Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations

Alexey S. Eremin, Nikolai A. Kovrizhnykh, Igor V. Olemskoy

Research output

Abstract

Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.

Original languageEnglish
Title of host publicationComputational Science and Its Applications – ICCSA 2019
Subtitle of host publication19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I
EditorsSanjay Misra, Elena Stankova, Vladimir Korkhov, Carmelo Torre, Eufemia Tarantino, Ana Maria A.C. Rocha, David Taniar, Osvaldo Gervasi, Bernady O. Apduhan, Beniamino Murgante
PublisherSpringer
Pages89-102
ISBN (Electronic)978-3-030-24289-3
ISBN (Print)9783030242886
DOIs
Publication statusPublished - 1 Jul 2019
Event19th International Conference on Computational Science and Its Applications, ICCSA 2019 - Saint Petersburg
Duration: 1 Jul 20194 Jul 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11619 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference19th International Conference on Computational Science and Its Applications, ICCSA 2019
CountryRussian Federation
CitySaint Petersburg
Period1/07/194/07/19

Fingerprint

Runge-Kutta Methods
System of Ordinary Differential Equations
Ordinary differential equations
Explicit Methods
Partitioning
Permutation
Unknown
Evaluation

Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Eremin, A. S., Kovrizhnykh, N. A., & Olemskoy, I. V. (2019). Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. In S. Misra, E. Stankova, V. Korkhov, C. Torre, E. Tarantino, A. M. A. C. Rocha, D. Taniar, O. Gervasi, B. O. Apduhan, ... B. Murgante (Eds.), Computational Science and Its Applications – ICCSA 2019 : 19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I (pp. 89-102). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11619 LNCS). Springer. https://doi.org/10.1007/978-3-030-24289-3_8
Eremin, Alexey S. ; Kovrizhnykh, Nikolai A. ; Olemskoy, Igor V. / Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. Computational Science and Its Applications – ICCSA 2019 : 19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I. editor / Sanjay Misra ; Elena Stankova ; Vladimir Korkhov ; Carmelo Torre ; Eufemia Tarantino ; Ana Maria A.C. Rocha ; David Taniar ; Osvaldo Gervasi ; Bernady O. Apduhan ; Beniamino Murgante. Springer, 2019. pp. 89-102 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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title = "Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations",
abstract = "Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.",
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author = "Eremin, {Alexey S.} and Kovrizhnykh, {Nikolai A.} and Olemskoy, {Igor V.}",
note = "Eremin A.S., Kovrizhnykh N.A., Olemskoy I.V. (2019) Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. In: Misra S. et al. (eds) Computational Science and Its Applications – ICCSA 2019. ICCSA 2019. Lecture Notes in Computer Science, vol 11619. Springer, Cham",
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booktitle = "Computational Science and Its Applications – ICCSA 2019",
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Eremin, AS, Kovrizhnykh, NA & Olemskoy, IV 2019, Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. in S Misra, E Stankova, V Korkhov, C Torre, E Tarantino, AMAC Rocha, D Taniar, O Gervasi, BO Apduhan & B Murgante (eds), Computational Science and Its Applications – ICCSA 2019 : 19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11619 LNCS, Springer, pp. 89-102, Saint Petersburg, 1/07/19. https://doi.org/10.1007/978-3-030-24289-3_8

Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. / Eremin, Alexey S.; Kovrizhnykh, Nikolai A.; Olemskoy, Igor V.

Computational Science and Its Applications – ICCSA 2019 : 19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I. ed. / Sanjay Misra; Elena Stankova; Vladimir Korkhov; Carmelo Torre; Eufemia Tarantino; Ana Maria A.C. Rocha; David Taniar; Osvaldo Gervasi; Bernady O. Apduhan; Beniamino Murgante. Springer, 2019. p. 89-102 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11619 LNCS).

Research output

TY - GEN

T1 - Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations

AU - Eremin, Alexey S.

AU - Kovrizhnykh, Nikolai A.

AU - Olemskoy, Igor V.

N1 - Eremin A.S., Kovrizhnykh N.A., Olemskoy I.V. (2019) Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. In: Misra S. et al. (eds) Computational Science and Its Applications – ICCSA 2019. ICCSA 2019. Lecture Notes in Computer Science, vol 11619. Springer, Cham

PY - 2019/7/1

Y1 - 2019/7/1

N2 - Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.

AB - Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.

KW - explicit Runge–Kutta

KW - Multischeme methods

KW - Order conditions

KW - Partitioned methods

KW - Structural partitioning

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U2 - 10.1007/978-3-030-24289-3_8

DO - 10.1007/978-3-030-24289-3_8

M3 - Conference contribution

AN - SCOPUS:85069154587

SN - 9783030242886

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 89

EP - 102

BT - Computational Science and Its Applications – ICCSA 2019

A2 - Misra, Sanjay

A2 - Stankova, Elena

A2 - Korkhov, Vladimir

A2 - Torre, Carmelo

A2 - Tarantino, Eufemia

A2 - Rocha, Ana Maria A.C.

A2 - Taniar, David

A2 - Gervasi, Osvaldo

A2 - Apduhan, Bernady O.

A2 - Murgante, Beniamino

PB - Springer

ER -

Eremin AS, Kovrizhnykh NA, Olemskoy IV. Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations. In Misra S, Stankova E, Korkhov V, Torre C, Tarantino E, Rocha AMAC, Taniar D, Gervasi O, Apduhan BO, Murgante B, editors, Computational Science and Its Applications – ICCSA 2019 : 19th International Conference, Saint Petersburg, Russia, July 1–4, 2019, Proceedings, Part I. Springer. 2019. p. 89-102. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-24289-3_8