© 2001 by Institute of Mathematics and its Applications
Global errors of numerical ODE solvers and Lyapunov's theory of stability
1 Departments of Computer Science and Mathematics, University of Chicago, USA, e-mail: divakar{at}cs.uchicago.edu
The error made by a numerical method in approximating the solution of the initial value problem 
(t) = f (t, x), x (0) = x0, t 
0, x (t) 
Rd varies with the time of integration. The increase of the global error ||
(t; h) – x (t)||, where (t, h) is an approximation derived by a numerical method with time step h, with time t determines the feasibility of approximating the solution accurately for increasing t. However, the best available theoretical bounds involve the Lipshitz constant and are exponential in t for some problems where the actual increase of global error is only linear in time.
Using techniques from Lyapunov's theory of stability, we prove that the increase of global errors is linear in time for trajectories of dynamical systems which fall into a hyperbolic and attracting cycle or into a hyperbolic and attracting torus, with the flow on the torus being quasi-periodic. The increase is linear for non-linear problems when certain stability properties of the solution can be verified. The error analysis uses a conditioning function E(t) associated with the exact solution, which captures the propagation and accumulation of global errors.
Received 4 January 1999. Accepted 15 December 1999.