Some linear stability results for iterative schemes for implicit Runge-Kutta methods

This article examines stability properties of some linear iterative schemes that have been proposed for the solution of the nonlinear algebraic equations arising in the use of implicit Runge-Kutta methods to solve a differential system x' = f(x). Each iteration step requires the solution of a set of linear equations, with constant matrix I - h lambda J, where J is the Jacobian of f evaluated at some fixed point. It is shown that the stability properties of a Runge-Kutta method can be preserved only if lambda is an eigenvalue of the coefficient matrix A. Suppose A has minimal polynomial (x-lambda)(m)p(x), p(lambda) not equal 0. Then stability can be preserved only if the order of the method is at most m + 2 (at most m + 1 except for one case).