Exact Riemann Solver: Dry Bed

InDry bed Vacuum Wet/dry front this Chapter we solve the Riemann problem for the split one-dimensional shallow water equations for the case in which the solution is adjacent to dry regions, or vacuum regions, that is when $$h(x,y,t)=0$$ h ( x , y , t ) = 0 . The boundary separating regions of water and no water, called the wet/dry front, emerges from the exact solution of the Riemann problem as the edge (tail) of a strong rarefaction wave, which is the only wave present in the solution structure. This wet/dry front is a very fast wave of speed $$S=u_{L}+2a_{L}$$ S = u L + 2 a L , for the case of a dry right data state, and is the source of considerable challenges in computational practice. We solve the Riemann problem exactly under two conditions: (i) vacuum is present in one of the initial states at time $$t=0$$ t = 0 and (ii) vacuum appears from the interaction of two non-vacuum states, generating two wet/dry fronts. The solution strategy is then extended to the split two-dimensional shallow water equations and for the case of pollutant transport models. A computer-programme listing is given in Chap. 8 , for the complete exact Riemann solver that can deal with both wet and dry bed conditions. Useful background is found in Chaps. 1 – 6 .