Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations
Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations
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DOI：
10.1016/j.advwatres.2010.08.005
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发表时间：
2010-12
影响因子：
4.7
通讯作者：
Y. Xing;Xiangxiong Zhang;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
Y. Xing;Xiangxiong Zhang;Chi-Wang Shu
具有非平底地形的浅水方程已被广泛应用于河流和海岸地区的水流模拟。在这些模拟中出现的一个重要困难是没有水的干燥地区的出现，因为标准的数值方法可能会失败，在这些地区的存在。这些方程也有静水稳定状态的解决方案，其中的通量梯度是非零的，但完全平衡的源项。本文提出了一种高阶间断Galerkin方法，该方法既能精确地保持静水定常状态，又能在不损失质量守恒的前提下保持水高的非负性。在一维引入一个简单的保正限制器，在适当的CFL条件下是有效的，然后用矩形网格扩展到二维。数值试验验证了该方法的保正性、良好平衡性、高阶精度以及对光滑解和间断解的良好分辨。
Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas where no water is present, as standard numerical methods may fail in the presence of these areas. These equations also have still water steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. A simple positivity-preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity-preserving property, well-balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.