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.

A shock-pinned random projection method( SRPM) is proposed. To overcome difficulty of multidirectional propagation,local random projection is applied by introducing a shock indicator along with proper rules for choosing local projection zones around shocks in each reaction step. Numerical experiments are presented. It shows that SRPM is appropriate for detonation waves with multi-directional propagation,which is high-efficient in solving stiff reactive equations and extensible to 3D situations.

A split--step muhisympleetic scheme is proposed for a kind of fourth-order Schrödinger equations with eubie nonlinear term.Ihe basic idea is to combine muhisymplectic integrator with split-step method.The method not only preserves muhisymplectic structure of multisympleetie integrators,but also has the virtue of efficiency and flexibility of split-step method in computation.Numerical experiments show that the split-step muhisympleetic method is more economic in computational time and computer memory than traditional muhisymplectic integrator.

Symplectic Fourier pseudo-spectral integrators for Klein-Gordon-Schrödinger equations(KGS) are investigated.A Hamiltonian formulation is presented.Fourier pseudo-spectral discretization is applied to the space approximation which leads to a finite-dimensional Hamiltonian system.Symplectic integrators,including Störmer/Verlet method and midpoint rule,are adopted in the time direction which leads to symplectic integrators for KGS.It suggests that the Störmer/Verlet method is explicit which can be coded effciently,and the midpoint rule captures mass of the original system exactly.Numerical experiments show that symplectic integrator can simulate various solitary well over a long period.