An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces
An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces
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DOI：
10.1016/j.jcp.2019.04.035
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发表时间：
2019-08
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影响因子：
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通讯作者：
Xin Liu;Alina Chertock;A. Kurganov
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文献类型：
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作者：
Xin Liu;Alina Chertock;A. Kurganov
本文考虑二维带科里奥利力的Saint-Venant浅水方程组。我们专注于一个低弗劳德数的情况下，在该系统是刚性和传统的显式数值方法是非常低效的，往往是不切实际的。我们的目标是设计一个渐近保持（AP）计划，这是一致渐近一致和稳定的广泛的（低）弗劳德数。使用[Haack等人，Commun. Comput.物理、12（2012），pp. 955-980]中的等熵欧拉和Navier-Stokes方程。我们分裂成刚性和非刚性部分的通量，然后使用隐式显式的方法：应用显式双曲型求解器（我们使用二阶中心迎风格式）的非刚性部分的系统，同时处理它的刚性部分隐式。此外，通量的刚性部分是线性的，因此我们将所提出的方法的隐式阶段减少到求解Poisson型椭圆方程，该方程使用标准的二阶中心差分格式离散。这表明，开发的AP计划实现了理论的二阶收敛速度和时间，阶跃稳定性约束与弗劳德数无关。这使得所提出的AP计划的一个有效的和强大的替代全显式数值方法。
We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting proposed in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955–980] in the context of isentropic Euler and Navier-Stokes equations. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver (we use the second-order central-upwind scheme) to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme.We conduct a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.