Directional approximation of the Jacobians in row-methods (Q2640326)

This paper deals with Rosenbrock-type methods. Attempts are made in the literature [see \textit{T. Steihaug} and \textit{A. Wolfbrandt}, Math. Comp. 33, 521-534 (1979; Zbl 0451.65055)] to replace the Jacobian \(f'_ n\) by a suitable approximation. A new approximation \(A_ n\) satisfying the condition \(A_ nf_ n=f'_ nf_ n+O(h)\) is proposed with the effect that the number of order conditions is reduced. The main results are 2- stage methods of order 3 and families of imbedded 4-stage methods of order 4(3). Numerical tests are enclosed.