THE MODIFIED NEWTON METHOD IN THE SOLUTION OF STIFF ORDINARY DIFFERENTIAL-EQUATIONS

This paper presents an analysis of the modified Newton method as it is used in codes implementing implicit formulae for integrating stiff ordinary differential equations. We prove that near a smooth solution of the differential system, when the Jacobian is essentially negative dominant and slowly varying, the modified Newton iteration is contractive, converging to the locally unique solution-whose existence is hereby demonstrated-of the implicit equations. This analysis eliminates several common restrictive or unrealistic assumptions, and provides insight for the design of robust codes.