Publication detail

Explicit and Implicit Taylor Series Based Computations

KUNOVSKÝ, J. SEHNALOVÁ, P. ŠÁTEK, V.

Original Title

Explicit and Implicit Taylor Series Based Computations

English Title

Explicit and Implicit Taylor Series Based Computations

Type

conference paper

Language

en

Original Abstract

This paper deals with computer simulations of continuous systems. The research group "High performance computing" has been working on extremely exact and fast solutions of homogenous differential equations, nonlinear ordinary and partial differential equations, stiff systems, large systems of algebraic equations, real time simulations and corresponding software and hardware (parallel) implementations since 1980. The Modern Taylor Series Method (MTSM) developed at our university is an original mathematical method which uses the Taylor series method for solving differential equations in a non-traditional way. Unfortunately, it is easier said than done as there are some peculiar systems of differential equations, which cannot be solved by commonly used (explicit) methods - the stiff systems.While the definition of this kind of systems is intuitively clear to the mathematicians the exact definition has not been specified yet. Often unnoticed, stiff systems showed up too often in practice such as in simulations of electrical circuits, chemical reactions and so on. To solve this kind of problems we can use for example multiple arithmetic or implicit numerical methods. In this paper we compare explicit and implicit Taylor series method in solution of the well-known Dahlquist's equation. We focus on stability and convergence of corresponding computations. Both explicit and implicit Taylor series methods give very successful results for the Dahlquist's equation and it can be expected that similar results will be obtained for sophisticated problems.

English abstract

This paper deals with computer simulations of continuous systems. The research group "High performance computing" has been working on extremely exact and fast solutions of homogenous differential equations, nonlinear ordinary and partial differential equations, stiff systems, large systems of algebraic equations, real time simulations and corresponding software and hardware (parallel) implementations since 1980. The Modern Taylor Series Method (MTSM) developed at our university is an original mathematical method which uses the Taylor series method for solving differential equations in a non-traditional way. Unfortunately, it is easier said than done as there are some peculiar systems of differential equations, which cannot be solved by commonly used (explicit) methods - the stiff systems.While the definition of this kind of systems is intuitively clear to the mathematicians the exact definition has not been specified yet. Often unnoticed, stiff systems showed up too often in practice such as in simulations of electrical circuits, chemical reactions and so on. To solve this kind of problems we can use for example multiple arithmetic or implicit numerical methods. In this paper we compare explicit and implicit Taylor series method in solution of the well-known Dahlquist's equation. We focus on stability and convergence of corresponding computations. Both explicit and implicit Taylor series methods give very successful results for the Dahlquist's equation and it can be expected that similar results will be obtained for sophisticated problems.

Keywords

Differential equations, Taylor series method, TKSL, Stiff systems

RIV year

2010

Released

30.09.2010

Publisher

American Institute of Physics

Location

Tripolis

ISBN

978-0-7354-0831-9

Book

8th International Conference of Numerical Analysis and Applied Mathematics

Edition

NEUVEDEN

Edition number

NEUVEDEN

Pages from

587

Pages to

590

Pages count

4

BibTex


@inproceedings{BUT34856,
  author="Jiří {Kunovský} and Pavla {Sehnalová} and Václav {Šátek}",
  title="Explicit and Implicit Taylor Series Based Computations",
  annote="This paper deals with computer simulations of continuous systems. The research
group "High performance computing"
has been working on extremely exact and fast solutions of homogenous differential
equations, nonlinear ordinary and partial
differential equations, stiff systems, large systems of algebraic equations, real
time simulations and corresponding software
and hardware (parallel) implementations since 1980.
The Modern Taylor Series Method (MTSM) developed at our university is an original
mathematical method which uses
the Taylor series method for solving differential equations in a non-traditional
way.
Unfortunately, it is easier said than done as there are some peculiar systems of
differential equations, which cannot be
solved by commonly used (explicit) methods - the stiff systems.While the
definition of this kind of systems is intuitively clear
to the mathematicians the exact definition has not been specified yet. Often
unnoticed, stiff systems showed up too often in
practice such as in simulations of electrical circuits, chemical reactions and so
on. To solve this kind of problems we can use
for example multiple arithmetic or implicit numerical methods.
In this paper we compare explicit and implicit Taylor series method in solution
of the well-known Dahlquist's equation.
We focus on stability and convergence of corresponding computations. Both
explicit and implicit Taylor series methods give
very successful results for the Dahlquist's equation and it can be expected that
similar results will be obtained for sophisticated
problems.",
  address="American Institute of Physics",
  booktitle="8th International Conference of Numerical Analysis and Applied Mathematics",
  chapter="34856",
  edition="NEUVEDEN",
  howpublished="print",
  institution="American Institute of Physics",
  year="2010",
  month="september",
  pages="587--590",
  publisher="American Institute of Physics",
  type="conference paper"
}