Lee, H. and Lee, N., 2023. An implicit quadrature-free modal Discontinuous Galerkin (DG) scheme for shallow water equations. In: Lee, J.L.; Lee, H.; Min, B.I.; Chang, J.-I.; Cho, G.T.; Yoon, J.-S., and Lee, J. (eds.), Multidisciplinary Approaches to Coastal and Marine Management. Journal of Coastal Research, Special Issue No. 116, pp. 91-95. Charlotte (North Carolina), ISSN 0749-0208.

Even though the Discontinuous Galerkin (DG) method has gained current status as an effective numerical tool for hyperbolic conservation laws (Euler equations, shallow water equations, etc.), its use is limited due to computational burden and algorithmic complexity. As an alternative to the conventional and standard approaches based on nodal basis schemes, an implicit modal discontinuous Galerkin scheme was developed for shallow water equations. The developed scheme employs a quadrature-free approach with the orthogonal basis functions on a triangular element and flux integrals for Riemann solvers with an edge coordinate system. It is believed to be simpler and more efficient compared to the conventional discrete quadrature method. In addition, the use of implicit algorithm made the code robust and allowed larger time steps. The model was applied to some benchmark problems (including channel contraction, partial dam-break flow, and curved channel flow) and good agreements were observed.