Discontinuous Galerkin Methods with Nodal and Hybrid Modal/Nodal Triangular, Quadrilateral, and Polygonal Elements for Nonlinear Shallow Water Flow
Discontinuous Galerkin Methods with Nodal and Hybrid Modal/Nodal Triangular, Quadrilateral, and Polygonal Elements for Nonlinear Shallow Water Flow
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DOI：
10.1016/j.cma.2013.11.006
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发表时间：
2014-03
影响因子：
7.2
通讯作者：
D. Wirasaet;E. Kubatko;C. Michoski;Seizo Tanaka;J. Westerink;C. Dawson
中科院分区：
文献类型：
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作者：
D. Wirasaet;E. Kubatko;C. Michoski;Seizo Tanaka;J. Westerink;C. Dawson
我们对一系列非结构化网格上的节点和混合模态/节点不连续伽辽金 (DG) 有限元解进行了全面评估，以平滑解决非线性浅水流。考虑三角形上的节点 DG 方法和四边形上的张量积节点基础。混合模态/节点DG方法利用两种不同的基于多边形的协同多项式来实现DG离散化；由 Gram-Schmidt 过程构建的正交基函数用作 DG 弱公式中的试验和测试函数；节点基础作为区域整合的有效手段。这些是在三角形、四边形和多边形元素上实现的。此外，我们还讨论了为了实现所谓的良好平衡特性而需要考虑的方面，即在空间变化的床中保持静止状态的稳态。通过对具有制造解决方案的非线性问题以及具有平床和非平床的非线性 Stommel 问题进行 h 和 p 收敛研究，证明了精度和计算成本方面的性能。为了评估四边形和多边形单元与三角形单元相比的性能，我们考虑一种设置，其中四边形网格、混合的三角形-四边形网格和多边形网格是从给定的三角形网格导出的，反之亦然。进行的测试揭示了使用四边形单元在单位精度的计算成本和计算时间方面的优点。更重要的是，数值结果清楚地表明，在给定精度水平下，高阶方案显着提高了成本性能，与线性和二次插值法相比，三次或双三次插值法在精度方面尤其取得了显着的提高，并且当 p> 3 时，效益逐渐递减。
We present a comprehensive assessment of nodal and hybrid modal/nodal discontinuous Galerkin (DG) finite element solutions on a range of unstructured meshes to nonlinear shallow water flow with smooth solutions. The nodal DG methods on triangles and a tensor-product nodal basis on quadrilaterals are considered. The hybrid modal/nodal DG methods utilize two different synergistic polynomial bases on polygons in realizing the DG discretization; orthogonal basis functions constructed by the Gram–Schmidt process are used as trial and test functions in a DG weak formulation; and a nodal basis is used as an efficient means for area integration. These are implemented on triangular, quadrilateral, and polygonal elements. In addition, we discuss aspects to be considered in order to achieve the so-called well-balanced property that preserves steady state at rest with a spatially varying bed. The performance in terms of accuracy and computational cost is demonstrated using h and p convergence studies on a nonlinear problem with a manufactured solution and the nonlinear Stommel problem with flat and non-flat beds. To assess the performance of quadrilateral and polygonal elements in comparison to triangular elements, we consider a setting in which a quadrilateral mesh, a mixed triangular–quadrilateral mesh, and polygonal mesh are derived from a given triangular mesh and vice versa. The tests conducted reveal the merit of using the quadrilateral elements in terms of computational cost per accuracy and computing time. More importantly, the numerical results clearly show that high order schemes significantly improve the cost performance for a given level of accuracy, with cubic or bi-cubic interpolants particularly achieving dramatic improvements in accuracy as compared to linear and quadratic interpolants, with diminishing benefit as p> 3.