptionally well in maintaining hydrostatic balance, achieving high-order accuracy, and resolving discontinuous solutions.