A finite difference Hermite RBF-WENO scheme for hyperbolic conservation laws

One of the best ideas for controlling the Gibbs oscillations in hyperbolic conservation laws is to apply weighted essentially non-oscillatory (WENO) schemes. The traditional WENO and Hermite WENO (HWENO) schemes are based on the polynomial interpolation. In this research, a non-polynomial HENO and HWENO finite difference scheme in order to enhance the local accuracy and convergence is proposed. The idea of the reconstruction in HWENO schemes is derived from traditional WENO schemes, with the difference that in traditional WENO schemes only function values are evolved and used, while in HWENO schemes both the values of the function and the first derivative of the function are evolved in time and used. The Hermite radial basis functions (RBFs) interpolation offers the flexibility to control the local error by optimizing the free parameter. The numerical results demonstrate that the developed Hermite RBF-ENO and RBF-WENO schemes enhance the local accuracy and give sharper solution profile than the Hermite ENO and WENO schemes based on the polynomial interpolation.