Global error control with explicit peer methods
27/03/2014 Thursday 27th March 2014, 14:00 (Room P3.10, Mathematics Building)
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Ruediger Weiner, Institut für Mathematik, University of Halle, Germany
Step size control in the numerical solution of initial value problems
\[y'=f(t,y),\quad y(t_0)=y_0\]
is usually based on the control of the local error.
We present numerical tests showing that this
may lead to high global errors, i.e. the real error is much larger than the prescribed tolerance.
There is a tolerance proportionality, with more stringent tolerances also the global error is reduced. However,
tolerance and achieved global error may differ by several magnitudes.
A very simple idea to overcome this problem is to use two methods of different orders with same step size sequences and local error
control for the lower order method. Then the difference of the numerical approximations of both methods
is an estimate of the global error of the lower order method.
This strategy was implemented for pairs of explicit peer methods in Matlab . Numerical tests show the reliability of this
approach. The numerical costs are comparable with those of ode45, but in contrast to ode45
the required accuracy is achieved.
