High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.

翻译：高阶格式在存在激波间断或分辨率不足的解特征时，对非线性守恒律而言已知是不稳定的。熵稳定格式通过确保物理相关解独立于离散化参数满足半离散熵不等式来消除这种不稳定性。本文将高阶熵稳定格式推广至准一维浅水方程和准一维可压缩欧拉方程，这些方程描述了通过变截面渠道或喷嘴的一维流动。我们针对两组准一维方程引入了新的非对称熵守恒有限体积通量，并将熵守恒条件推广至非对称通量。当与熵稳定界面通量结合时，所得格式具有高阶精度、守恒性及半离散熵稳定性。对于准一维浅水方程，该格式同时具有平衡保持特性。