High Order Entropy Stable Schemes for the Quasi-One-dimensional Shallow Water and Compressible Euler Equations.

High order schemes are known to be unstable in the presence of shockdiscontinuities or under-resolved solution features for nonlinear conservationlaws. Entropy stable schemes address this instability by ensuring thatphysically relevant solutions satisfy a semi-discrete entropy inequalityindependently of discretization parameters. This work extends high orderentropy stable schemes to the quasi-1D shallow water equations and the quasi-1Dcompressible Euler equations, which model one-dimensional flows throughchannels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes forboth sets of quasi-1D equations, as well as a generalization of the entropyconservation condition to non-symmetric fluxes. When combined with an entropystable interface flux, the resulting schemes are high order accurate,conservative, and semi-discretely entropy stable. For the quasi-1D shallowwater equations, the resulting schemes are also well-balanced.