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

For discontinuous solutions of hyperbolic conservation laws, a Hybrid scheme, based on the high order nonlinear characteristic-wise weighted essentially non-oscillatory conservative finite difference (WENO) scheme and the high resolution spectral-like linear compact finite difference (Compact) scheme, is developed for capturing shock 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 partial sum and its derivative are investigated for its applicability as a shock detector due to its unique property, namely, the conjugate Fourier partial sum converges to the location and strength of an isolated jump. For a non-periodic problem, the data is first evenly extended before the derivative of the conjugate Fourier partial sum and its mean are computed. The mean allows one to partition the domain into subdomains containing strong gradients or smooth solutions. The locations of shocks are then accurately identified and flagged for special treatment using the WENO scheme. The matrix-matrix multiply (MXM), Even-Odd decomposition (EOD) and Cosine/Sine fast transform (CFT) algorithms of the conjugate Fourier (cF) analysis are derived, and their advantages and disadvantages in their implementations, usage and technical issues are discussed in detail. The conjugate Fourier shock detector and its iterative version for detecting jumps of large difference in scales are presented. The Hybrid-cF scheme is applied to 1D shock-density wave interaction problem, 2D Riemann IVP problems, and 2D Mach 10 double Mach reflection problems. The preliminary results are in good agreement with those obtained in the literature. They demonstrate the spatial and temporal adaptivity of the Hybrid-cF scheme for problems containing strong shocks, multiple developing shocklets, and high frequency waves. A speedup of the CPU times with factor up to 2-3 is obtained showing the potential efficiency of the Hybrid-cF scheme over the pure WENO-Z scheme.