Some secant approximations for Rosenbrock W-methods

Here we present a class of W-methods for stiff ODEs based on some special approximations of the Jacobian matrices that allow to reduce the number of order conditions. The approximations are considered modifications of some rank-1 updates, known from quasi-Newton methods. Two new embedded W-methods of order 3 and 4 are constructed and tested on some classical stiff equations arising from the semidiscretization of parabolic problems.