High Order Hybrid Weighted Compact Nonlinear Schemes for Hyperbolic Conservation Laws

High order weighted compact nonlinear scheme (WCNS) has become an alternative method of finite difference weighted essentially non-oscillatory (WENO) scheme in many different research areas due to its better spectral properties. However, its heavy computational time even more expensive than the classical WENO scheme is still a bottleneck problem. To relieve it in a sense, a framework of high order hybrid WCNS (HWCNS) combining the weighted nonlinear interpolations proposed in [Deng et al., JCP, 165] or [Zhang et al., JCP, 227] in the non-smooth stencils with corresponding linear compact interpolations in the smooth stencils respectively is designed for solving the hyperbolic conservation laws in this work. A newly developed high order shock detector based on the radial basis function, which can capture the locations of shocks and high gradients accurately and sharply, is used to measure the smoothness of the solution at each grid point. The HWCNS demonstrates higher resolution, fewer dissipation/dispersion errors, less computational time in the extensive one- and two-dimensional classical examples by comparing with the WCNS.