Order reduction phenomenon for general linear methods

Abstract

The order reduction phenomenon for general linear methods (GLMs) for stiff differential equations is investigated. It turns out that, similarly as for standard Runge–Kutta methods, the effective order of convergence for a large class of GLMs applied to stiff differential systems, is equal to the stage order of the method. In particular, it is demonstrated that the global error ‖e[n]‖ of GLMs of order p and stage order q applied to the Prothero–Robinson test problem y′(t)=λ(y(t)−φ(t))+φ′(t), t∈[t0,T], y(t0)=φ(t0), is O(hq)+O(hp) as h→0 and hλ→−∞. Moreover, for GLMs with Runge–Kutta stability which are A(0)-stable and for which the stability function R(z) of the underlying Runge–Kutta methods, (i.e., the corresponding RK methods which have the same absolute stability properties as the GLMs), is such that R(∞)≠1, the global error satisfies ‖e[n]‖=O(hq+1)+O(hp) as h→0 and hλ→−∞. These results are confirmed by numerical experiments.