### 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 language | English (US) |
---|---|

Pages (from-to) | 121-137 |

Number of pages | 17 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 45 |

Issue number | 1-2 |

DOIs | |

State | Published - Apr 8 1993 |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Journal of Computational and Applied Mathematics*,

*45*(1-2), 121-137. https://doi.org/10.1016/0377-0427(93)90269-H

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

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 45, no. 1-2, pp. 121-137. https://doi.org/10.1016/0377-0427(93)90269-H

}

TY - JOUR

T1 - Time-point relaxation Runge-Kutta methods for ordinary differential equations

AU - Bellen, A.

AU - Jackiewicz, Zdzislaw

AU - Zennaro, M.

PY - 1993/4/8

Y1 - 1993/4/8

N2 - 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.

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.

KW - Ordinary differential equations

KW - Runge-Kutta method

KW - stability analysis

KW - time-point relaxation

UR - http://www.scopus.com/inward/record.url?scp=38249001865&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249001865&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(93)90269-H

DO - 10.1016/0377-0427(93)90269-H

M3 - Article

AN - SCOPUS:38249001865

VL - 45

SP - 121

EP - 137

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -