Explicit solution to the exact Riemann problem and application in nonlinear shallow-water equations
The Riemann solver is the fundamental building block in the Godunov-type formulation of many nonlinear fluid-flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical Solution to file exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth-order term. For the nonlinear shallow-water equations, it sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite-volume model with it Godunov-type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two-dimensional darn-break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright (c) 2007 John Wiley & Sons, Ltd.
riemann problem shallow-water equations riemann solver homotopy analysis method dam-break problem long-wave equations