Probabilistic and deterministic convergence proofs for software for initial value problems (Q1370361)

For the numerical solution of the initial value problem a system of differential equations, an adaptive algorithm based on two explicit Runge-Kutta methods is constructed. Several basic results concerning this adaptive algorithm are proved. The main theorem is concerned with the convergence of the algorithm. This theorem is used to prove that, with probability one, the adaptive algorithm converges on a general class of nonlinear problems. In the last section a modified version of the algorithm is presented which allows the global error to be controlled.