Starting BDF and Adams codes at optimal order

We present an algorithm based on Runge—Kutta formulas with interpolation to start BDF and Adams multistep codes at optimal order and step size. It first finds automatically and reliably an ‘on scale’ initial step size. Besides being convenient for the user, this results in a more robust and efficient start, especially for stiff problems, because a starting step is taken with an efficient RK formula of moderate order before switching to the multistep formula. Starting the multistep formula at optimal order and step size makes the integration more efficient and reduces difficulties due to mesh distortion. We present the results of a variety of numerical experiments which demonstrate the efficacy of our technique.