The Effect of Changing the Stepsize in Linear Multistep Codes

In the usual convergence theory for linear multistep methods, a constant stepsize h is used. For a method of order p, the discretization error is proportional to $h^{p + 1} $. In a variable step code, it is necessary to predict what the discretization error would be if the stepsize were changed to $rh$. It is usual to say that the observed error will be altered by a factor of $r^{p + 1} $. Unfortunately this is not correct for multistep methods. The discrepancy arises in the fact that the usual theory does not model the way variable stepsize codes actually work. In this paper the correct behavior is determined for important classes of formulas and ways of changing stepsize.