A high-resolution hybrid scheme for hyperbolic conservation laws

SummaryA new hybrid scheme is proposed, which combines the improved third‐order weighted essentially non‐oscillatory (WENO) scheme presented in this paper with a fourth‐order central scheme by a novel switch. Two major steps have been gone through for the construction of a high‐performance and stable hybrid scheme. Firstly, to enhance the WENO part of the hybrid scheme, a new reference smoothness indicator has been devised, which, combined with the nonlinear weighting procedure of WENO‐Z, can drive the third‐order WENO toward the optimal linear scheme faster. Secondly, to improve the hybridization with the central scheme, a hyperbolic tangent hybridization switch and its efficient polynomial counterpart are devised, with which we are able to fix the threshold value introduced by the hybridization. The new hybrid scheme is thus formulated, and a set of benchmark problems have been tested to verify the performance enhancement. Numerical results demonstrate that the new hybrid scheme achieves excellent performance in resolving complex flow features, even compared with the fifth‐order classical WENO scheme and WENO‐Z scheme. Copyright © 2015 John Wiley & Sons, Ltd. The high‐resolution hybrid scheme HWENO‐N3 has been proposed in this article, accompanied by its WENO counterpart WENO‐N3. It is demonstrated by benchmark numerical tests that HWENO‐N3 outperforms WENO‐JS5 and WENO‐Z5 in resolving the complicated flow structures involved with discontinuities, even employing less stencil points.