APPLICATION OF AN EXPLICIT RIEMANN SOLVER TO ONE-DIMENSIONAL NONLINEAR SHALLOW-WATER EQUATIONS

摘要

This paper reports the derivation and application of an explicit series solution to the exact Riemann problem for the nonlinear shallow-water equations in one dimension. A Riemann solver is the fundamental building block in the Godunov-type formulation of many nonlinear problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, the present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the governing equations. A sensitivity analysis shows fast convergence of the series solution and the first two or three terms can provide highly accurate results. The proposed Riemann solver is implemented with a Godunov-type scheme for the nonlinear-shallow water equations. Numerical experiments show that the model correctly describes the formation of shocks and rarefaction fans and produces excellent agreement with exact solutions. The proposed Riemann solver is explicit and thus can be easily implemented into existing finite-volume models for long-wave propagation and runup calculations.