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. 1. Background, results, significance 1.1. Prototype stiff problems. Their salient properties. Note the structure of the solution of the model differential equation (1.1) y = Xy + cos t, A<c-1, namely, y(t) = eXt (y(0) + —?—) + —^(sin t ¿cost). V l+k1) l+k2 There is an initial transient of duration 0(|/l_1|log|A|), after which the term e ' is not active and the solution is as smooth as cos t. Under suitable conditions, see [29] and infra, solutions of the stiff timevarying linear system (1.2) y' = B(t)y + g(t), yeRN, have the same structure: y(t) is the sum of a smooth particular solution ys(t) and a transient v(t). The transient, a solution of the homogeneous equation v'(t) = B(t)v(t), v(0) = y(0)-ys(0), expires after a short time. Meanwhile, ys(t) and its derivative have bounds expressible in terms of (1.3) B~l{t)77~' B~l{t)W' I/ = 0,1' Received September 28, 1989; revised August 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65L05, 65H10.