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 a new type of methods which avoids the use of commutators, but which has a much lower number of flow computations than the Crouch-Grossman methods. We argue that the new methods may be particularly useful when applied to problems on homogeneous manifolds with large isotropy groups, or when used for stiff problems. Numerical experiments verify these claims when applied to a problem on the orthogonal Stiefel manifold, and to an example arising from the semidiscretisation of a linear inhomogeneous heat conduction problem.

We present a new class of integration methods for differential 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 the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

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We consider interpolation in Lie groups. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of Lie-algebra actions, we use the truncated inverse of the differential of the exponential mapping and the truncated Baker-Campbell-Hausdorff formula to relatively cheaply construct an interpolation polynomial.Much effort has lately been put into research on geometric integration, i.e., the process of integrating differential equations in such a way that the configuration space of the true solution is respected by the numerical solution. Some of these methods may be viewed as generalizations of classical methods, and we investigate the construction of intrinsic dense output devices as generalizations of the continuous Runge-Kutta methods.

During the last few years, different approaches for integrating ordinary differential equations on manifolds have been published. In this work, we consider two of these approaches. We present some numerical experiments showing benefits and some pitfalls when using the new methods. To demonstrate how they work, we compare with well known classical methods, e.g. Newmark and Runge-Kutta methods