We extend a systematic method for the derivation of high order schemes for affinely controlled nonlinear systems to a larger class of systems in which the control variables are allowed to appear nonlinearly in multiplicative terms. Using an adaptation of the stochastic Taylor expansion to control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-...

A one-dimensional mathematical model was developed for the simulation of limestone calcination in rotary kilns. The model is comprised of ordinary differential equations derived from mass and energy conservation principles. The system of equations is solved by a fourth-order Runge-Kutta method. The model predicts, in the axial direction, temperature profiles for the bed of particles, the gas ph...

This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analys...

The subject matter of this paper is the recovery of invariants and conservation laws of ordinary diierential systems by numerical methods. We prove that the most likely candidates for this task, Runge{Kutta schemes, fail to stay on manifolds deened by r-tensors with r 3. As an alternative, we suggest diieomorphically mapping complicated man-ifolds to simpler ones. This procedure allows for reco...

Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case i...

The role of the Killing–Yano tensors in the construction of the Dirac-type operators is pointed out. The general results are applied to the case of the four-dimensional Euclidean Taub–Newman–Unti–Tamburino space. Three new Dirac-type operators, equivalent to the standard Dirac operator, are constructed from the covariantly constant Killing–Yano tensors of this space. Finally the Runge–Lenz oper...

Conditions on Runge-Kutta algorithms can be obtained which ensure smooth stepsize selection when stability of the algorithm is restricting the stepsize. Some recently derived results are shown to hold for a more general test problem.

This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for diierential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in [9, 10]. The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher’s a...