Boundary Closures For Fourth-order Energy Stable Weighted Essentially Non-Oscillatory Finite Difference Schemes

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

“…The details of the generally applicable correction procedure are detailed below. The full implementation details including the WENO stencil biasing algorithm throughout the domain are available in Fisher et al [10] and Carpenter et al [30] for (2-4-2) and (3-6-3) operators, respectively. 1 The first step to construct a WENO finite difference operator is to cast the difference operator in flux form, Qf = ∆f .…”

“…All WENO schemes including those used herein [10,28,29], incorporate elaborate stencil biasing mechanics to achieve nearly monotone solutions in the vicinity of shocks and other discontinuities. It is critically important that the additional dissipation provided by the entropy stabilization terms does not contaminate the desirable attributes of the baseline WENO scheme.…”

“…A linearlystable WENO scheme, termed ESWENO, was developed in Fisher et al [10] with full SBP boundary closures. This scheme is constructed such that when the WENO operator is cast in flux form,…”

“…There are a variety of mechanisms to achieve this, but in this work WENO finite difference methods are used. The implementation uses unique formal boundary closures from Fisher et al [10] that satisfy the SBP condition. Stencil biasing mechanics follow two papers by Yamaleev and Carpenter [28,29].…”

“…We also show how schemes developed using linear stability can fail in the presence of a shock by comparing the newly developed entropy stable schemes with the energy stable WENO scheme of Fisher et al [10].…”