Finite difference modified WENO schemes for hyperbolic conservation laws with non‐convex flux

Abstract

AbstractThe Weighted Essentially Non‐Oscillatory (WENO) reconstruction provides higher‐order accurate solutions to hyperbolic conservation laws for convex flux. But it fails to capture composite structure in the case of non‐convex flux and converges to the wrong solution (Qiu & Shu SIAM J Sci Comput. 2008;31:584‐607). In this article, we have developed a Modified WENO (MWENO) scheme in the finite difference framework, which can resolve the composite structure and ensures the entropic convergence. The MWENO reconstruction procedure involves the identification of the troubled‐cells, followed by the use of first‐order monotone modification in the troubled‐cells and employ the fifth‐order WENO reconstruction in the non‐troubled cells. A new troubled‐cell indicator is developed using the information of the smoothness indicator of the WENO reconstruction. Numerical experiments are performed for 1D and 2D test cases, which ensure the entropic convergence of the proposed schemes.