A General Approach for Producing Hamiltonian Numerical Schemes for Fluid Equations

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-Poincar\'e 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.