On implicity Runge-Kutta methods with a stability function having distinct real poles (Q1120963)

As distinct from the situation in serial machines, where singly implicit Runge-Kutta (RK) methods are computationally more advantageous than other implicit RK formulae, a parallel environment may suggest the use of RK methods whose stability function has pairwise distinct poles. These are called multiply implicit RKs (MIRKs). Here the attention is concentrated on MIRKs with real poles (real MIRKs). Complex poles would allow a higher classical order but not a higher stiff order. It is proved that a real MIRK with q stages and order \(q+1\) cannot be \(A\)-stable unless \(q=1,2,3,5\). The author construct for each \(q\) a \(q\)-stage, \(A_ 0\)-stable, order \(q+1\) real MIRK and a q-stage, I-stable, order \(q\) real MIRK. Algebraically stable real MIRKs are also given.