Order theory for discrete gradient methods
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We present a subclass of the discrete gradient methods, which are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of the methods, we derive conditions for arbitrarily high order. We devote considerable space to the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented and applied to the Hénon-Heiles system and a Lotka-Volterra system.
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