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 $$h(x,t)>0$$ 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 $$h_{*}$$ h ∗ and velocity $$u_{*}$$ u ∗ are found in the Star Region; the solution $$h_{*}$$ h ∗ is the root of a non-linear algebraic equation $$f(h)=0$$ 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 $$u_{*}$$ u ∗ for velocity follows directly from $$h_{*}$$ 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 .