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

ハミルトン系偏微分方程式に対するシンプレクティック数値積分におけるポアンカレ積分不変量と運動量保存

佐々 成正  

Sasa, Narimasa

ハミルトン系偏微分方程式の時間発展問題を考える。特に元の偏微分方程式系で運動量保存則が成り立っている時、時間発展手法にシンプレクティック数値積分法を用いると運動量保存則が数値計算においても保存されることを一般的に証明した。この証明には、ハミルトン系の相空間における保存則を満たすポアンカレ不変量(相空間におけるLouvilleの定理の最低次元版)を用いた。

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.