The rate of error growth in Hamiltonian-conserving integrators |
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Authors: | Donald J Estep Andrew M Stuart |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, 30332 Atlanta, GA;(2) Scientific Computing and Computational Mathematics Program, Division of Applied Mechanics, Stanford University, Durand 252, 94305-4040 Stanford, California, USA |
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Abstract: | In this note, we consider numerical methods for a class of Hamiltonian systems that preserve the Hamiltonian. We show that the rate of growth of error is at most linear in time when such methods are applied to problems with period uniquely determined by the value of the Hamiltonian. This contrasts to generic numerical schemes, for which the rate of error growth is superlinear. Asymptotically, the rate of error growth for symplectic schemes is also linear. Hence, Hamiltonian-conserving schemes are competitive with symplectic schemes in this respect. The theory is illustrated with a computation performed on Kepler's problem for the interaction of two bodies. |
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