Control of local error stabilizes integrations

Codes for the solution of the initial value problem for a system of ordinary differential equations adjust their step sizes so as to keep the estimated local errors smaller than a given tolerance and, at the same time, solve the problem as efficiently as possible. Step sizes providing the desired accuracy may be too large for stability, yet this does not cause the popular codes to become unstable. The theory explaining why this is so holds only in very restricted circumstances. This paper broadens and deepens the theory for explicit Runge-Kutta methods.