Some linear stability results for iterative schemes for implicit Runge-Kutta methods
This paper is concerned with the solution of the algebraic equations which arise in implicit Runge-Kutta methods by means of quasi-Newton type methods. Since implicit methods are used to satisfy the stability requiremets imposed by stiff problems, the author analyzes whether these stability requirements on the method are preserved by some iterative schemes usually employed in the solution of the nonlinear algebraic equations after a fixed number of iterations. In this context, after defining the stability preservation, two main results on this property have been obtained depending on the starting value of the iteration. In the first one, assuming arbitrary starting values, it is proved that some conditions on a parameter of the iteration process, the matrix of coefficients of the Runge-Kutta method and the order of the method are necessary in order to preserve the stability. In the second one a similar result is proved in the case of a fixed starting value. Finally, as remarked by the author, these stability preservations restrict severely the class of Runge-Kutta methods that can preserve the stability, but it must be observed that in practice the starting values of the iteration are usually obtained by using back information of the integration process and therefore they are neither arbitrary nor fixed for all problems.
scientific article; zbMATH DE number 881184