Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes (Communications in Computational Physics, 2020)
In this paper, we introduce a new type of troubled-cell indicator to improvehybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-orderlinear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore,they can reduce computational cost while maintaining non-oscillatory properties innon-smooth regions. Reliable troubled-cell indicators are essential for efficient hybridWENO methods. Most of troubled-cell indicators require proper parameters to detectdiscontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theoremthat does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations areconducted to demonstrate the performance of the proposed indicator. The results withthe proposed troubled-cell indicator are in good agreement with pure WENO schemes.Also the new indicator has advantages in the computational cost compared with theother indicators.
Keywords: WENO approximation; up-wind linear approximation; troubled-cell indicator; hyperbolic conservation laws; hybrid schemes.