Аннотация:

We study the accuracy of shock-capturing schemes for hyperbolic systems of conservation laws theoretically and numerically. Based on the method of differential approximation of the numerical schemes, we give a theoretical justification for the fact that high order shock-capturing schemes can have significantly different accuracy in the various smoothness regions of calculated weak solutions. We also show that in the case of the hyperbolic system with invariants, different invariants of this system can be calculated with significantly different accuracy in the same smoothness region of the exact solution, in particular, in the shock influence areas or inside the centered rarefaction waves. To illustrate these theoretical results, we use test calculations by the RBM (Rusanov-Burstein-Mirin), TVD and WENO5 finite-difference schemes of the special shallow water Cauchy problem, which exact solution contains a shock and centered rarefaction waves.