A 2×799²-order nonlinear Hamiltonian system of two-dimensional non-stationary Sine-Gordon equation is introduced when the five point difference scheme is used to discretize the differential operator L=(ə²)/(əx²)+(ə²)/(əy²) in the rectangle [-a,a]×[-a,a]. An iterative method is designed to solve the nonlinear system, which is formed by using the centered Euler scheme for the Hamiltonian system. The condition and the velocity of convergence for this method are given. Numerical examples for evaluating one-soliton and two-soliton of the Sine-Gordon equation show that the symplectic method is an efficient algorithm.

By using the central difference quotient operator for (∂²)/(∂x²) and the diagonal Padé approximation of exp t, two kinds of symplectic schemes which have accuracy O(Δx²+ Δt^2l) and O(Δx⁴+ Δt^2l), respectively, can be attained for wave partial differential equation. Two iterative methods are described for the linear systems formed from the above schemes. Their conditions of convergence are given for l=1,2,3,4. The numerical experiments demonstrate that the symplectic algorithm have efficiency and both methods are convergent.