Abstract
In the past few years, a number of Lie-group methods based on Runge—Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge—Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie-group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness.
In this paper we discuss Lie-group methods and their dependence on centering coordinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge—Kutta schemes and symmetric coordinates, based on geodesic or on flow midpoint. As result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.
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A. Zanna, K. Engø, and H. Z. Munthe-Kaas, Adjoint and selfadjoint Lie-group methods, Technical Report NA 1999/ 02, DAMTP, University of Cambridge, UK, 1999.
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Zanna, A., Engø, K. & Munthe-Kaas, H.Z. Adjoint and Selfadjoint Lie-group Methods. BIT Numerical Mathematics 41, 395–421 (2001). https://doi.org/10.1023/A:1021950708869
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DOI: https://doi.org/10.1023/A:1021950708869