Modern Taylor series method in numerical integration

• Autorzy: Chaloupka J. , Necasová G. , Veigend P. , Kunovský J. , Šátek V.

Warianty tytułu

CS

Moderní metoda Taylorovy řady v numerické integraci

• Języki publikacji: EN

Abstrakty

EN

The paper deals with extremely exact, stable, and fast numerical solutions of systems of differential equations. It also involves solutions of problems that can be reduced to solving a system of differential equations. The approach is based on an original mathematical method, which uses the Taylor series method for solving differential equations in a non-traditional way. Even though this method is not much preferred in the literature, experimental calculations have verified that the accuracy and stability of the Taylor series method exceed the currently used algorithms for numerically solving differential equations. The Modern Taylor Series Method (MTSM) is based on a recurrent calculation of the Taylor series terms for each time interval. Thus, the complicated calculation of higher order derivatives (much criticised in the literature) need not be performed but rather the value of each Taylor series term is numerically calculated. An important part of the method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires. The aim of our research is to propose the extremely exact, stable, and fast numerical solver for modelling technical initial value problems that offers wide applications in many engineering areas including modelling of electrical circuits, mechanics of rigid bodies, control loop feedback (controllers), etc.

CS

Clánek se zabývá presným, stabilním a rychlým rešením soustav diferenciálních rovnic. Soustavou diferenciálních rovnic lze reprezentovat velké množství reálných problému. Numerické rešení je založeno na unikátní numerické metode, která netradicne využívá Taylorovu radu. I presto, že tato metoda není v literature príliš preferována, experimentální výpocty potvrdily, že presnost a stabilita této metody presahuje aktuálne používané numerické algoritmy pro numerické rešení diferenciálních rovnic. Moderní metoda Taylorovy rady je založena na rekurentním výpoctu clenu Taylorovy rady v každém casovém intervalu. Derivace vyšších rádu nejsou pro výpocet prímo využity, derivace jsou zahrnuty do clenu Taylorovy rady, které se pocítají rekurentne numericky. Duležitou vlastností metody je automatická volba rádu metody v závislosti na velikosti integracního kroku, tzn. je využito tolik clenu Taylorovy rady, kolik vyžaduje zadaná presnost výpoctu. Cílem výzkumu je navrhnout velmi presný, stabilní a rychlý nástroj pro modelování technických pocátecních problému využitých v praxi pri modelování elektrických obvodu, mechaniky tuhých teles, problematiky zpetnovazebního rízení a další.

Słowa kluczowe

EN

Taylor series; ordinary differential equations; technical initial value problems

PL

szereg Taylora; równanie różniczkowe zwyczajne

• Wydawca: STE GROUP
• Czasopismo: Systemy Wspomagania w Inżynierii Produkcji
• Rocznik: 2017
• Tom: Vol. 6, iss. 4
• Strony: 263--273
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Chaloupka J.

• Brno University of Technology, Faculty of Information Technology, Božetechova 2, 612 66, Brno, Czech Republic

autor

Necasová G.

• Brno University of Technology, Faculty of Information Technology, Božetechova 2, 612 66, Brno, Czech Republic

autor

Veigend P.

• Brno University of Technology, Faculty of Information Technology, Božetechova 2, 612 66, Brno, Czech Republic

autor

Kunovský J.

• Brno University of Technology, Faculty of Information Technology, Božetechova 2, 612 66, Brno, Czech Republic

autor

Šátek V.

• IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33, Ostrava-Poruba, Czech Republic

Bibliografia

• 1. R. Barrio. Performance of the Taylor series method for ODEs/DAEs. In Applied Mathematics and Computation, volume 163, pages 525-545, 2005. ISSN 00963003.
• 2. R. Barrio, F. Blesa, and M. Lara. VSVO Formulation of the Taylor Method for the Numerical Solution of ODEs. In Computers and Mathematics with Applications, volume 50, pages 93-111, 2005.
• 3. R. Barrio, M. Rodríguez, A. Abad, and F. Blesa. TIDES: A free software based on the Taylor series method. Monografías de la Real Academia de Ciencias de Zaragoza, 35:83-95, 2011.
• 4. M. Berz. COSY INFINITY version 8 reference manual. Technical Report MSUCL-1088, National Superconducting Cyclotron Lab., Michigan State University, East Lansing, Mich., 1997.
• 5. J. Chaloupka, F. Kocina, P. Veigend, G. Necasová, V. Šátek, and J. Kunovský. Multiple integral computations. In 14th International Conference of Numerical Analysis and Applied Mathematics, 2016.
• 6. Y. F. Chang and G. Corliss. Atomf: solving odesand daes using Taylor series. Computers Math. Applic., 28:209-233, 1994.
• 7. E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. vol. Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 1987. ISBN 3-540-56670-8.
• 8. A. Jorba and M. Zou. A software package for the numerical integration of ODE by means of high-order Taylor methods. In Exp. Math., volume 14, pages 99-117, 2005.
• 9. Filip Kocina, Jirí Kunovský, Martin Marek, Gabriela Necasová, Alexander Schirrer, and Václav Šátek. New trends in Taylor series based computations. In 12th International Conference of Numerical Analysis and Applied Mathematics, number 1648. American Institute of Physics, 2014.
• 10. J. Kunovský. Modern Taylor Series Method. FEI-VUT Brno, 1994. Habilitation work.
• 11. Jirí Kunovský, Václav Šátek, Gabriela Necasová, Petr Veigend, and Filip Kocina. The Positive Properties of Taylor Series Method. In Proceedings of the 13th International Conference Informatics’ 2015, pages 156-160. Institute of Electrical and Electronics Engineers, 2015.
• 12. A. Martinkovicová, J. Chaloupka, J. Kunovský, V. Šátek, and P. Veigend. Numerical integration of multiple integrals using Taylor’s polynomial. In Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pages 163-171. SciTePress - Science and Technology Publications, 2015.
• 13. Nedialko S. Nedialkov and J. Pryce. Solving differential algebraic equations by Taylor series III. the DAETS code. JNAIAM J. Numer. Anal. Ind. Appl. Math., 3:61-80, 2008.
• 14. TKSL software. High Performance Computing, [cit. 21-3-2017]. URL: http://www.fit.vutbr.cz/ kunovsky/TKSL/index.html.en [online].
• 15. Václav Valenta, Gabriela Necasová, Jirí Kunovský, Václav Šátek, and Filip Kocina. Adaptive solution of the wave equation. In Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pages 154-162. SciTePress - Science and Technology Publications, 2015.
• 16. Wolfram. MATHEMATICA software, [cit. 21-3-2017].