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：高阶格式在处理非线性守恒律中的激波间断或未充分分辨的解特征时易出现不稳定性。熵稳定格式通过确保物理相关解满足与离散参数无关的半离散熵不等式来解决这一不稳定性问题。本研究将高阶熵稳定格式推广至准一维浅水方程与准一维可压缩欧拉方程，这两类方程分别描述了流经变宽度渠道或喷管的一维流动。针对两组准一维方程，我们引入了一种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形式。当与熵稳定界面通量相结合时，所建立的格式具有高阶精度、守恒性及半离散熵稳定性。对于准一维浅水方程，所得格式同时具备平衡态保持特性。