Algebraic stability and the existence of solutions of implicit Runge- Kutta equations (Q1069676)

It is shown that an irreducible Runge-Kutta method whose Runge-Kutta matrix, A, is nonsingular is (k,\(\ell)\)-algebraically stable on (- \(\infty,a)\), \(a>0\) with \(k(0)=1\) iff the method is algebraically stable and there exists a diagonal matrix \(R>0\) such that \(RA+A^ TR>0\). This last condition guarantees the existence of unique solutions for the internal approximations when a Runge-Kutta method is applied to a monotonic problem.