A Godunov's type method based on an exact solution to the Riemann problem for the shallow water equations

Explicit front-capturing finite-difference schemes, designed for modeling fluid flows described by the shallow water equations with any bottom relief, are constructed and analysed. The values of functions are calculated on discrete cell boundaries by invoking Stoker's exact solution (1948) to the hydrodynamic Riemann problem. An efficient iterative algorithm is developed for computing the exact solution to the Riemann problem, and the error of the first iterative approximation is estimated. Modified variants of the method, designed to ensure second-order accuracy in the domains of smooth solution, are presented. Examples of 1D and 2D computations are given to illustrate the effectiveness of the method.