WENO smoothness indicator based troubled-cell indicator for hyperbolic conservation laws

Abstract

Hybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve low-cost high-order accurate schemes in smooth regions and non-oscillatory shock-capturing schemes in the vicinity of discontinuities. Troubled-cell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubled-cell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubled-cell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of high-order accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENO-Z reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENO-Z scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with non-convex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENO-Z schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENO-Z schemes while retaining the advantages of WENO-Z schemes.