##
**A variable step implicit block multistep method for solving first-order ODEs.**
*(English)*
Zbl 1183.65094

Summary: A new four-point implicit block multistep method is developed for solving systems of first-order ordinary differential equations (ODEs) with variable step size. The method computes the numerical solution at four equally spaced points simultaneously. The stability of the proposed method is investigated. The Gauss-Seidel approach is used for the implementation of the proposed method in the \(PE(CE)^m\) mode. The method is presented in a simple form of Adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients. Numerical examples are given to illustrate the efficiency of the proposed method.

### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

### Keywords:

variable step size; ordinary differential equations; Adams-type method; four-point implicit block multistep method; stability Gauss-Seidel approach; divided difference; numerical examples
PDF
BibTeX
XML
Cite

\textit{S. Mehrkanoon} et al., J. Comput. Appl. Math. 233, No. 9, 2387--2394 (2010; Zbl 1183.65094)

Full Text:
DOI

### References:

[1] | Burrage, K., Efficient block predictor – corrector methods with a small number of corrections, J. comput. appl. math., 45, 139-150, (1993) · Zbl 0782.65088 |

[2] | Milne, W.E., Numerical solution of differential equations, (1953), Wiley New York · Zbl 0007.11305 |

[3] | Rosser, J.B., A runge – kutta for all seasons, SIAM rev., 9, 417-452, (1967) · Zbl 0243.65041 |

[4] | Shampine, L.F.; Watts, H.A., Block implicit one-step methods, Math. comp., 23, 731-740, (1969) · Zbl 0187.40202 |

[5] | P.J. Houwen, P.B. Sommeijer, block Runge-Kutta methods on parallel computers, Report NM-R8906, Center for Mathematics and Computer Science, Amsterdam, 1989 |

[6] | Z. Omar, Developing Parallel Block Methods for Solving Higher Order ODEs Directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999 |

[7] | Z.A. Majid, Parallel block methods for solving ordinary differential equations, Ph.D. Thesis, University Putra Malaysia, 2004 |

[8] | Majid, Z.A.; Suleiman, M., Performance of 4-point diagonally implicit block method for solving ordinary differential equations, Matematika, 22, 2, 137-146, (2006) |

[9] | Majid, Z.A.; Suleiman, M., Implementation of four-point fully implicit block method for solving ordinary differential equations, Appl. math. comput., 184, 514-522, (2007) · Zbl 1114.65080 |

[10] | Majid, Z.A.; Suleiman, M.; Omar, Z., 3-point implicit block method for solving ordinary differential equations, Bull. malays. math. sci. soc. (2), 29, 1, 23-31, (2006) · Zbl 1123.65072 |

[11] | Johnson, A.I.; Barney, J.R., Numerical solution of large systems of stiff ordinary differential equations, (), 97-124 |

[12] | Hairer, E.; Norsett, S.P.; Wanner, G., Solving ordinary differential equations I, () · Zbl 1185.65115 |

[13] | Cong, N.H.; Xuan, L.N., Twostep-by-twostep PIRK-type PC methods with continuous output formulas, J. comput. appl. math., 221, 165-173, (2008) · Zbl 1156.65070 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.