Hydrostatic relaxation scheme for the 1D shallow water - Exner equations in bedload transport

•We construct a hydrostatic reconstruction scheme with relaxation solver for shallow water-Exner equations.•The numerical sheme works well even in non-homogeneous problem.•We obtain the good accuracy of the scheme correspond to the analytical solution.•The comparisons of numerical results have a good agreement with data experiments in dam-break simulation. This paper describes the numerical scheme of the one-dimensional Exner - shallow water equations for sediment modeling. Here, our novel numerical scheme is called hydrostatic relaxation scheme which is a robust and straightforward scheme. This scheme is given as the hydrostatic reconstruction of the relaxation solver. Some numerical tests are presented such as analytical solution, transcritical flow over a granular bump, and erodible bottom due to dam-break. The comparisons of numerical and data experiments are also given in dam-break over granular bed simulation. The results of our simulation have a good agreement with analytical solution and the data experiments. Moreover, satisfactory convergence rate in L1-norm is obtained.