Abstract:
A new finite-difference (NFD) scheme of the fifth order in space and the third order in time which preserves increased accuracy in the regions of shock wave influence is constructed. A comparative analysis of the NFD scheme accuracy with the RBM (Rusanov–Burstein–Mirin) and A-WENO (Alternative Weighted Essentially Non-Oscillatory) schemes is performed in the calculation of a special Cauchy problem with smooth periodic initial data for shallow water equations, in the exact solution of which shock waves arise inside the calculated region as a result of gradient catastrophes. It is shown that in smooth parts of the approximated solution, outside the regions of influence of shock waves, the NFD scheme is significantly more accurate than the third-order RBM scheme, and on sufficiently coarse numerical grids it is more accurate than the fifth-order A-WENO scheme in space and third-order in time; on smaller numerical grids, the NFD and A-WENO schemes have approximately the same accuracy in these parts of the calculated solution. In areas of impact of shock waves, where the RBM scheme becomes significantly more accurate than the A-WENO scheme, the NFD scheme has a higher accuracy than the RBM schema.
Key words:high-precision numerical schemes, shallow water equations, local and integral convergence of numerical solutions.
