AI 中文总结
该研究针对色散浅水矩方程，通过将压力方程重写为类似泊松问题的形式，用投影型分裂格式求解，给出线性模型计算及非线性情况讨论，还引入混合方法并展示了相关非定常数值结果
AI 中文摘要
著名的浅水方程（SWE）用于模拟不可压缩自由表面流，当浅度允许垂直平均时，即垂直效应与水平效应相比可忽略不计。但垂直平均会导致沿垂直轴信息丢失。浅水流动的矩模型虽维度降低但包含垂直速度和压力剖面信息。一类包含非静水压力的此类模型此前已作为色散浅水矩模型（DSM）引入。然而，尚未提出求解非定常方程的方法，主要是因为不清楚如何以无散约束形式计算压力方程。我们将DSM模型的压力方程重写为类似泊松问题的形式，以便用投影型分裂格式求解。对于线性方程，给出广义模型的计算并讨论非线性情况。给出前两个线性模型及其相应非线性对应模型。最后，引入混合有限体积有限差分方法并讨论具有周期性边界和不平底部地形实验的非定常数值结果。
英文摘要
The well-known Shallow Water Equations (SWE) are used for modeling incompressible free-surface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM). However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.
