多孔浅水方程的高阶平衡保正有限差分AWENO格式
High-Order Well-Balanced and Positivity-Preserving Finite-Difference AWENO Scheme with Hydrostatic Reconstruction for Porous Shallow Water Equations
摘要: 本文提出了求解具有不连续孔隙度和底部地形的一维多孔浅水方程的高阶平衡保正有限差分AWENO格式，所提出的格式保持了静水稳态的良好平衡特性。在这个数值框架中，采用静水重构方法具有两个主要优点：1) 使用任意单调通量和重新表述源项的方法获得了良好平衡特性。2) 采用Lax-Friedrichs (LF)通量的一阶方案在适当的时间步长内保持了水高保正性。通过大量的数值算例验证了该格式具有高阶精度和良好平衡特性，所有算例的数值结果与解析解一致。
Abstract: In this paper, we propose a higher-order well-balanced and positivity-preserving finite-difference AWENO scheme for solving the one-dimensional porous shallow water equation with discontinuous porosity and bottom topography. The proposed format maintains the well-balanced property of the hydrostatic steady state. In this numerical framework, the hydrostatic reconstruction (HR) method is employed with two main advantages: 1) The method using arbitrary monotone fluxes and reformulated source terms obtains the well-balanced property. 2) The first-order scheme using Lax-Friedrichs (LF) fluxes and the HR method maintains the water height preserving properties with an appropriate time step. We verify that the scheme has high-order accuracy and well-balanced properties through a large number of numerical examples, and the numerical results of all cases agree with the analytical solutions.
文章引用：刘仁迪, 钱守国, 李刚. 多孔浅水方程的高阶平衡保正有限差分AWENO格式[J]. 应用数学进展, 2024, 13(10): 4629-4641. https://doi.org/10.12677/aam.2024.1310444

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