Third-order Energy Stable WENO scheme

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

“…The nearly skew-symmetric matrix, Q, is an undivided differencing operator where all rows sum to zero and the first and last column sum to −1 and 1, respectively. The difference operator, D, approximates the first derivative as 28) where T p 2p p is the truncation error of the approximation. The nomenclature 2p refers to the interior accuracy and p refers to the accuracy at the left and right boundaries.…”

“…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.…”

“…Then, spatial operators are derived that enable these continuous properties to be mimicked, which is shown through the semi-discrete entropy analysis. The entropy-entropy flux pair for the Navier-Stokes equations is 28) and the potential-potential flux pair is…”

“…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

“…[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.…”