The effect of the stopping of the Newton iteration in implicit linear multistep methods Original Research Article

This paper concerns the application of implicit, linear multistep methods in the numerical solution of stiff, nonlinear initial value problems. Liniger (1971) studied the (modified) Newton iteration for coping with the nonlinear equations connected with the implicitness of the multistep methods. He gave formulas for the order of the corresponding Newton stopping errors. Dorsselaer and Spijker (1994) showed, for a class of strongly nonlinear initial value problems, that estimates for the Newton stopping errors are in force with an order that is lower than that of Liniger. In the present paper we find that the actual effect of the stopping of the Newton iteration differs remarkably from what one might expect in view of the paper by Dorsselaer and Spijker: This effect turns out to be of a higher order than suggested by a naive application of the orders of Dorsselaer and Spijker. We find that Linigerʹs order is reliable for stiff problems that are mildly nonlinear. Moreover, for stiff problems that are strongly nonlinear, the order of the accumulated effect of all Newton stopping errors turns out to be greater (by one) than the order which one may expect.