Entropy Stable High Order SBP for Finite Domain Compressible Flows

Mark Carpenter
NASA Langley Research Center
Hampton, Virginia, USA


High order methods often exhibit unstable behavior when simulating underresolved gradients or shocks. Summation-by-parts finite difference operators applied in an energy stable fashion have been used to overcome some types of instability, but the energy analysis relies on a linearization of the governing equations. This type of analysis is not appropriate when discontinuities are admitted in the solution. Using Burgers equation as a model, a nonlinear entropy analysis has been used to construct entropy stable WENO finite difference operators on bounded domains. These operators are provably stable even for discontinuous solutions. This methodology is extended to the Euler and Navier Stokes equations. New entropy stable WENO finite differences have been constructed along with narrow stencil, high order entropy stable viscous terms. The new schemes are applied to various structured multiblock configurations and are shown effectively simulate unsteady flows in moderately complex configurations that exhibit significant vortical and/or shock structures. Additionally, for smooth problems these methods do not exhibit a degraded order of accuracy on generalized curvilinear grids.