Comparative analysis of the accuracy of three different schemes in the calculation of shock waves

We perform a comparative accuracy study of the weighted essentially non-oscillatory (WENO), compact high-order weak approximation (CWA) and central-upwind (CU) schemes used to compute discontinuous solutions containing shocks propagating with variable velocity. We demonstrate that the accuracy of the formally high-order WENO and CU schemes, which are constructed using nonlinear flux correction mechanisms, reduces to the first order after the formation of shocks. This is done by measuring the integral convergence on intervals containing the region of shock wave influence. At the same time, the CWA scheme, which is designed to be high-order in the weak sense and does not rely on any nonlinear flux corrections, retains approximately the second order of integral convergence even in the regions of shock wave influence. As a result, in these areas, the accuracy of the WENO and CU schemes is significantly lower than the accuracy of the CWA scheme. We provide a theoretical justification of these numerical results.