Higham, D.J. (1996) *Runge-Kutta type methods for orthogonal integration.* Applied Numerical Mathematics, 22. pp. 217-223. ISSN 0168-9274

## Abstract

A simple characterisation exists for the class of real-valued, autonomous, matrix ODEs where an orthogonal initial condition implies orthogonality of the solution for all time. Here we present first and second order numerical methods for which the property of orthogonality-preservation is always carried through to the discrete approximation. To our knowledge, these are the first methods that guarantee to preserve orthogonality, without the use of projection, whenever it is preserved by the flow. The methods are based on Gauss-Legendre Runge-Kutta formulas, which are known to preserve orthogonality on a restricted problem class. In addition, the new methods are linearly-implicit, requiring only the solution of one or two linear matrix systems (of the same dimension as the solution matrix) per step. Illustrative numerical tests are reported.

Item type: | Article |
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ID code: | 191 |

Keywords: | Geometric integration, Implicit midpoint rule, ODEs on manifolds, Orthogonality, Structure preservation, Runge-Kutta methods, mathematics, Mathematics |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Related URLs: | |

Depositing user: | Ms Sarah Scott |

Date Deposited: | 01 Mar 2006 |

Last modified: | 12 Mar 2012 10:35 |

URI: | http://strathprints.strath.ac.uk/id/eprint/191 |

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