Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience

• Published in: Computer methods in applied mechanics and engineering 2018-12, Vol.342, p.710-741
• Main Authors: Kesserwani, Georges, Ayog, Janice Lynn, Bau, Domenico
• Format: Article
• Language: English
• Subjects: Basis functions; Cartesian coordinates; Coefficients; Comparison and verification; Complexity; Computational fluid dynamics; Computer simulation; Drying; Finite volume method; Fluid flow; Fluid mechanics; Galerkin method; Hydrodynamics; Interdependence; Mathematical models; Modal discontinuous Galerkin; Quadrilateral elements; Quadrilaterals; Second-order formulations; Shallow water equations; Topography; Two dimensional models; Well-balancedness; Wetting; Wetting and drying
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2018.08.003

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. •Second-order Discontinuous Galerkin (DG2) models on quadrilateral elements for modelling 2D hydrodynamics.•Standard 2D-DG2 form simplified to a so-called “slope-decoupled” form for robustness and efficiency purposes.•Well-balancedness of the slope-decoupled 2D-DG2 planar solutions is theoretically studied.•Both the standard and the slope-decoupled 2D-DG2 schemes deliver comparable accuracy for analytical tests.•The slope-decoupled 2D-DG2 scheme has attractive conse