Stability of Collocation at Gaussian Points

Abstract

Symmetric Runge–Kutta schemes are particularly useful for solving stiff two-point boundary value problems. Such A-stable schemes perform well in many cases, but it is demonstrated that in some instances the stronger property of algebraic stability is required. A characterization of symmetric, algebraically stable Runge–Kutta schemes is given. The class of schemes thus defined turns out not to be very rich: The only collocation schemes in it are those based on Gauss points, and other schemes in the class do not seem to offer any advantage over collocation at Gaussian points.