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Runge-Kutta type methods for orthogonal integration

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

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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
ID code: 191
Keywords: Geometric integration, Implicit midpoint rule, ODEs on manifolds, Orthogonality, Structure preservation, Runge-Kutta methods, mathematics, Mathematics, Computational Mathematics, Applied Mathematics, Numerical Analysis
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
    Depositing user: Ms Sarah Scott
    Date Deposited: 01 Mar 2006
    Last modified: 04 Sep 2014 09:57
    URI: http://strathprints.strath.ac.uk/id/eprint/191

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