Detail publikace

ŠÁTEK, V. KUNOVSKÝ, J. KOPŘIVA, J.

Originální název

Anglický název

Jazyk

en

Originální abstrakt

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.

Anglický abstrakt

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.

Dokumenty

BibTex

``````
@inproceedings{BUT76444,
author="Václav {Šátek} and Jiří {Kunovský} and Jan {Kopřiva}",
annote="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.",
address="Faculty of Electrical Engineering and Informatics, University of Technology Košice",
booktitle="Proceedings of the Eleventh International Scientific Conference on Informatics",
chapter="76444",
edition="NEUVEDEN",
howpublished="print",
institution="Faculty of Electrical Engineering and Informatics, University of Technology Košice",
year="2011",
month="november",
pages="208--212",
publisher="Faculty of Electrical Engineering and Informatics, University of Technology Košice",
type="conference paper"
}``````