Jialin Hong

Splitting multisymplectic integrators for Maxwell's equations

In the paper, we describe a novel kind of multisymplectic method for three-dimensional (3-D) Maxwells equations. Splitting the 3-D Maxwells equations into three local one-dimensional (LOD) equations, then applying a pair of symplectic Runge-Kutta methods to discretize each resulting LOD equation, it leads to splitting multisymplectic integrators. We say this kind of schemes to be LOD multisymplectic scheme (LOD-MS). The discrete conservation laws, convergence, dispersive relation, dissipation and stability are investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, non-dissipative, and of first order accuracy in time and second order accuracy in space. As a reduction, we also consider the application of LOD-MS to 2-D Maxwells equations. Numerical experiments match the theoretical results well. They illustrate that LOD-MS is not only efficient and simple in coding, but also has almost all the nature of multisymplectic integrators.

Exponential dichotomy and trichotomy for difference equations

Abstract In this paper, a roughness theorem of exponential dichotomy and trichotomy of linear difference equations is proved. It is also shown that if an almost periodic difference equation has an exponential dichotomy on a sufficiently long finite interval, then it has one on (−∞, +∞).

Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations

In this paper, we propose explicit multi-symplectic schemes for Klein-Gordon-Schrodinger equation by concatenating suitable symplectic Runge-Kutta-type methods and symplectic Runge-Kutta-Nystrom-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments.

The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences

Abstract The existence of almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions of differential equations with piecewise constant argument is characterized in terms of almost periodic, asymptotically, and pseudo almost periodic sequences. Thus Meisterss and Opials theorems are extended.

High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations

In this paper, we develop a new kind of multisymplectic integrator for the coupled nonlinear Schrodinger (CNLS) equations. The CNLS equations are cast into multisymplectic formulation. Then it is split into a linear multisymplectic formulation and a nonlinear Hamiltonian system. The space of the linear subproblem is approximated by a high-order compact (HOC) method which is new in multisymplectic context. The nonlinear subproblem is integrated exactly. For splitting and approximation, we utilize an HOC-SMS integrator. Its stability and conservation laws are investigated in theory. Numerical results are presented to demonstrate the accuracy, conservation laws, and to simulate various solitons as well, for the HOC-SMS integrator. They are consistent with our theoretical analysis.

Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients

In this paper, we consider Runge-Kutta-Nystrom (RKN) methods applied to nonlinear Schrodinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nystrom methods in spatial direction and symplectic Runge-Kutta methods in temporal direction for NLSEvc leads to multi-symplectic integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. It is shown that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. We present a (4-order) multi-symplectic RKN method and use it in numerical simulation of quasi-periodically solitary waves for NLSEvc, and we compare the multi-symplectic RKN method with a non-multi-symplectic RKN method on the errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge. The precise conservation of discrete charge under the multi-symplectic RKN discretizations is attested numerically. Some numerical superiorities of the multi-symplectic RKN methods are revealed.

A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients

We describe a novel numerical approach to simulations of nonlinear Schrodinger equations with varying coefficients, based on the discovery of a new and intrinsic conservation law for varying coefficient nonlinear Schrodinger equations. The approach is shown to preserve some crucial classical conservations, such as the spatial ergodicity, and utilized in numerical simulations of periodically and quasi-periodically solitary waves for nonlinear Schrodinger equations with periodic or quasi-periodic coefficients. Some numerical experiments are presented to illustrate the conservative property.

Long-term numerical simulation of the interaction between a neutron field and a neutral meson field by a symplectic-preserving scheme

In this paper, we propose a family of symplectic structure-preserving numerical methods for the coupled Klein–Gordon–Schrodinger (KGS) system. The Hamiltonian formulation is constructed for the KGS. We discretize the Hamiltonian system in space first with a family of canonical difference methods which convert an infinite-dimensional Hamiltonian system into a finite-dimensional one. Next, we discretize the finite-dimensional system in time by a midpoint rule which preserves the symplectic structure of the original system. The conservation laws of the schemes are analyzed in succession, including the charge conservation law and the residual of energy conservation law, etc. We analyze the truncation errors and global errors of the numerical solutions for the schemes to end the theoretical analysis. Extensive numerical tests show the accordance between the theoretical and numerical results.

Predictor-corrector methods for a linear stochastic oscillator with additive noise

The predictor-corrector methods P(EC)^k with equidistant discretization are applied to the numerical integration of a linear stochastic oscillator. Their ability in preserving the symplecticity, the linear growth property of the second moment, and the oscillation property of the solution of this stochastic system is studied. Their mean-square orders of convergence are discussed. Numerical experiments are performed.