Step Size Adjustment at Discontinuities for Fourth Order Runge-Kutta Methods

With discontinuities in the differential equations of an initial value problem it is in general necessary to alter the effective step size to evaluate the variables at the point of discontinuity. Since this latter point is often determined by the values of some or all of the dependent variables it would normally be observed by a checking procedure just prior to updating these dependent variables. A method of calculating the fraction of the interval to the point of discontinuity and of updating the values of the dependent variables to this point is given. The method which will be subsequently referred to as the Alpha Method is third order, and no new evaluation of the function is required. It compares favourably both for computational time and programming with the normal method of continuing the tabulation beyond the discontinuity and employing inverse interpolation.