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Approximately preserving symmetries in the numerical integration of ordinary differential equations

Published online by Cambridge University Press:  01 October 1999

ARIEH ISERLES ,
ROBERT McLACHLAN  and
ANTONELLA ZANNA
ARIEH ISERLES
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, UK
ROBERT McLACHLAN
Affiliation:
Mathematics, Massey University, Private Bag 11-222, Palmerston North, New Zealand
ANTONELLA ZANNA
Affiliation:
Newnham College, University of Cambridge, Cambridge CB3 9DF, UK
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Abstract

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 10 , Issue 5 , October 1999 , pp. 419 - 445
Copyright
1999 Cambridge University Press

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