Runge-Kutta type methods for orthogonal integration

D.J. Higham

Research output: Contribution to journalArticle

14 Citations (Scopus)

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.
LanguageEnglish
Pages217-223
Number of pages6
JournalApplied Numerical Mathematics
Volume22
DOIs
Publication statusPublished - 1996

Fingerprint

Runge-Kutta
Orthogonality
Discrete Approximation
Numerical methods
Legendre
Preservation
Gauss
Initial conditions
Linearly
Numerical Methods
Projection
First-order
Imply
Class

Keywords

  • Geometric integration
  • Implicit midpoint rule
  • ODEs on manifolds
  • Orthogonality
  • Structure preservation
  • Runge-Kutta methods
  • mathematics

Cite this

@article{5d25f4de0f86418bb318bc19778f0649,
title = "Runge-Kutta type methods for orthogonal integration",
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.",
keywords = "Geometric integration, Implicit midpoint rule, ODEs on manifolds, Orthogonality, Structure preservation, Runge-Kutta methods, mathematics",
author = "D.J. Higham",
year = "1996",
doi = "1016/S0168-9274(96)00033-5",
language = "English",
volume = "22",
pages = "217--223",
journal = "Applied Numerical Mathematics",
issn = "0168-9274",

}

Runge-Kutta type methods for orthogonal integration. / Higham, D.J.

In: Applied Numerical Mathematics, Vol. 22, 1996, p. 217-223.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Runge-Kutta type methods for orthogonal integration

AU - Higham, D.J.

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

KW - Geometric integration

KW - Implicit midpoint rule

KW - ODEs on manifolds

KW - Orthogonality

KW - Structure preservation

KW - Runge-Kutta methods

KW - mathematics

UR - http://dx.doi.org/1016/S0168-9274(96)00033-5

U2 - 1016/S0168-9274(96)00033-5

DO - 1016/S0168-9274(96)00033-5

M3 - Article

VL - 22

SP - 217

EP - 223

JO - Applied Numerical Mathematics

T2 - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -