High Order Positivity- and Bound-Preserving Hybrid Compact-WENO Finite Difference Scheme for the Compressible Euler Equations

Abstract

Based on the same hybridization framework of Don et al. (SIAM J Sci Comput 38:A691–A711 2016), an improved hybrid scheme employing the nonlinear 5th-order characteristic-wise WENO-Z5 finite difference scheme for capturing high gradients and discontinuities in an essentially non-oscillatory manner and the linear 5th-order conservative compact upwind (CUW5) scheme for resolving the fine scale structures in the smooth regions of the solution in an efficient and accurate manner is developed. By replacing the 6th-order non-dissipative compact central scheme (CCD6) with the CUW5 scheme, which has a build-in dissipation, there is no need to employ an extra high order smoothing procedure to mitigate any numerical oscillations that might appear in an hybrid scheme. The high order multi-resolution algorithm of Harten is employed to detect the smoothness of the solution. To handle the problems with extreme conditions, such as high pressure and density ratios and near vacuum states, and detonation diffraction problems, we design a positivity- and bound-preserving limiter by extending the one developed in Hu et al. (J Comput Phys 242, 2013) for solving the high Mach number jet flows, detonation diffraction problems and detonation passing multiple obstacles problems. Extensive one- and two-dimensional shocked flow problems demonstrate that the new hybrid scheme is less dispersive and less dissipative, and allows a potential speedup up to a factor of more than one and half times faster than the WENO-Z5 scheme.