Existence and smoothness of continuous and discrete solutions of a two-dimensional shallow water problem over movable beds

We consider a two-dimensional shallow water system over movable beds. We begin with a continuous system and prove the existence of the solutions, and then we investigate their smoothness. Then, we employ a Galerkin method to obtain a finite-dimensional problem which is solved using a Brouwer fixed point theorem. Therefore, we show that the limits of the resulting solution sequences satisfy the model equations.After solving the continuous problem, we focus on the corresponding discrete problem. We employ a local discontinuous Galerkin scheme for numerical solution of the discrete system and conduct an error analysis of the numerical scheme. We prove that the method is convergent and that the error is bounded according to a specific norm defined herein.