Abstract: A symplectic schemes with high order accuracy is proposed for solving the high order schrödinger type equation (əu)/(ət)=1(-1)^m(ə^2mu)/(əx^2m) via the third type of generating function method. At first, the equation is written into the canonical Hamilton system; secondly, overcoming successfully the essential difficulty on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type of generating function method. Furthermore the discretization of the related modified equation of original equation are obtained. Finally, arbitrary order accuracy symplectic scheme is obtained. Numerical results are also presented to show the effectiveness of the scheme and its high order accuracy and properties of excellent long-time numerical behavior.

Key words: variation, high order accuracy symplectic scheme, high order schrödinger type equation