Moving mesh version of wave propagation algorithm based on augmented Riemann solver

Abstract

Shallow water equations are a nonlinear system of hyperbolic conservation laws. This system admits discontinuous solutions. In addition, a discontinuous bathymetry can lead to the discontinuous solutions. The accuracy of numerical methods depends on the accuracy of the numerical solutions around these discontinuities.In order to increase the accuracy, the adaptive mesh methods are more cost-effective than the methods based on the global refinement, In terms of computational cost. Moving mesh method is an adaptive mesh method. In this method, an increase of the number of mesh points is not needed for increasing the accuracy and efficiency of numerical solutions. Highly accurate solutions only can be obtained by concentrating points in areas where more accuracy is needed and decreasing the number of points in smooth areas.In this paper, LeVeque wave propagation algorithm is improved by the moving mesh method, for shallow water equations with variable bathymetry. Moreover, augmented Riemann solver is used for solving Riemann problems in each time step. Finally, moving mesh version of wave propagation algorithm based on augmented Riemann solver is proposed. The numerical solutions based on the moving mesh method and fixed mesh method have a significant difference and application of a moving mesh method yields highly accurate solutions around shocks and discontinuities.