An algorithm for the computer simulation of very large dynamic systems

Abstract

An algorithm for integrating high dimensional stiff nonlinear differential equations of the type x ̇ = Ax+f(x , t)+ Bu(t) , x( t 0)=x 0, where u( t) is a specified time function, f(x, t) is a nonlinear function with a small Lipschitz constant and A is a matrix whose eigenvalues are widely distributed is given. The proposed algorithm has a truncation error of 0( h 5) where h is the step-size, is numerically stable for any h provided the original system is stable and the Lipschitz constant is small enough, will give exact steady-state solutions for constant input systems for any h, and is especially suited to those systems in which the order n of the system is large, for example, n ⪢ 10. Some numerical examples varying from 10th to 80th order are included and a comparison of the computation time required by the proposed method is made with other algorithms—the Runge-Kutta method, and Gear's method. It is found that the proposed algorithm is approximately 10 times faster than Gear's method for the 80th order example.