基于边界值方法的微分动力系统数值计算方法

Numerical computation of differential dynamic systems using boundary value methods

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。

For the high dimensional nonlinear initial value problem,the differential quadrature method( DQM)can be used to solve a higher dimensional nonlinear equations in the integration process of each step,so its computation workload is huge. Based on the relationship between DQM and the boundary value methods,the generalized backward difference formulae( GBDF) and the extended implicit trapezoidal rules of the second kind(ETR2) can be regarded as the sparse representation of classical DQM. In this paper,the GBDF methods and ETR2 are applied to the numerical solution of the differential dynamic systems,and a new numerical method is proposed. Theoretical analysis and numerical examples show that,the proposed numerical method has higher computational efficiency than classical DQM for the numerical solution of the nonlinear differential initial value problem with high dimensions.