TY - JOUR

T1 - Order reduction phenomenon for general linear methods

AU - Braś, Michał

AU - Cardone, Angelamaria

AU - Jackiewicz, Zdzislaw

AU - Welfert, Bruno

N1 - Funding Information:
The authors wish to express their gratitude to Willem Hundsdorfer for pointing out the reference [23]. The work of the first author (M. Braś) was supported by the National Science Center under grant DEC-2011/01/N/ST1/02672 and the Polish Ministry of Science and Higher Education. The work of second author (A. Cardone) was supported by GNCS-INdAM.
Publisher Copyright:
© 2017 IMACS

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

KW - General linear methods

KW - Linear stability analysis

KW - Order conditions

KW - Order reduction phenomenon

KW - Prothero–Robinson problem

KW - Stiff differential systems

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U2 - 10.1016/j.apnum.2017.04.001

DO - 10.1016/j.apnum.2017.04.001

M3 - Article

AN - SCOPUS:85017499439

SN - 0168-9274

VL - 119

SP - 94

EP - 114

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -