Automatic step size and order control in implicit one-step extrapolation methods

A theory is presented for implicit one-step extrapolation methods for ordinary differential equations. The computational schemes used in such methods are based on the implicit Runge–Kutta methods. An efficient implementation of implicit extrapolation is based on the combined step size and order control. The emphasis is placed on calculating and controlling the global error of the numerical solution. The aim is to achieve the user-prescribed accuracy in an automatic mode (ignoring round-off errors). All the theoretical conclusions of this paper are supported by the numerical results obtained for test problems.