A row-type method for stiff differential equations (Q1901528)

A modified Rosenbrock method of order three is proposed for the numerical solution of stiff differential systems. The methods under consideration include in the usual Rosenbrock scheme a semiexplicit additional term which requires the solution of a linear system with the same matrix of coefficients used in stages of the current step. This term introduces new free parameters into the formula which can be chosen to improve the accuracy and/or stability of the method. It is shown that with three steps, it is possible to construct a family of \(A\)-stable formulas of order three. Then the available parameters are chosen so that the local truncation error is minimized and an optimal formula is given. Finally, some statistics comparing the new formula with a third-order formula of \textit{P. Kaps} and \textit{P. Rentrop} [Numer. Math. 33, 55-68 (1979; Zbl 0436.65047)] are presented.