Modified Stencil Approximations for Fifth-Order Weighted Essentially Non-oscillatory Schemes

In this paper, a modified fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme is presented. The quadratic polynomial approximation of numerical flux on each candidate stencil of the traditional WENO-JS scheme is modified by adding a form of cubic terms such that the resulting stencil approximation achieves fourth-order accuracy. And the corresponding smoothness indicators are calculated. The modified candidate fluxes and local smoothness indicators, when used in the WENO-JS scheme, can make the resulting new scheme (called WENO-MS) achieve fifth-order convergence in smooth regions including first-order critical points. A series of one- and two-dimensional numerical examples are presented to demonstrate the performance of the new scheme. The numerical results show that the proposed WENO-MS scheme provides a comparable or higher resolution of fine structures compared with the WENO-M, WENO-Z and P-WENO schemes, while it increases only 7% of CPU time compared with the traditional WENO-JS scheme.