Automatic detection and treatment of oscillatory and/or stiff ordinary differential equations

Abstract

The next generation of ODE software can be expected to detect special problems and to adapt to their needs. This paper is principally concerned with the low-cost, automatic detection of oscillatory behavior, the determination of its period, and methods for its subsequent efficient integration. It also discusses stiffness detection. In the first phase, the method for oscillatory problems discussed examines the output of any integrator to determine if the output is nearly periodic. At the point this answer is positive, the second phase is entered and an automatic, nonstiff, multirevolutionary method is invoked. This requires the occasional solution of a nearly periodic initial-value problem over one period by a standard method and the re-determination of its period. Because the multirevolutionary method uses a very large step, the problem has a high probability of being stiff in this second phase. Hence, it is important to detect if stiffness is present so an appropriate stiff, multirevolutionary method can be selected. Stiffness detection uses techniques proposed by a number of authors. The same technique can be used to switch to a standard stiff method if necessary for a non-oscillatory problem, in the first phase of an oscillatory problem, or in the standard integration over one period of an oscillatory problem.