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

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

“…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]. The details of the generally applicable correction procedure are detailed below.…”

“…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.…”

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

“…We now develop using mimetic techniques (see Ref. [6] or Ref. [10]) a class of discrete spatial operators for which the discrete energy is bounded from above by the continuous target estimate provided by equations (2) and (3).…”

“…[5] and [6]. The new interior/boundary ESWENO schemes (henceforth referred to as "finite-domain ESWENO") retain all the salient features of the original periodic schemes, including: 1) conservation and L 2 -energy stability for constant coefficient (linear) hyperbolic systems, including those with discontinuous initial or boundary data, 2) design order accuracy throughout the entire domain, especially regions near the boundaries or near smooth extrema, and 3) full stencil biasing mechanics (Left, Central, Right) at all possible points.…”

“…This task is accomplished using the summationby-parts (SBP) [6,9] framework (as was done in the periodic case). The SBP framework requires the construction of accurate discrete schemes that satisfy basic matrix conditions, that if met guarantee discrete stability and conservation.…”