Error estimation and control in O.D.E. integration

The global error at any point in the step by step numerical solution of an initial value problem is the sum of local errors introduced at every step, each multiplied by an amplification factor due to the stability or instability of the differential equation. If these factors were unity (as, for example, in a quadrature problem) it is possible to keep the global error e(t) over the interval [a, b] less than Δ by keeping the local error in a step of length h less than hΔ/(b-a). This is called error per unit step. Unfortunately, theory shows that the error strategy which minimizes the amount of work for a given global error Δ is to make the local error equal to g(Δ) where, for a pth order method, g(Δ) = q(t)Δ(p+1)/P+ higher terms, and q(t) depends on the differential equation. The talk will present experimental evidence of the increased efficiency of error per step versus the smoother dependence of the global error on the error control parameter for error per unit step. The existing theory will be summarized, and the question of how the local error estimate can be justified theoretically will be examined.