On error bounds forG-stable methods

Abstract

Error bounds for numerical solutions of the initial value problem

$$y' = f(y), y(0) = \bar y \in R^d ,$$

are derived. The methods (ϱ,σ) are assumed to beG-stable [2], andf satisfies for someμ teR and for some inner product 〈, 〉 the relation

$$\left\langle {u - v,f(u)} \right\rangle \leqq \mu \left\| {u - v} \right\|^2 ,u,v \in R^d $$

As corollaries of the bounds we get, forμ=0, the result that whenever the local errors {q _n } ∈l ¹ then the global errors {z _n } ∈l ^∞. Forμ<0, assuming in addition that the zeros ofσ(ζ) lie inside the unit circle, {q _n } tel ^p implies {z _n } tel ^p forp ≧ 2.