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 discretization 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.
Highlights
IntroductionStandard estimates may include various sources contributing to the global error, such as discretization errors, accounting for the use of finite time steps, quadrature errors, accounting for the approximation of the right-hand side f by a particular quadrature rule, and data errors, accounting for the approximation of the initial value u0
We consider the numerical solution of general initial value problems for systems of ordinary differential equations (ODE), u(t) = f (u(t), t), t ∈ (0, T ], u(0) = u0, (1)
For the numerical results presented at the end of this work, we have used a particular time-stepping method formulated as a Galerkin finite element method, which, for any particular choice of finite element basis and quadrature, will correspond to a particular implicit Runge–Kutta method
Summary
Standard estimates may include various sources contributing to the global error, such as discretization errors, accounting for the use of finite time steps, quadrature errors, accounting for the approximation of the right-hand side f by a particular quadrature rule, and data errors, accounting for the approximation of the initial value u0. We extend these 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, since it is typically much smaller than the contribution from the data or discretization error. When the system (1) is very sensitive to perturbations, when the time interval [0, T ] is very long, or when a solution is sought with very high accuracy, the effect of numerical round-off errors as a result of finite numeric precision can and will be the dominating error source, which limits the computability of a given problem
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