High Order Runge { Kutta Methods on Manifolds

High Order Runge { Kutta Methods on Manifolds Hans

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.

Runge{kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for di erentiable manifolds as proposed by Crouch and Grossman. The di erential system is phrased by means of a system of frame vector elds E1; : : : ; En on the manifold. The numerical approximation is obtained by composing ows of certain vector elds in the linear span of E1; : : : ; En that are tangent to the di erential system at various points. The ...

Numerical Solution of a System SEIR Nonlinear ODEs by Runge-Kutta Fourth Order Method

In this paper, we introduce the numerical solution of the system of SEIR nonlinear ordinary differential equations, which are studied the effect of vaccine on the HIV (Human Immunology virus). We obtained the numerical solutions on stable manifolds by Runge-Kutta fourth order method.

Lie-butcher Theory for Runge-kutta Methods

Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees . In the present formulation they appear as commutators between vector fiel...

Invariant manifolds in differential algebraic equations of index 3 and in their Runge-Kutta discretizations