Experiments with Quasi-Newton Methods in Solving Stiff ODE Systems

Abstract

A nonlinear algebraic system must be solved at each step of the integration of a stiff system of ordinary differential equations by methods based on backward differentiation formulas. Quasi-Newton methods are of potential benefit in solving these algebraic problems. Three types of quasi-Newton methods are studied for this purpose—Doolittle LU updates, and Broyden's first and second methods performed implicitly. Detailed algorithms are given. Tests on some large stiff systems show that significant benefits can be obtained for some problems.