### Abstract

Original language | English |
---|---|

Pages (from-to) | 1189-1212 |

Number of pages | 24 |

Journal | BIT Numerical Mathematics |

Volume | 56 |

Issue number | 4 |

Early online date | 22 Mar 2016 |

DOIs | |

Publication status | Published - 1 Dec 2016 |

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### Keywords

- Time-symmetric general linear methods
- G-symplectic methods · Multivalue methods
- Conservative methods

### Cite this

*BIT Numerical Mathematics*,

*56*(4), 1189-1212. https://doi.org/10.1007/s10543-016-0613-1

**Symmetric general linear methods.** / Butcher, J. C; Hill, A. T.; Norton, T. J. T.

Research output: Contribution to journal › Article

*BIT Numerical Mathematics*, vol. 56, no. 4, pp. 1189-1212. https://doi.org/10.1007/s10543-016-0613-1

}

TY - JOUR

T1 - Symmetric general linear methods

AU - Butcher, J. C

AU - Hill, A. T.

AU - Norton, T. J. T.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.

AB - The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.

KW - Time-symmetric general linear methods

KW - G-symplectic methods · Multivalue methods

KW - Conservative methods

U2 - 10.1007/s10543-016-0613-1

DO - 10.1007/s10543-016-0613-1

M3 - Article

VL - 56

SP - 1189

EP - 1212

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 4

ER -