A Conditioning Function for the Convergence of Numerical ODE Solvers and Lyapunov’s Theory of Stability

For the ordinary differential equation (ODE) ẋ(t) = f(t, x), x(0) = x0, t ≥ 0, x ∈ R , assume f to be at least continuous in t and locally Lipshitz in x, and if necessary, several times continuously differentiable in t and x. We associate a conditioning function E(t) with each solution x(t) which captures the accumulation of global error in a numerical approximation in the following sense: if x̃(t;h) is an approximation derived from a single step method of time step h and order r then ‖x̃(t;h) − x(t)‖ < K(E(t) + ǫ)h for 0 ≤ t ≤ T , any ǫ > 0, sufficiently small h, and a constant K > 0. Using techniques from the stability theory of differential equations, this paper gives conditions on x(t) for E(t) to be upper bounded linearly or by a constant for t ≥ 0. More concretely, these techniques give constant or linear bounds on E(t) when x(t) is a trajectory of a dynamical system which falls into a stable, hyperbolic fixed point; or into a stable, hyperbolic cycle; or into a normally hyperbolic and contracting manifold with quasiperiodic flow on the manifold.