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Many n umerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker-Campbell-Hausdorr formula and the recently developed Lie group methods for integration of diierential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case(More)
RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present(More)
Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these " Generalized Runge–Kutta processes " was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, recently rediscovered by Cox and Matthews. The deficit(More)
We deene the Newton iteration for solving the equation f y = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two v ersions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian(More)
We show that while Runge-Kutta methods cannot preserve polynomial invariants in general , they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods preserve arbitrary quadratic invariants [4]. However, no RK method(More)
We present a new class of integration methods for diierential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves(More)
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time averaged version of the invariant is developed for systems of partial differential equations with polynomial nonlin-earities. The framework is rather general and(More)
Continuous Explicit Runge-Kutta methods with the minimal number of stages are considered. These methods are continuously diierentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for order 3,4 and 5. It is shown how the free parameters of these methods can be used either to minimize(More)