Pollutant transport by shallow water equations on unstructured meshes: Hyperbolization of the model and numerical solution via a novel flux splitting scheme
Pollutant transport by shallow water equations on unstructured meshes: Hyperbolization of the model and numerical solution via a novel flux splitting scheme
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DOI：
10.1016/j.jcp.2016.05.023
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发表时间：
2016-09
期刊：
影响因子：
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通讯作者：
D. Vanzo;A. Siviglia;E. Toro
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文献类型：
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作者：
D. Vanzo;A. Siviglia;E. Toro
本文的目的是双重的。首先，使用 Cattaneo 松弛方法，我们将浅水流在非平坦地形上的污染物迁移和各向异性扩散的控制方程组重新表述为具有刚性源项的双曲平衡定律。所提出的弛豫系统规避了标准平流扩散模型中固有的无限波速悖论。事实证明，这为时间步长的选择提供了更大的稳定性范围。其次，按照通量分裂方法，我们推导出一种新颖的数值方法来离散所产生的问题。特别是，我们提出了一种新的通量分裂，并研究了相关的两个微分方程组，分别称为“流体动力学”和“松弛扩散”系统。对于所提出的分裂，我们分析了所得的两个微分方程组，并提出了两种戈杜诺夫型离散化方案。与现有方法相比，这些方案实现简单、稳健、准确且快速。由此产生的方法在非结构化网格上实现，并系统地评估了一系列精心挑选的测试问题（包括非平坦地形以及润湿和干燥问题）的准确性、鲁棒性和效率。正式的二阶精度是通过收敛率研究来评估的。
The purpose of this paper is twofold. First, using the Cattaneo's relaxation approach, we reformulate the system of governing equations for the pollutant transport by shallow water flows over non-flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms. The proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection–diffusion models. This turns out to give a larger stability range for the choice of the time step. Second, following a flux splitting approach, we derive a novel numerical method to discretise the resulting problem. In particular, we propose a new flux splitting and study the associated two systems of differential equations, called the “hydrodynamic” and the “relaxed diffusive” system, respectively. For the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the Godunov-type. These schemes are simple to implement, robust, accurate and fast when compared with existing methods. The resulting method is implemented on unstructured meshes and is systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems including non-flat topography and wetting and drying problems. Formal second order accuracy is assessed through convergence rates studies.