Meng-Zhao Qin

Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation

The Hamiltonian and the multi-symplectic formulations of the nonlinear Schrodinger equation are considered. For the multi-symplectic formulation, a new six-point difference scheme which is equivalent to the multi-symplectic Preissman integrator is derived. Numerical experiments are also reported

Construction of symplectic schemes for wave equations via hyperbolic functions sinh(x), cosh(x) and tanh(x)

Abstract Hamiltonian systems are canonical systems on phase space endowed with symplectic structures. The dynamical evolutions, i.e., the phase flow of the Hamiltonian systems are symplectic transformations which are area-preserving. The importance of the Hamiltonian systems and their special property require the numerical algorithms for them should preserve as much as possible the relevant symplectic properties of the original systems. Feng Kang [1–3] proposed in 1984 a new approach to computing Hamiltonian systems from the view point of symplectic geometry. He systematically described the general method for constructing symplectic schemes with any order accuracy via generating functions. A generalization of the above theory and methods for canonical Hamiltonian equations in infinite dimension can be found in [4]. Using self-adjoint schemes, we can construct schemes of arbitrary even order [5]. These schemes can be applied to wave equation [6,7] and the stability of them can be seen in [7,8]. In this paper, we will use the hyperbolic functions sinh( x ), cosh( x ) and tanh( x ) to construct symplectic schemes of arbitrary order for wave equations and stabilities of these constructed schemes are also discussed.

Multisymplectic Geometry and Multisymplectic Scheme for the Nonlinear Klein Gordon Equation

The multisymplectic structure of the Nonlinear Klein Gordon equation is presented directly from the variational principle. In the numerical aspect, we give a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissmann scheme. A series of numerical results are reported to show the effectiveness of the scheme.

Volume-Preserving Schemes and Numerical Experiments☆

Abstract In this paper, we discuss the conditions for Euler midpoint rule to be volume-preserving and present explicit volume preserving schemes. Some numerical experiments are done to test these schemes.

Canonical Runge-Kutta-Nyström (RKN) methods for second order ordinary differential equations

Abstract In this paper, we combine the order conditions with the canonical conditions to get the “canonical order conditions” for RKN methods, and then we use these conditions to get two three-stage canonical RKN methods of order 4.

Lie-Poisson integration for rigid body dynamics☆

Abstract In this paper, the splitting midpoint rule is presented and proved to be the Lie-Poisson integrators to the rigid body systems. Further discussions are also given. Numerical experiments show that this method has well properties comparing with the Runge Kutta method and ordinary midpoint rule.

Poisson Schemes for Hamiltonian Systems on Poisson Manifolds

Abstract In this paper, we construct Poisson difference schemes of any order accuracy based on Pade approximation for linear Hamiltonian systems on Poisson manifolds with constant coefficients. For nonlinear Hamiltonian systems on Poisson manifolds, we point out that symplectic diagonal implicit Runge-Kutta methods are also Poisson schemes. The preservation of distinguished functions and quadratic first integrals of the original Hamiltonian systems of these schemes are also discussed.

Explicit symplectic schemes to solve vortex systems

Abstract Here we propose a family of explicit symplectic schemes for vortex systems, which are corresponding to the symmetry of these systems.

A note for Lie-Poisson Hamilton-Jacobi equation and Lie-Poisson integrator

Abstract In this paper, a clear Lie-Poisson Hamilton-Jacobi theory is presented. How to construct a Lie-Poisson integrator by generating function methods is also given, which is different from the Ge-Marsden methods [1]. An example on a rigid body has been given to illustrate this point.

Order conditions of two kinds of canonical difference schemes

Abstract The main purpose of this paper is to develop and simplify the general conditions for an s -stage explicit canonical difference scheme to be of q th order while the simplified order conditions for canonical RKN methods, which are applied to special kinds of second order ordinary differential equations, are also obtained here.