### 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) |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Journal of Computational and Applied Mathematics*,

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