Poincar integral invariant and momentum conservation in symplectic time integration of Hamiltonian partial differential equations

Sasa, Narimasa  

The momentum conservation law is investigated in numerical time evolutions of discretized Hamiltonian partial differential equations using symplectic integrators. Our investigation is based on the equivalence between the total momentum and the Poincare integral invariant of the system following our previous work. By introducing a Fourier interpolation method that is a canonical transformation, the total momentum of the system is shown to be an exact conserved quantity if the Hamiltonian possesses a discrete translational invariance and aliasing errors do not appear in the time evolutions. In addition, we perform numerical simulations that demonstrate the validity of our theoretical results under several conditions. Consequently, all symplectic integrators for discretized Hamiltonian partial differential equations are shown to inherit the property of the simultaneous conservation of the total momentum and approximated energy.