Numerical integration of ordinary differential equations on manifolds
This article considers systems of differential equations \(\dot z = \sum^ n_{i=1} f_ i(z) E_ i(z)\), where \(f_ i(z)\) are scalar functions and \(E_ i(z)\) are vector fields on \(\mathbb{R}^ N\). Typically the solutions are constrained to some submanifold of \(\mathbb{R}^ N\). When the coefficients \(f_ i(z)\) are replaced by constants (``frozen coefficients'') it is assumed that the problem can be solved analytically. Runge-Kutta methods and linear multistep methods are generalized in such a way that only evaluations of the \(f_ i(z)\) are needed and that the numerical approximation evolves in the solution manifold. The main part of this paper studies the order of the methods. It is shown that, besides the usual order conditions, additional relations have to be satisfied by the parameters of the method. New algorithms of order 3 are presented.
scientific article; zbMATH DE number 563604
scientific article