High Order Compact Splitting Multisymplectic Schemes for 1D Gross-Pitaevskii Equation

Abstract

We construct two high order compact schemes for 1D Gross-Pitaevskii (GP) equation. These schemes possess properties of multi-symplectic integrators, splitting method and high order compact method. It improves greatly computational efficiency of multisymplectic integrators. Firstly, 1D GP equation is reformulated into multisymplectic formulation. Then, it is split into a linear multisymplectic Hamiltonian and a nonlinear Hamiltonian system. The nonlinear sub-problem can be solved exactly based on new pointwise mass conservation law. The linear problem is discretized by high order compact multi-symplectic integrator. With different composition of the two sub-problems, we obtain two numerical schemes. These schemes have characters of multisymplectic integrators, splitting method and high order compact schemes, and they are mass-preserving as well. Numerical results are reported to illustrate performance of our methods.