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.