Comparing numerical methods for stiff systems of O.D.E:s

Abstract

Abstract.This paper describes a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations. The basis of a fair comparison is discussed in detail. Measurements of cost and reliability are made over a collection of 25 carefully selected problems. The problems have been designed to show how certain major factors affect the performance of a method.The technique is applied to five methods, of which three turn out to be quite good, including one based on backward differentiation formulas, another on second derivative formulas, aaud a third on extrapolation. However, each of the three has a weakness of its own, which can be identified with particular problem characteristics.

Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

“…Our second example is given by the van der Pol equation (4.2) y" -fi(X -y2)y'+ y = 0. For ß = 5, this is problem E2 from the test set of Enright et al [8]. However, as reported there, on the interval [0,1] the spectral radius of the Jacobian does not exceed 15, so that the problem is not really stiff.…”

Implementation of Rosenbrock Methods

“…Also, the conversion changes a linear to a nonlinear problem. It is interesting to note that the fa:nous set of test problems [8] did precisely this with the Liniger-Willoughby problem Dl.…”

“…The co:ie is DSGRK with the pair given in section 6 replaced by GRK4A. The B family of prob.lem.s in tho= test set [8] Ha•rine evalua:te::l f, fy' fx' a··~ the initial p~?inl:i, we are in a positio::~ to ta:-ce a. "virtual" step rv.lth a Taylor series ·r(l,2) pair: .…”

An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations

“…They have been compared with Gear's methods and with Enright's second derivative multistep methods, using, respectively, the routines EPISODE [2] and SECDER [1]. The comparison has been carried out on some significant test problems picked out from those proposed by Hull [5], [7] and Krogh [8]. These problems, listed in the Appendix, have been classified in the following classes:…”