Abstract
This paper describes a computation method for time-responses of stiff, nonlinear ordinary differential equations, which are frequently adopted as models of real systems but are difficult to solve by numerical methods. Applying explicit methods to these equations does not always yield stable solutions because of the numerical instability induced by the system stiffness. On the other hand, application of implicit methods makes the computational algorithm very complicated and does not always assure numerical stability.
The new method presented here matches well with equations as such. In it, the stiff, nonlinear equation X=g(X, F) is represented as x=Ax+Bf+ue by separating g(X, F) into the stiff linear term Ax+Bf and the nonlinear term ue. The approximate solutions x0k+1 of xk+1 is obtained by applying an implicit method to the equation x=Ax+Bf+ue during the interval [kτ, (k+1)τ], where ue is at first regarded as a constant value uek. Then an implicit method is iteratively applied to compensate for x0k+1. Here, the approximate value of ue can be estimated by using the approximate solution of xk+1 at each iteration of the compensation process. The proposed iterative method solves equations of a broad stepwidth with high speed.
Finally, this paper evaluates the numerical performance, attesting the performance by computing the time-responses of a chemical process and a mass-damper nonlinear stiffness system.
