An improved WENO method based on Gauss-kriging reconstruction with an optimized hyper-parameter

An adaptive finite-difference WENO method with Gauss-kriging reconstruction (we call it WENO-K) is proposed to reduce dissipation in smooth regions of flow while preserving high-resolution around discontinuities for hyperbolic system of conservation laws. The method adopts a kriging model with non-polynomial Gauss exponential function to obtain new reconstruction coefficients that contain a hyper-parameter. By adaptively optimizing the hyper-parameter and automatically identifying troubled cells using newly developed indicators, the accuracy in the smooth region is obviously improved. Compared with the classical WENO-JS method, the proposed WENO-K method provides more accurate reconstructions and sharper solution profiles near discontinuities. Furthermore, the WENO-K method is easy to implement in an existing classical WENO code with less than 13%–16% of additional computational cost. Numerical results demonstrate that the proposed method outperforms the WENO-JS method for a broad range of problems. This method is supposed to be applied to other variants of WENO scheme and offers the potential of improving their accuracy.