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 Yamaleev and Carpenter (AIAA 2008–2876, 2008) 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 upwindbiased ESWENO schemes; ESWENO schemes up to eighth order are presented in the appendix. New weight functions are also developed that provide (1) formal consistency, (2) much faster convergence for smooth solutions with an arbitrary number of vanishing derivatives, and (3) improved resolution near strong discontinuities.