Radial Basis Function ENO and WENO Finite Difference Methods Based on the Optimization of Shape Parameters

We present adaptive finite difference ENO/WENO methods with infinitely smooth radial basis functions (RBFs). These methods slightly perturb the polynomial reconstruction coefficients with RBFs as the reconstruction basis and enhance accuracy in the smooth region by locally optimizing the shape parameters. Compared to the classical ENO/WENO methods, the RBF-ENO/WENO methods provide more accurate reconstructions and sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in the existing classical ENO/WENO code. The numerical results in 1D and 2D presented in this paper show that the proposed finite difference RBF-ENO/WENO methods perform better than the classical ENO/WENO methods.