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.
|Titolo:||Order reduction phenomenon for general linear methods|
|Data di pubblicazione:||2017|
|Appare nelle tipologie:||1.1.2 Articolo su rivista con ISSN|