On Time-Discretization of the 2d Euler Equation by a Symplectic Crouch-Grossman Integrator
This work provides a rigorous numerical analysis for a symplectic integrator applied to the 2D Euler equation, which is of interest to computational fluid dynamics researchers.
The authors propose a symplectic Crouch-Grossman integrator for time-discretizing the 2D Euler equation in vorticity form, proving first-order convergence with error estimates in Sobolev norms.
We consider time discretizations of the two-dimensional Euler equation written in vorticity form. The discretization method uses a Crouch-Grossman integrator that proceeds in two stages: first freezing the velocity vector field at the beginning of the time step, and then solve the resulting elementary transport equation by using a symplectic integrator to discretize in time the flow of the associated Hamiltonian differential equation. We prove that these schemes yield order one approximations of the exact solutions, and provide error estimates in Sobolev norms.