Global error control with explicit peer methods
Step size control in the numerical solution of initial value
problems 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.
