On convergence of finite-difference shock-capturing schemes in regions of shock waves influence

We perform a comparative accuracy study of the Rusanov, CABARETM, and WENO5 difference schemes used to compute the dam break problem for shallow water theory equations. We demonstrate that all three schemes have the first order of convergence inside the region occupied by a centered rarefaction wave, and the Rusanov scheme has the second order of convergence in the area of constant flow between the shock and the rarefaction wave, while in the CABARETM and WENO5 schemes there is no local convergence in this area. This is due to the fact that the numerical solutions obtained by the CABARETM and WENO5 schemes have undamped oscillations in the region of influence of the shock, the amplitude of which does not decrease with decreasing of the difference grid steps. As a result, taking into account the Lax-Wendroff theorem, the numerical solutions obtained by the conservative schemes CABARETM and WENO5 converge only weakly to the exact constant solution in the region of influence of the shock wave, in contrast to the Rusanov scheme, which locally converges with the second order to the exact solution in this region.