A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

TL;DR

This paper develops sharp, computable error estimates for numerical solutions of ODEs, explicitly accounting for round-off errors, and demonstrates how finite precision limits the accuracy of long-term computations.

Contribution

It introduces a new error estimate that includes round-off error propagation, extending previous models and highlighting the impact of numeric precision on solution accuracy.

Findings

Round-off errors grow inversely with the square root of step size.

Numeric precision limits the achievable accuracy in long-term ODE solutions.

Theoretical estimates are validated with numerical experiments on Lorenz and van der Pol systems.

Abstract

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.