An a priori error estimate for the local discontinuous Galerkin method applied to two‐dimensional shallow water and morphodynamic flow

Abstract

The application of discontinuous Galerkin (DG) methods to the numerical solution of the two‐ and three‐dimensional shallow water equations has seen increased interest over the past decade. In this article, previous work by the second author and several collaborators on the application and analysis of the DG method is extended to coupled shallow water/bed morphology dynamics, also called morphodynamics. Morphodynamic boundary value problems are used to simultaneously model shallow water hydrodynamics and sediment transport processes in estuarine and coastal systems. The governing equations of interest arise when the two‐dimensional Saint‐Venant equations are tightly coupled to the corresponding Exner equation. The tight coupling of these two processes presents numerous modeling and analytical challenges. The resulting nonlinear system is incompletely parabolic, and contains a nonconservative product. In this work, some of these analytical challenges are tackled by applying a local discontinuous Galerkin method to the morphodynamic system; the quantities of interest and their gradients are solved separately. A general, semidiscrete finite element formulation is presented, and after justifying some mild assumptions, a new a priori error estimate for the system is derived. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 397–421, 2015