Convergence of numerical ODE solvers and Lyapunov's theory of stability

Abstract

For the ordinary differential equation (ODE) $\dot{x}(t)=f(t,x)$, $x(0)=x_0$, $t\ge 0$, $x\in R^d$, 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 $\tilde{x}(t;h)$ is an approximation derived from a single step method of time step $h$ and order $r$ then $\norm{\tilde{x}(t;h) - x(t)} < K(E(t)+ε)h^r$ for $0\le t\le T$, any $\epsilon >0$, 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 $t\ge 0$. 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.