Exact Riemann Solver: Wet Bed

This Chapter presents a solver for computing the exact solution through the entire wave structure of the Riemann problem for the non-linear shallow water equations for the case of a wet bed, that is for \(h(x,t)>0\) (no vacuum). The computing algorithm is based on a detailed study of elementary waves in the Riemann problem performed in Chap. 5 . There are two steps in computing the complete solution. First, the water depth \(h_{*}\) and velocity \(u_{*}\) are found in the Star Region; the solution \(h_{*}\) is the root of a non-linear algebraic equation \(f(h)=0\) , which is solved iteratively with a Newton-Raphson method. A detailed study of the behaviour of the depth function f(h) is carried out to determine existence and uniqueness of the solution, and for selecting the best solution method. The solution \(u_{*}\) for velocity follows directly from \(h_{*}\) . The second step is concerned with finding the solution through the entire wave system, which includes identifying the type of waves present and a solution sampling procedure to determine the vector Q(x, t) at any point (x, t) in the x-t half plane. The solution will be used in successive chapters for constructing upwind numerical methods for the general initial-boundary value problem, for applying boundary conditions and for assessing the accuracy of numerical approximations. Useful background is found in Chaps. 1 – 5 . The exact solution for the Riemann problem in the presence ofWet/dry front wet/dry fronts is presented in Chap. 7 . The listing of a computer programme for the exact Riemann solver, including both wet and dry-bed conditions, is given in Chap. 8 .