Tests with Exact Solution

In this Chapter we present a suite of carefully selected test problems with exact solution for the non-linear shallow water equations on a horizontal bottom, admitting both wet-bed and dry-bed conditions. The purpose of these test problems is to rigorously assess numerical methods for the general initial-boundary value problem, against exact solutions for the kind of physical situations that are bound to be challenging to the numerical modeller. Each of the test problems has been carefully designed and offers a specific challenge to numerical methods. The exact solutions will be used to asses the suitability of approximate Riemann solvers and the accuracy of approximate numerical solutions in successive chapters. A computer programme listing for the Riemann solvers is given at the end of the chapter, for both wet and dry-bed conditions. Useful background is found in Chaps. 6 and 7 , as well as in Refs. [1, 2] for the wet-bed case and [3] for the dry-bed case.