四级四阶对角隐式辛Runge-Kutta方法参数计算

ON COMPUTE OF PARAMETERS FOR 4-STAGE 4-ORDER DIAGONALLY IMPLICIT SYMPLECTIC RUNGE-KUTTA METHODS

1.引 言设有Hamilton系统这坐 H（p1,…,pn,q1,…,qn）是 Hamilton函数,它和t无关.记z=（p1,…;Pn,q1,…,qn）T和Hamilton系统（1）的右端项为f（z）,则（1）可表示为dz/dt=f（z）. 冯康用辛几何的观点提出了计算Hamilton系统的辛差分格式[1].Runge-Kutta方法是求非线性常微分方程（组）数值解的重要单步方法.若能找到具有辛性的Runge-Kutta方法,对于求解非线性 Hamilton系统数值解将具有非常重要的意义.J.M.

A nonlinear system with 3 equations and 3 unknowns was got by using sym-plectic conditions to reduce the system with 8 equations and 4 unknowns, which the coefficients of 4-stage and 4-order diagonally implicit symplectic Runge-Kutta methods must satisfy. An optimal problem was constructed from the nonlinear system. We investigated on the minimum points of the optimal problem and obtained 9 approximate of them. The 9 computational solutions are obtaind respectively, when Broyden-Flecher-Shanno quasi-Newton methods for solve nonlinear equations was used. These solutions can be regarded as the coefficients of fourth-stage and fourth-order diagonally implicit Runge-Kutta methods respectively.