Advanced Stiff Systems Detection

Stiff systems, Taylor series terms, Modern Taylor Series Method, TKSL

The paper deals with stiff systems of differential equations. To solve this sort
of system numerically is a difficult task. 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 ordinary differential equations (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. On the other hand a very
interesting and promising 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. 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.