Abstract

An ordinary differential equation (ODE) can be split into simpler sub equations and each of the sub equations is solved subsequently by a numerical method. Such a procedure involves splitting error and numerical error caused by the time stepping methods applied to sub equations. The aim of the paper is to present an integral formula for the global error expansion of a splitting procedure combined with any numerical ODE solver.

Highlights

  • MotivationThe local error (le) of a numerical method un+1 = R∆t(un) with step size ∆t for the initial value problem du dt

  • We propose to approximate the global error in terms of the local errors and the discrete flow by a Riemann integral

  • The interaction of the errors caused by splitting procedure and time stepping methods applied to sub problems should be considered because the interaction might lead to order reduction in the long time run

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Summary

Motivation

The local error (le) of a numerical method un+1 = R∆t(un) with step size ∆t for the initial value problem du dt. Consider the initial value problem y′ = f (t, y), y(t0) = y0,. When a small perturbation is introduced to the initial value y0, for the perturbed solution y(t), the error e(t) = y(t)−y(t), evolves with [3,8]. (7) where Ψ(t) is the solution of variational equation of the corresponding initial value problem. (8) fore the first order difference equation for global error is given by en+1 =enΨ(tn+1)Ψ−1(tn) + ∆tr+1le(u(tn)). Assuming ei = 0, the solution of the linear difference equation is f −1 en =∆tr+1Ψ(tn) Ψ−1(tk+1)(le(u(tk)). That predicts the global error at t = tf in terms of initial value ui at t = ti and step size ∆t

Global error of Lie Trotter Splitting
Numerical Example
Remarks and Conclusion

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