Hybrid Compact-WENO Finite Difference Scheme with Radial Basis Function Based Shock Detection Method for Hyperbolic Conservation Laws

Abstract

A hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the spectral-like linear compact finite difference scheme, has been developed for capturing shocks and strong gradients accurately and resolving fine scale structures efficiently for hyperbolic conservation laws. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm which is capable of determining the smoothness of the solution at any given grid point and time. An improved iterative adaptive multiquadric radial basis function (IAMQ-RBF-Fast) method [W. S. Don, B. S. Wang, and Z. Gao, J. Sci. Comput, 75 (2018), pp. 1016--1039], which employed the $O(N^2)$ recursive Levinson--Durbin method and the Sherman--Morrison--Woodbury method for solving the perturbed Toeplitz matrix system, has been successfully developed as an efficient and accurate edge detector of the piecewise smooth functions. In this study, the method, together with Tukey's boxplot method and the domain segmentation technique, is extended to serve as a novel shock detection algorithm for solving the Euler equations. The applicability and performance of the RBF edge detection method as the shock detector in the hybrid scheme in terms of accuracy, robustness, efficiency, resolution, and other implementation issues are given. Several one- and two-dimensional benchmark problems in shocked flow demonstrate that the proposed hybrid scheme can reach a speedup of the CPU times by a factor up to 2--3 compared with the pure fifth order WENO-Z scheme.