On the existence of solution of stage equations in implicit Runge-Kutta methods

This paper is concerned with the unique solvability of stage equations which arise when implicit Runge-Kutta methods apply to nonlinear stiff systems of differential equations y' = f(t, y). Denoting by A the matrix of coefficients of the Runge-Kutta method and by mu(2)[J] the logarithmic norm of the matrix J associated with the l(2)-norm, several authors (Crouzeix et al., BIT 23 (1983) 84-91; Hundsdorfer and Spijker, SIAM J. Numer. Anal. 24 (1987) 583-594; Kraaijevanger and Schneid, Numer. Math. 59 (1991) 129-157; Liu and Kraaijevanger, BIT 28(4) (1988) 825-838) have obtained conditions on A that ensure, for a given a, the unique solvability of stage equations for all stepsize h and stiff system with h mu(2)[f'(t, y)] < lambda, where f'(t, y) is the jacobian matrix of f with respect to y. The aim of this paper is to study the unique solvability of stage equations in the frame of the l(infinity)- and l(1)-norms. For a given real lambda it will be proved that the condition mu(infinity)[(lambda I - A(-1))D] < 0, for some positive-definite diagonal matrix D, implies that the stage equations are uniquely solvable for all stepsize h and function f such that h mu(infinity)[f'(t, y)] less than or equal to lambda. Further, it is shown that these conditions also imply the BSI-stability i.e. the stability of stage equations under non uniform pertubations. Applications to some well-known families of Runge-Kutta methods are included. (C) 1999 Elsevier Science B.V. All rights reserved.