On convergence of numerical schemes when calculating Riemann problems for shallow water equations

We investigate the convergence of four high order numerical schemes when calculating Riemann problems for shallow water equations. We compare a couple of the NFC (Nonlinear Flux Correction) schemes: the second order TVD (Total Variation Diminishing) and the fifth-order in space, the third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) with a couple of the third-order QL (Quasi-Linear) schemes: RBM (Rusanov–Burstein–Mirin) and CWA (Compact high order Weak Approxiation), where nonlinear flux correction is not applied. It is shown that inside the shock influence areas for the NFC schemes, unlike the QL schemes, there is no uniform local convergence of the numerical solution to the exact constant one. At the same time, inside the centered rarefaction waves, solutions of these schemes with different orders converge to the different invariants of the exact solution: with the first order to the invariant that transferred along the characteristics outgoing from the center of the rarefaction wave; and with the order not lower than the second to the invariant that is constant inside the rarefaction wave. For numerical solutions of the studied schemes we perform the classification of various types of convergence to the corresponding exact solutions of the calculated Riemann problems.