Analysis Of The Dynamics Of Local Error Control Via A Piecewise Continuous Residual

. Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for certain minor modifications of some well-known step-size selection algorithms. Using the smallness of the residual, it follows that at any finite time, the continuous interpolant converges to the true solution as the error tolerance tends to zero. Furthermore, by studying the perturbed differential equation it is possible to prove discrete analogs of the long-time dynamical properties of the equation---dissipative, contractive and gradient systems are analysed in this way. Such results establish forms of nonlinear stabili..