Fifth-Order Finite-Difference Scheme in Space with Increased Accuracy in Shock Influence Areas

A new finite-difference (NFD) scheme of fifth order in space and third order in time that maintains high accuracy in shock influence areas is constructed. The accuracy of the NFD scheme is compared with the Rusanov–Burstein–Mirin (RBM) and alternative weighted essentially non-oscillatory (A-WENO) schemes as applied to the shallow water equations for a special Cauchy problem with smooth periodic initial data and with an exact solution involving shock waves that arise in the computational domain due to gradient catastrophes. It is shown that the NFD scheme is much more accurate than the third-order RBM scheme in smooth parts of the approximated solution outside the shock influence areas and is more accurate than the A-WENO scheme (which is of the fifth order in space and third order in time) on rather coarse numerical grids; on finer grids, the NFD and A-WENO schemes have nearly identical accuracy in these parts of the computed solution. In shock influence areas, where the RBM scheme becomes much more accurate than A-WENO, the NFD scheme has higher accuracy than RBM.