流体方程哈密顿数值格式的通用构造方法
A General Approach for Producing Hamiltonian Numerical Schemes for Fluid Equations
AI总结：
提出一种基于拉格朗日粒子位置和动量的通用半离散化方法，将流体方程转化为正则哈密顿系统，并通过辛积分器进行时间积分，适用于浅水方程、EP-Diff方程和三维可压缩欧拉方程等。
摘要：
给定一个具有约化拉格朗日量 $l$ 的流体方程，其中 $l$ 是欧拉坐标下速度 $\MM{u}$ 和平流密度 $D$ 的泛函，我们给出一种通用的半离散化方法，将其转化为正则哈密顿系统；该系统随后可使用辛积分器进行时间积分。该方法是拉格朗日型的，变量为一组拉格朗日粒子位置及其相关动量。所得的正则方程在投影到网格上时，给出了 $l$ 的欧拉-庞加莱方程的离散形式，并采用一种新的离散微积分来表示梯度算子和散度算子。针对一大类方程（包括浅水方程、EP-Diff方程和三维可压缩欧拉方程），我们提出了实用的辛时间积分器，并通过EP-Diff方程的数值实验展示了该技术。
英文摘要：
Given a fluid equation with reduced Lagrangian $l$ which is a functional of velocity $\MM{u}$ and advected density $D$ given in Eulerian coordinates, we give a general method for semidiscretising the equations to give a canonical Hamiltonian system; this system may then be integrated in time using a symplectic integrator. The method is Lagrangian, with the variables being a set of Lagrangian particle positions and their associated momenta. The canonical equations obtained yield a discrete form of Euler-Poincaré equations for $l$ when projected onto the grid, with a new form of discrete calculus to represent the gradient and divergence operators. Practical symplectic time integrators are suggested for a large family of equations which include the shallow-water equations, the EP-Diff equations and the 3D compressible Euler equations, and we illustrate the technique by showing results from a numerical experiment for the EP-Diff equations.
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