Enhanced Robustness of the Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws with Multi-resolution Analysis and Tukey’s Boxplot Method

The high order shock detection algorithm employing the high order multi-resolution (MR) analysis (Harten in J Comput Phys 49:357–393, 1983) for identifying the smooth and non-smooth stencils has been employed extensively in the hybrid schemes, such as the hybrid compact-weighted essentially non-oscillatory finite difference scheme, for solving hyperbolic conservation laws containing both discontinuous and complex fine scale structures. However, the accuracy and efficiency of the hybrid scheme depend heavily on the setting of a user defined ad hoc and problem dependent parameter, namely, MR tolerance ϵMR$$\epsilon _{\scriptscriptstyle {MR}}$$. In this study, we improve the outlier-detection algorithm based on the Tukey’s boxplot method as proposed by Vuik et al. (SIAM J Sci Comput 38:A84–A104, 2016) for solving the hyperbolic conservation laws with the hybrid scheme. By incorporating the global mean of the data in the definition of fences in each segmented subdomain, spurious false positive of the discontinuities can be eliminated. The improved outlier-detection algorithm essentially removes the need of specifying the MR Tolerance ϵMR$$\epsilon _{\scriptscriptstyle {MR}}$$, and thus greatly improves the robustness of the hybrid scheme. The accuracy, efficiency and robustness of the improved hybrid scheme with the shock detection algorithm using the high order MR analysis and the improved outlier-detection algorithm are demonstrated with numerous classical one- and two-dimensional examples of the shallow water equations and the Euler equations with discontinuous solutions.