The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

Abstract

The Shallow Water equations model is commonly used to reproduce a wide variety of environmental flows. Its applications include river hydraulics and flood forecasting.Therefore, an algorithm to solve the Riemann problem for the Shallow Water equations can be a considerable asset for the development of numerical schemes devoted to the solution of such equations. Nevertheless, for real world applications, it becomes mandatory to consider the influence of the irregular shape of the cross section of natural channels in the solution.This work presents an energy based exact Riemann solver for the Shallow Water equations for channels with an arbitrarily shaped cross section with the presence of a non flat bottom and dry states.The procedure for solving the Riemann problem involves the solution of a system of non linear algebraic equations, for which a CPU time efficient solution strategy based on nested Newton-Raphson methods is presented.The proposed solver can be used to develop Finite Volume or Discontinuous Galerkin schemes, based on the Godunov method. For a non flat bottom, the Shallow Water equations are a non conservative system, once the bottom elevation is included in the state variables. Many available schemes take advantage of this fact to use approximate path-conservative Riemann solvers. In other cases, the term related to the non flat bottom is treated as a source term. The use of this exact solver avoids the necessity of considering the bottom topography as a source term and of reverting to approximate solvers. These strategies can in facts pose numerical problems in situations such as transonic rarefactions and flows approaching abrupt changes in bottom topography.The solver is tested by reproducing the solution of Riemann problems with a Godunov finite volume scheme. An analysis of the performance of the solver shows that it requires less computational time than path-conservative schemes and that it has better stability and convergence properties, especially in situations involving resonant waves, such as transonic flows across a bottom step.