Dynamic behaviour of multi-component multi-stage systems. Numerical methods for the solution

Abstract

The dynamic behaviour of multistage systems is described by large sets of non-linear first-order differential equations. These are usually linearized in order to obtain a solution, but it is here shown that this in general leads to inconsistency with the physical requirements of the problem. The extreme stability of the physical system confers properties on the equations which cause instability in most of the standard numerical methods of solution, and the consequent error swamps the effects of non-linearity. A step-by-step procedure is proposed which makes use of exponential functions and avoids this instability. The non-linear effects then determine the permissible step lengths, and study of the behaviour of the approximating solution results in a method of correcting for finite step length, which is confirmed by numerical experiments. Step length criteria are discussed, and comparison with the Kutta-Simpson process shows that at least in some cases the step length can be up to 128 times greater for a given accuracy in the results. A review of standard numerical procedures is included for general interest.