A scheme for the implementation of implicit Runge-Kutta methods
The application of an s-stage implicit Runge-Kutta method to a stiff ordinary differential equation (dimension n) leads to a nonlinear system in \({\mathbb{R}}^{s\cdot n}\). For its solution, an iterative scheme is proposed which requires only one LU-decomposition of a real \(n\times n\) matrix (also in the case where the Runge-Kutta matrix has different and/or complex eigenvalues). Compared to the iteration proposed by \textit{G. J. Cooper} and \textit{J. C. Butcher} [IMA J. Numer. Anal. 3, 127-140 (1983; Zbl 0525.65052)] less matrix vector multiplications (with a cost of \({\mathcal O}(s^ 2n)\) operations) are needed. For the Gauss methods of order 2s optimal values for the free parameters in the iteration are presented. The convergence rates are confirmed by numerical examples.
scientific article; zbMATH DE number 4199515