Well-posedness analysis and numerical implementation of a linearized two-dimensional bottom sediment transport problem

A two-dimensional linearized model of coastal sediment transport due to the action of waves is studied. Up till now, one-dimensional sediment transport models have been used. The model under study makes allowance for complicated bottom relief, the porosity of the bottom sediment, the size and density of sediment particles, gravity, wave-generated shear stress, and other factors. For the corresponding initial-boundary value problem the uniqueness of a solution is proved, and an a priori estimate for the solution norm is obtained depending on integral estimates of the right-hand side, boundary conditions, and the norm of the initial condition. A conservative difference scheme with weights is constructed that approximates the continuous initial-boundary value problem. Sufficient conditions for the stability of the scheme, which impose constraints on its time step, are given. Numerical experiments for test problems of bottom sediment transport and bottom relief transformation are performed. The numerical results agree with actual physical experiments.