Directional approximation of the jacobians in ROW-methods

Abstract

Abstract.In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobians f~ are substituted by matrices A. satisfying a directional consistency condition Anf, = f'f. + O(h). In contrast to general W-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.

Some secant approximations for Rosenbrock W-methods

“…This is substantially the same approach proposed in [1], but since we are interested in creating reliable codes we must take into account of the stepsize selection that leads to variable stepsize methods. Therefore the update relation (6) has to be slightly modified in order to address this requirement.…”

“…The theoretical property given by (19) is fundamental to state the order conditions of the method, but the constant before the O(h) term can become very large as m increases. Moreover, if we investigate the order of the method, when h → 0 we have m = c/ h and hence W m = J (y m ) + O (1). For overcoming this problem, in practical implementation we adopt the restart of the methods.…”

“…Actually the idea is not new since it was firstly proposed in [1] where the authors used the so-called good Broyden's update [3] for approximating the Jacobian at each step, starting with a certain approximation of J (y 0 ). In this sense, we are particularly interested in stiff problems where a fixed approximation of the Jacobian (e.g.…”

“…W = J (y 0 )) does not represent a reliable approach. The first purpose of this paper is to extend this idea of [1] in order to make the update suitable for variable stepsize integration, that is necessary in order to create reliable codes based on the stepsize selection. The second one is to consider two other secant updates, the so-called bad Broyden's and Schubert's update.…”

“…Setting α 2 , α 3 , we get then b1 , b 2 , b 3 , b 4 from (R1 ), (R3a ), (W3 ), (R4a ). From (R4c ),(W4b )and (B3 ) we get β 32 , β 42 , β 43 .…”