Using a relative error test we would control the

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Unformatted text preview: the i th ODE does not exceed R jyn ij + A 50 may be speci ed by using the maximum norm and selecting = max( A and R) wi = ( R jyn ij + A )= : Present Runge-Kutta software controls: 1. the local error ~ kdnkw (3.5.8a) ~ kdnkw h (3.5.8b) 2. the local error per unit step 3. or the indirect (extrapolated) local error per unit step ~ kdnkw C h (3.5.8c) where C is a constant depending on the method. The latter two formulas are attempts to control a measure of the global error. Let us describe a step size selection process for controlling the local error per unit step in a p th order Runge-Kutta method. Suppose that we have just completed an ~ integration from tn 1 to tn. We have computed an estimate of the local error dn using ~ either Richardson's extrapolation or order embedding. We compare kdnkw with the prescribed tolerance and ; ~ 1. if kdnkw > we reject the step and repeat the integration with a smaller step size, 2. otherwise we accept the step and suggest a step size for the subsequent step. In either case, ~ kdnk...
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