A new adaptive weighted essentially non-oscillatory WENO-θ scheme for hyperbolic conservation laws

Abstract

A new adaptive weighted essentially non-oscillatory WENO-θ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter θ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator τθ measuring the smoothness of the large stencil is chosen among two candidates which are devised based on the possible highest-order variations of the reconstruction polynomials in L2 sense. In addition, a new set of smoothness indicators β̃k of the substencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around the point xj. Numerical results show that the new scheme outperforms other comparing 6th-order WENO schemes in terms of improving the resolution at critical regions of nonsmooth problems as well as maintaining symmetry in the solutions.