Stiff systems in theory and practice

Stiff systems, Modern Taylor Series Method, Differential equations, Continous system modelling

The words "stiff system" are used frequently in this work as it is the top topic of it. In particular the paper deals with stiff systems of differential equations. To solve this sort of system numerically is a diffult task. In spite of the fact that we come across stiff systems quite often in the common practice, it was real challenge even to find suitable articles or other bibliography that would discuss the matter properly. On the other hand a very interesting and promissing numerical method of solving systems of ordinary differential equations based on Taylor series has appeared. The question was how to harness the said "Modern Taylor Series Method" for solving of stiff systems. The potential of the Taylor series has been exposed by many practical experiments and a way of detection and solution of large systems of ordinary differential equations has been found. Generally speaking, a stiff system contains several components, some of them are heavily suppressed while the rest remain almost unchanged. This feature forces the used method to choose an extremely small integration step and the progress of the computation may become very slow. However, we often need to find out the solution in a long range. It is clear that the mentioned facts are troublesome and ways to cope with such problems have to be devised. There are many (implicit) methods for solving stiff systems of ODE's, from the most simple such as implicit Euler method to more sophisticated (implicit Runge-Kutta methods) and finally the general linear methods. The mathematical formulation of the methods often looks clear, however the implicit nature of those methods implies several implementation problems. Usually a quite complicated auxiliary system of equations has to be solved in each step. These facts lead to immense amount of work to be done in each step of the computation. These are the reasons why one has to think twice before using the stiff solver and to decide between the stiff and non-stiff solver.