Error bounding functions for Runge-Kutta methods (Q1121647)

This paper deals with Runge-Kutta methods applied to stiff differential equations. Consider \(x'(t)=f(t,x(t)),\) where f: \({\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) fulfills the one-sided Lipschitz condition \(<f(t,u)- f(t,v),u-v>\leq \lambda \| u-v\|^ 2\) for all \(u,v\in {\mathbb{R}}^ n\), \(t\in {\mathbb{R}}\), then two solutions u and v with the initial values \(u(t_ 0)=\mu\) and \(v(t_ 0)=\nu\) satisfy \(\| u(t_ 0+h)-v(t_ 0+h)\| \leq e^{\ell}\| \mu -\nu \| \quad (\ell =h\lambda).\) The numerical approximations y to \(u(t_ 0+h)\) and z to \(v(t_ 0+h)\) satisfy \(\| y-z\| \leq \phi (\ell)\| \mu -\nu \|,\) where \(\phi\) (\(\ell)\) reflects stability properties. On the other hand there exist algebraic conditions which guarantee stability properties. The author reviews the algebraic conditions associated with error bounding functions and presents a result of relevance to B-convergence theory. In particular, the algebraic condition for C-stability is obtained. The problem of deriving sharp error bounds is examined and examples are given.