Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience
Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience
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DOI：
10.1016/j.cma.2018.08.003
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发表时间：
2018-12
影响因子：
7.2
通讯作者：
G. Kesserwani;Janice Lynn Ayog;D. Baù
中科院分区：
文献类型：
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作者：
G. Kesserwani;Janice Lynn Ayog;D. Baù
在流体力学建模中，间断伽辽金（DG）方法构成了一个更复杂和现代的替代完善的有限体积法。后者保留了一些理想的实际功能，模拟流体动力学，如空间通量和陡峭的地形梯度之间的平衡，能够将润湿和干燥过程，计算负担得起的。在这种情况下，DG方法最初被设计用于求解具有不规则地形和润湿和干燥的二维（2D）浅水方程（SWE），尽管将公式的复杂性降低到通常是二阶精度（DG 2）。本文的目的是：（a）概述一个所谓的“斜率解耦”制定的标准2D-DG 2-SWE模拟器，其中理论的复杂性是故意降低;（B）突出所提出的斜率解耦模拟器的能力，在提供一个设置，简化的假设验证内的制定。标准和斜率解耦的2D-DG 2-SWE模型都采用2D模态基函数，通过沿笛卡尔坐标使用平均系数和沿着两个斜率系数，在四边形单元上形成局部平面DG 2解。在四边形单元上，简化了斜率解耦2D-DG 2公式的模板，以消除流量和地形近似的斜率系数的相互依赖性。从理论上研究了斜率解耦的2D-DG 2-SWE平面解的完全平衡性。后者的性能进行了比较，在经典的模拟测试与标准的2D-DG 2制定。其他测试进行诊断验证的保守属性的2D-DG 2-SWE方法在涉及急剧的地形梯度和潮湿和/或干燥区的情况下。进行的分析提供了强有力的证据表明，建议的斜率解耦的2D-DG 2-SWE模拟器是非常有吸引力的鲁棒洪水模型的发展。
In the modelling of hydrodynamics, the Discontinuous Galerkin (DG) approach constitutes a more complex and modern alternative to the well-established finite volume method. The latter retains some desired practical features for modelling hydrodynamics, such as well-balancing between spatial flux and steep topography gradients, ability to incorporate wetting and drying processes, and computational affordability. In this context, DG methods were originally devised to solve the two-dimensional (2D) Shallow Water Equations (SWE) with irregular topographies and wetting and drying, albeit at reduction in the formulation’s complexity to often being second-order accurate (DG2). The aims of this paper are: (a) to outline a so-called “slope-decoupled” formulation of a standard 2D-DG2-SWE simulator in which theoretical complexity is deliberately reduced; (b) to highlight the capabilities of the proposed slope-decoupled simulator in providing a setting where the simplifying assumptions are verified within the formulation. Both the standard and the slope-decoupled 2D-DG2-SWE models adopt 2D modal basis functions for shaping local planar DG2 solutions on quadrilateral elements, by using an average coefficient and two slope coefficients along the Cartesian coordinates. Over a quadrilateral element, the stencil of the slope-decoupled 2D-DG2 formulation is simplified to remove the interdependence of slope-coefficients for both flow and topography approximations. The fully well-balanced character the slope-decoupled 2D-DG2-SWE planar solutions is theoretically studied. The performance of the latter is compared with the standard 2D-DG2 formulation in classical simulation tests. Other tests are conducted to diagnostically verify the conservative properties of the 2D-DG2-SWE method in scenarios involving sharp topography gradients and wet and/or dry zones. The analyses conducted offer strong evidence that the proposed slope-decoupled 2D-DG2-SWE simulator is very attractive for the development of robust flood models.