Runge–Kutta Defect Control Using Hermite–Birkhoff Interpolation

Two techniques for reliably controlling the defect (residual) in the numerical solution of nonstifi initial value problems were given in [D. J. Higham, SIAM J. Numer. Anal., 26(1989), pp. 1175–1183]. This work describes an alternative approach based on Hermite–Birkhofi interpolation. The new approach has two main advantages—it is applicable to Runge–Kutta schemes of any order, and it gives rise to a defect of the optimum asymptotic order of accuracy. For a particular Runge–Kutta formula the asymptotic analysis is verified numerically.