高阶单步法求解非线性动力方程的研究

RESEARCH ON HIGH ORDER SINGLE STEP METHOD TO SOLVE NONLINEAR DYNAMIC EQUATIONS

应用高阶单步法来求解非线性动力方程Z(t)=G(Z,t)Z(t)+H(t)。根据泰勒展开的思想提出该文预估式,将已有的高阶单步法转为隐式方法,进行多次预估校正即可求解;另一种思路是把非线性部分看作非齐次项处理,在形式上化为线性系统Z(t)=GZ(t)+H(Z,t),推导了新的高阶单步法逐步积分算法,该文方法为隐式方法,同样进行多次预估校正即可求解。数值算例论证了这两种求解思路的有效性。该文扩大了高阶单步法的适用范围。

This paper conducts research on the applicability of high order step method to solve nonlinear dynamic equations Z.（t） = G（Z,t）Z（t） ＋ H（t）. Based on the Taylor series expansion, the prediction formulas are proposed, making the high order step method to an implicit method. Combining the implicit method and the predict-correct method, the nonlinear dynamic equations can be solved. Another way to solve the nonlinear dynamic equations is to transform the presented equations to the equations Z,（t） = GZ（t） ＋ H（Z, t）, from which a new implicit high order step method is derived. The nonlinear dynamic equations can be solved by the presented method. Numerical examples demonstrate the effectiveness of these two solutions. This paper extends the application scope of the high order single step method.