Numerical simulation of flows with shocks

Using the non-monotone Lax-Wendroff scheme, it is shown that the reduction in the accuracy (to the order not higher than the first) of TVD schemes in unsteady shock calculations is mainly due to decreasing smoothness of the difference flux operators, which is related to the use of minimax procedures. It is shown theoretically and numerically that the Lax-Wendroff scheme, in contrast to its TVD modifications, approximates with the second order the Hugoniot conditions on the fronts of unsteady shocks. At the same time, the Lax-Wendroff scheme reduces the order of convergence to the first in the vicinity of the point of gradient catastrophe, where the shock is forming. Such reduction in the order of convergence is related to the fact that this scheme, in contrast to the compact schemes with high-order divergent artificial viscosity, has only the first order of weak approximation for discontinuous solutions.