Global errors of numerical ODE solvers and Lyapunov's theory of stability

The error made by a numerical method in approximating the solutionof the initial value problem (t) = f (t,x), x (0) = x₀, t 0, x (t) R^d varies with the time of integration.The increase of the global error ||(t; h)– x (t)||, where (t, h) is an approximationderived by a numerical method with time step h, with time tdetermines the feasibility of approximating the solution accuratelyfor increasing t. However, the best available theoretical boundsinvolve the Lipshitz constant and are exponential in t for someproblems where the actual increase of global error is only linearin time. Using techniques from Lyapunov's theory of stability, we provethat the increase of global errors is linear in time for trajectoriesof dynamical systems which fall into a hyperbolic and attractingcycle or into a hyperbolic and attracting torus, with the flowon the torus being quasi-periodic. The increase is linear fornon-linear problems when certain stability properties of thesolution can be verified. The error analysis uses a conditioningfunction E(t) associated with the exact solution, which capturesthe propagation and accumulation of global errors. Received 4 January 1999. Accepted 15 December 1999.