Existence and smoothness of continuous and discrete solutions of a two-dimensional shallow water problem over movable beds with nonlinear sediment transport relationship

A two-dimensional shallow water system over movable beds with nonlinear sediment transport relationship is considered in this paper. The existence of the solutions for the continuous system is proved here and their smoothness is investigated. A Galerkin method is employed to obtain a finite-dimensional problem. A Brouwer fixed point theorem is employed for this problem and it is shown that the model equations are satisfied by the limits of the resulting solution sequences.We also consider the discretized problem using a local discontinuous Galerkin scheme. We perform an error analysis and show that the method is convergent and the error is bounded according to a specific norm defined herein.