On the Practical Value of the Notion of Bn-Stability

摘要

A standard implementation of diagonally implicit Runge-Kutta methods, having equal diagonal elements, is discussed. Under what circumstances unconditional nonlinear stability can be realized in practice is questioned. The oldest concept of unconditional stability of numerical integration methods for ordinary differential systems is that of A-stability. This concept is related to linear systems having constant coefficients. There has been much interest in unconditional stability properties of numerical integration methods when applied to nonlinear dissipative systems (G-stability, BN-stability, A-contractivity). Various classes of implicit Runge-Kutta methods have already been shown to be BN-stable. However, contrary to the property of A-stability, when implementing such a method for practical use this unconditional stability property may be lost. This is clarified for the class of diagonally implicit methods and it is shown at the same time that Rosenbrock's method is not BN-stable.