A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations
The global error for the numerical solution of ordinary differential equations is comprised of errors which originate from a variety of sources. In this work the authors extend these standard estimates by adding a new term accounting for the use of finite numeric precision in the computation of the numerical solution. This error is normally neglected due to its magnitude being much smaller than the contributions provided by the data or discretization error. Since general initial value problems for systems of ordinary differential equations can be very sensitive to perturbations when the time interval \([0, T]\) is very long, or when a solution is sought with very high accuracy, it is important that the effect of numerical round-off errors as a result of finite numeric precision is considered. This will inevitably be the dominating error source which will limit the computability of a given problem if not taken into account. Examples are investigated to support the analysis provided: the Lorenz system and Van der Pol oscillator. Both examples provide support for the work conducted by displaying competing convergence rates for discretisation errors, decreasing rapidly for smaller time steps, and computational errors (round-off error), increasing for smaller time steps. The authors note that ``the effect of round-off errors is mostly pronounced for large values of the stability factor, which includes both chaotic dynamical systems as well as long-time integration of systems which exhibit only a moderate growth of the stability factor.''
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