An embedded 4(3) Runge-Kutta pair

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

An embedded 4(3) Runge-Kutta pair, whose lower formula has order p=3p = 3, is stepping a comet's orbit near perihelion with current step size h=0.2 yrh = 0.2\,\mathrm{yr}. The embedded estimate of the local error over that step is err=2.5×10−6 AU\mathrm{err} = 2.5\times10^{-6}\,\mathrm{AU}, and the required local tolerance is tol=1×10−8 AU\mathrm{tol} = 1\times10^{-8}\,\mathrm{AU}. Using the step-size rule hnew=0.9h(tol/err)1/(p+1)h_{\rm new} = 0.9h\left(\mathrm{tol}/\mathrm{err}\right)^{1/(p+1)}, find the new step the controller should propose, and state whether the current step should be accepted or rejected.

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