Multiple Arithmetic in Dynamic System Simulation

Abstract

A very interesting and promising numerical method of solving systems of ordinary differential equations based on Taylor series has appeared. 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. 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. 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.