Global error control with explicit peer methods

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.