Time-point relaxation Runge-Kutta methods for ordinary differential equations

A. Bellen, Zdzislaw Jackiewicz, M. Zennaro

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or Gauss-Seidel modes denoted by TRGJRK(k) or TRGSRK(k), respectively, where k stands for the number of Picard-Lindelöf iterations. As k → ∞, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta (DSRK) method. It is proved that these methods are convergent with order min{k, p}, where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK(k), TRGSRK(k) and DSRK methods to the two-dimensional test system u′ = λu - μν, ν′ = μu + λν, t ≥ 0, where λ and μ are real parameters, are derived and stability regions in the (λ, μ)-plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindelöf iterations k becomes larger.

Original languageEnglish (US)
Pages (from-to)121-137
Number of pages17
JournalJournal of Computational and Applied Mathematics
Volume45
Issue number1-2
DOIs
StatePublished - Apr 8 1993

Fingerprint

Runge Kutta methods
Relaxation Method
Runge-Kutta Methods
Ordinary differential equations
Ordinary differential equation
Picard Iteration
Stability Region
One-step Method
Gauss-Seidel
Convergence Order
Convergence of numerical methods
Two-dimensional Systems
Test System
Recurrence relation
System of Ordinary Differential Equations
Jacobi
Gauss
Locus
Tend

Keywords

  • Ordinary differential equations
  • Runge-Kutta method
  • stability analysis
  • time-point relaxation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

Time-point relaxation Runge-Kutta methods for ordinary differential equations. / Bellen, A.; Jackiewicz, Zdzislaw; Zennaro, M.

In: Journal of Computational and Applied Mathematics, Vol. 45, No. 1-2, 08.04.1993, p. 121-137.

Research output: Contribution to journalArticle

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AB - We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or Gauss-Seidel modes denoted by TRGJRK(k) or TRGSRK(k), respectively, where k stands for the number of Picard-Lindelöf iterations. As k → ∞, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta (DSRK) method. It is proved that these methods are convergent with order min{k, p}, where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK(k), TRGSRK(k) and DSRK methods to the two-dimensional test system u′ = λu - μν, ν′ = μu + λν, t ≥ 0, where λ and μ are real parameters, are derived and stability regions in the (λ, μ)-plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindelöf iterations k becomes larger.

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