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 μ2[J] the logarithmic norm of the matrix J associated with the ℓ2-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 λ, the unique solvability of stage equations for all stepsize h and stiff system with hμ2[f′(t,y)]<λ, 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∞- and ℓ1-norms. For a given real λ it will be proved that the condition μ∞[(λI−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 hμ∞[f′(t,y)]⩽λ. Further, it is shown that these conditions also imply the BSI-stability i.e. the stability of stage equations under non uniform perturbations. Applications to some well-known families of Runge–Kutta methods are included.