Abstract In the present work, a novel DOT ADER numerical solver capable of handling porosity and bottom discontinuities in the framework of the 1D porous Shallow Water Equations (SWEs) is presented. In order to ensure the preservation of the water at rest condition, a new set of well-balanced governing equations based on the isotropic porosity parameter is derived. The effects exerted by the bed slope and porosity variation source terms are accurately accounted for inside the Riemann solver: to this purpose, an augmented Riemann problem is created by adding two fictitious equations stating the invariance of porosity and bottom in time to the SWEs system. With the aim of computing the non-conservative fluxes, which in the augmented system replace the original source terms, meanwhile ensuring robustness, stability and accuracy, a novel approximate numerical scheme, based on the entropy-satisfying DOT family, is introduced. The extension of the novel Riemann solver, which strictly conserves mass, to a second order of accuracy in both space and time is addressed in the ADER framework. The fulfillment of the C-property condition (i.e. the exact preservation of an initial quiescent flow) in the presence of a discontinuous porosity field and over a non-flat bottom with abrupt variation is theoretically proved and numerically verified. The capability of the proposed numerical scheme to simulate some Riemann problems developing across porosity discontinuities and bed steps is finally assessed.

中文翻译：

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基于 DOT ADER 增强黎曼求解器的多孔浅水方程的二阶数值格式

摘要 在目前的工作中，提出了一种能够在一维多孔浅水方程 (SWE) 框架中处理孔隙度和底部不连续性的新型 DOT ADER 数值求解器。为了保证静止状态下水的保存，基于各向同性孔隙度参数推导出了一套新的平衡良好的控制方程。Riemann 求解器内部准确地考虑了床坡度和孔隙度变化源项所施加的影响：为此，通过向 SWEs 系统添加两个虚拟方程来说明孔隙度和底部的时间不变性，从而创建了一个增强的黎曼问题。以计算非保守通量为目的，在增广系统中取代原始源项，同时确保鲁棒性、稳定性和准确性，介绍了一种基于熵满足 DOT 族的新近似数值方案。ADER 框架解决了将严格守恒质量的新型黎曼求解器扩展到二阶空间和时间精度的问题。在存在不连续孔隙度场和在具有突然变化的非平坦底部的情况下，C 性质条件的满足（即初始静态流动的精确保持）在理论上得到证明并在数值上得到验证。最终评估了所提出的数值方案模拟一些跨越孔隙度不连续性和床阶发展的黎曼问题的能力。ADER 框架解决了在空间和时间上达到二阶精度的问题。在存在不连续孔隙度场和在具有突然变化的非平坦底部的情况下，C 性质条件的满足（即初始静态流动的精确保持）在理论上得到证明并在数值上得到验证。最后评估了所提出的数值方案模拟在孔隙度不连续性和床阶上发展的一些黎曼问题的能力。ADER 框架解决了在空间和时间上达到二阶精度的问题。在存在不连续孔隙度场和在具有突然变化的非平坦底部的情况下，C 性质条件的满足（即初始静态流动的精确保持）在理论上得到证明并在数值上得到验证。最终评估了所提出的数值方案模拟一些跨越孔隙度不连续性和床阶发展的黎曼问题的能力。