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 systemx =f(x). Each iteration step requires the solution of a set of linear equations, with constant matrixI –hJ, whereJ is the Jacobian off evaluated at some fixed point. It is shown that the stability properties of a Runge-Kutta method can be preserved only if is an eigenvalue of the coefficient matrixA. SupposeA has minimal polynomial (x – ) ^m p(x),p() 0. Then stability can be preserved only if the order of the method is at mostm + 2 (at mostm + 1 except for one case).This work was partially supported by a grant from the Science and Engineering Research Council.