A systematic methodology for constructing high-order energy stable WENO schemes

A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme developed by the authors of this paper [N.K Yamaleev, M.H. Carpenter, Third-order energy stable WENO scheme, J. Comput. Phys. 228 (2009) 3025-3047] was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables "energy stable" modifications for existing WENO schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO schemes including one sixth-order central scheme; ESWENO schemes up to eighth order are presented in the Appendix. We also develop new weight functions and derive constraints on their parameters, which provide consistency, much faster convergence of the high-order ESWENO schemes to their underlying linear schemes for smooth solutions with arbitrary number of vanishing derivatives, and better resolution near strong discontinuities than the conventional counterparts. (C) 2009 Elsevier Inc. All rights reserved.