Top

Journal of Scientific Computing

Published in:

28-03-2015

A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control

Author: Pak-Wing Fok

Published in: Journal of Scientific Computing | Issue 1/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

To integrate large systems of locally coupled ordinary differential equations with disparate timescales, we present a multirate method with error control that is based on the Cash–Karp Runge–Kutta formula. The order of multirate methods often depends on interpolating certain solution components with a polynomial of sufficiently high degree. By using cubic interpolants and analyzing the method applied to a simple test equation, we show that our method is fourth order linearly accurate overall. Furthermore, the size of the region of absolute stability is increased when taking many “micro-steps” within a “macro-step.” Finally, we demonstrate our method on three simple test problems to confirm fourth order convergence.

previous article Numerical Analysis of AVF Methods for Three-Dimensional Time-Domain Maxwell’s Equations

next article Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

We use the word “interpolation” to describe a method to construct a smooth approximation to the numerical solution between two times \(t_n\) and \(t_{n+1}\). However, the function that we derive does not pass through the numerical solution at \(t_{n+1}\). Strictly speaking, it is not an interpolant. Nevertheless, we still refer to these approximating functions as “interpolants.”

Andrus, J.F.: Numerical solution of systems of ordinary differential equations separated into subsystems. SIAM J. Numer. Anal. 16, 605–611 (1979)MathSciNetCrossRefMATH

Bartel, A., Günther, M.: A multirate W-method for electrical networks in state-space formulation. J. Comput. Appl. Math. 147(2), 411–425 (2002)MathSciNetCrossRefMATH

Bartel, A., Günther, M., Kvaerno, A.: Multirate methods in electrical circuit simulation. Prog. Ind. Math. ECMI 2000, 258–265 (2000)

Constantinescu, E.M., Sandu, A.: Extrapolated multirate methods for differential equations with multiple time scales. J. Sci. Comput. 56(1), 28–44 (2013)

Constantinescu, E.M., Adrian, Sandu: Multirate timestepping methods for hyperbolic conservation laws. J. Sci. Comput. 33, 239–278 (2007)MathSciNetCrossRefMATH

Constantinescu, E.M., Sandu, Adrian: Multirate explicit Adams methods for time integration. J. Sci. Comput. 38, 229–249 (2009)MathSciNetCrossRefMATH

Enright, W.H., Jackson, K.R., Nørsett, S.P., Thomsen, P.G.: Interpolants for Runge–Kutta formulas. ACM Trans. Math. Softw. 12(3), 193–218 (1986)CrossRefMATH

Gear, C.W., Wells, D.R.: Multirate linear multistep methods. BIT 24, 484–502 (1984)MathSciNetCrossRefMATH

Günther, M., Kværnø, A., Rentrop, P.: Multirate partitioned Runge–Kutta methods. BIT 41(3), 504–514 (2001)MathSciNetCrossRefMATH

10.

Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer (1987)

11.

Horn, M.K.: Fourth and fifth order, scaled Runge–Kutta algorithms for treating dense output. SIAM J. Numer. Anal. 20(3), 558–568 (1983)MathSciNetCrossRefMATH

12.

Hundsdorfer, W., Savcenco, V.: Analysis of a multirate theta-method for stiff ODEs. Appl. Numer. Math. 59, 693–706 (2009)MathSciNetCrossRefMATH

13.

Kato, T., Kataoka, T.: Circuit analysis by a new multirate method. Electr. Eng. Jpn. 126(4), 1623–1628 (1999)CrossRef

14.

Logg, A.: Multi-adaptive time integration. Appl. Numer. Math. 48, 339–354 (2004)MathSciNetCrossRefMATH

15.

Makino, J., Aarseth, S.: On a Hermite integrator with Ahmad-Cohen scheme for gravitational many-body problems. Publ. Astron. Soc. Japan 44, 141–151 (1992)

16.

Press, W.H., Teukolsky, S.A., Vetterling, W.V., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)

17.

Savcenco, V.: Construction of a multirate RODAS method for stiff ODEs. J. Comput. Appl. Math. 225, 323–337 (2009)MathSciNetCrossRefMATH

18.

Savcenco, V., Hundsdorfer, W., Verwer, J.G.: A multirate time stepping strategy for stiff ordinary differential equations. BIT 47, 137–155 (2007)MathSciNetCrossRefMATH

19.

Shampine, L.F.: Some practical Runge-Kutta formulas. Math. Comput. 173, 135–150 (1986)MathSciNetCrossRef

20.

Skelboe, S.: Stability properties of backward differentiation multirate formulas. Appl. Numer. Math. 5, 151–160 (1989)MathSciNetCrossRefMATH

21.

Waltz, J., Page, G.L., Milder, S.D., Wallin, J., Antunes, A.: A performance comparison of tree data structures for \({N}\)-body simulation. J. Comput. Phys. 178, 1–14 (2002)CrossRefMATH

Title: A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control
Author: Pak-Wing Fok
Publication date: 28-03-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0017-4

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2016

High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations

Towards Optimized Schwarz Methods for the Navier–Stokes Equations

Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

A Spectral Analysis of Subspace Enhanced Preconditioners

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

Premium Partner