Methods for the integration of initial value problems for the ordinary differential equation-^-= f(x,y) which are a combination of one step procedures (eg., Runge-Kutta) and multistep procedures (eg., Adams' Method) (3) are discussed. A generalization of a theorem from Henrici proves that these methods converge under suitable conditions of stability and consistency. This, incidentally, is also a proof that predictor-corrector methods using a finite number of iterations converge. Four specific methods of order k,6,ti and 10 have been found. Numerical comparisons of the first three of these have been made with Adams', Nordsieck's and the Runge-Kutta methods.

T he best available descriptions of systems often come at a fine level (atomistic, stochastic, microscopic, agent based), whereas the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization, and control) are at a much coarser, macroscopic level. Traditional modeling approaches start by deriving macroscopic evolution equations from microscopic models, and then bringing an arsenal of computational tools to bear on these macroscopic descriptions. Over the last few years with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the "equation-free" approach, since it circumvents the step of obtaining accurate macroscopic descriptions. The backbone of this approach is the design of computational "experiments". In traditional numerical analysis, the main code "pings" a subroutine containing the model, and uses the returned information (time derivatives, etc.) to perform computer-assisted analysis. In our approach the same main code "pings" a subroutine that runs an ensemble of appropriately initialized computational experiments from which the same quantities are estimated. Traditional continuum numerical algorithms can, thus, be viewed as protocols for experimental design (where "experiment" means a computational experiment set up, and performed with a model at a different level of description). Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation -through matrix-free numerical analysis, and systems theory tools like estimation-bridge microscopic simulation with macroscopic modeling. If enough control authority exists to initialize laboratory experiments "at will" this computational enabling technology can lead to experimental protocols for the equation-free exploration of complex system dynamics. The Equation-Free ApproachA persistent feature of many complex systems is the emergence of macroscopic, coherent behavior from the interactions of microscopic agents such as molecules, cells, or individuals in a population. The implication is that macroscopic rules (a description of the system at a coarse-grained, high level) can somehow be deduced from microscopic ones (a description at a much finer level). For laminar Newtonian fluid mechanics, a successful coarse-grained description (the Navier-Stokes equations) was known on a phenomenological basis long before its approximate derivation from kinetic theory. Today, we must frequently study systems for which the physics can be modeled at a microscopic, fine scale; yet, it is practically impossible to derive a good macroscopic description from the microscopic rules. Hence, we look to the computer to explore the macroscopic behavior, based on the microscopic description.

An integration technique for the automatic solution of an initial value problem for a set of ordinary differential equations is described. A criterion for the selection of the order of approximation is proposed. The objective of the criterion is to increase the step size so as to reduce solution time. An option permits the solution of "stiff" differential equations. A program embodying the techniques discussed appears in Algorithm 407.

We show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters, one containing the "stiff," or fast, components, and one containing the slow components. These methods have large average step sizes relative to the fast components. Conventional implicit methods involve the solution of nonlinear equations at each step, which for large problems requires significant communication between processors on a multiprocessor machine. For such problems the methods proposed here have significant potential for speed improvement.

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