We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics charge transfer by intrinsic localized modes in nonlinear crystal lattice models. consider, without loss generality, one-dimensional models described classical dynamics, whereas the (electron or hole) is modeled as a quantum particle within tight-binding approximation. Canonica...

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects: conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes tn for mid-long time and demonstrate the symp...

Time-symmetric integration schemes share with symplectic schemes the property that their energy errors show a much better behavior than is the case for generic integration schemes. Allowing adaptive time steps typically leads to a loss of symplecticity. In contrast, time symmetry can be easily maintained, at least for a continuous choice of time step size. In large-scale N-body simulations, how...

A reduction method is presented for systems of conservation laws with boundary energy flow. It is stated as a generalized pseudo-spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port-variables. The symplecticity of this spatial reduction method is proved when used for the reduction of...

Discrete variational mechanics: A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator: A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such ...

The Crank–Nicolson scheme as well as its modified schemes is widely used in numerical simulations for the nonlinear Schrödinger equation. In this paper, we prove the multisymplecticity and symplecticity of this scheme. Firstly, we reconstruct the scheme by the concatenating method and present the corresponding discrete multisymplectic conservation law. Based on the discrete variational principl...

Conformal symplecticity is generalized to forced-damped multi-symplectic PDEs in 1+1 dimensions. Since a conformal multi-symplectic property has a concise form for these equations, numerical algorithms that preserve this property, from a modified equations point of view, are available. In effect, the modified equations for standard multi-symplectic methods and for space-time splitting methods s...