A General Approach for Producing Hamiltonian Numerical Schemes for Fluid Equations
نویسنده
چکیده
Given a fluid equation with reduced Lagrangian l which is a functional of velocity 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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