Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws

Abstract

For discontinuous solutions of hyperbolic conservation laws, a hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the high resolution spectral-like linear compact finite difference (Compact) scheme, is developed for capturing shocks and strong gradients accurately and resolving smooth scale structures efficiently. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm that is capable of determining the smoothness of the solution at any given grid point. The conjugate Fourier (cF) partial sum and its derivative are investigated for its applicability as a shock detector due to its unique property, namely, the cF partial sum converges to the location and strength of an isolated jump. For a nonperiodic problem, the data are first evenly extended before the derivative of the cF partial sum and its mean are computed. The mean allows one to par...