Taylor Remainder-Based Discontinuity Detection Method with Application to Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws

Abstract

In this study, the Taylor Remainder-based Discontinuity Detection Method (TRDDM) is designed to identify shocks in numerical solutions of hyperbolic conservation laws. The TRDDM leverages the fact that Taylor remainder is minimal in smooth regions but significantly large at discontinuities. This method reduces the need for threshold parameters, making it less complex. Pre-processing the target dataset involves normalization, detection of constants, and identification of high-frequency waves. After pre-processing, Taylor remainder is calculated for the remaining dataset that contains discontinuous patterns. An appropriate partition point separates the error components, with larger errors indicating discontinuities. TRDDM analyzes local features of numerical solutions and is capable of capturing even minor discontinuities. The accuracy and efficiency of the hybrid compact-WENO scheme with TRDDM are demonstrated through classical one- and two-dimensional shocked problems.