The velicity formulation of the Euler equation and the symplectic integration

In this paper we study Buttke's velicity reformulation of the incompressible Euler equation, and the symplectic integration of the Hamiltonian system obtained by discretizing the velicity equation.;We show that the linearized velicity equation is hyperbolic only in the weak sense and we analyze the characteristics of the linearized velicity equation for the variable coefficient case.;We will briefly describe the symplectic schemes; one is the implicit midpoint scheme, and the other one is a fourth-order implicit scheme. We apply the implicit midpoint scheme (IM) to our Hamiltonian system for large time, with the fixed time step (;Our work builds on recent progress of the Hamiltonian formulation of the incompressible Euler equation and the Lagrangian numerical methods for incompressible Euler's equation, particularly the work of Buttke.