Stability of the Richardson Extrapolation applied together with the θ-method

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:235

Article

Zlatev, Zahari¹  Farago, Istvan²  Havasi, Agnes³ 

[1] Aarhus Univ, Natl Environm Res Inst, DK-4000 Roskilde, Denmark

[2] Eotvos Lorand Univ, Dept Appl Anal & Computat Math, H-1117 Budapest, Hungary

[3] Eotvos Lorand Univ, Dept Meteorol, H-1117 Budapest, Hungary

关键词: Systems of ordinary differential equations;    Numerical methods;    Richardson Extrapolation;    Stability;    Atmospheric chemistry scheme;   

DOI  :  10.1016/j.cam.2010.05.052

来源: Elsevier

【 摘 要 】

Consider a system of ordinary differential equations (ODEs) dy/dt = f(t, y) where (a) t is an element of [a, b] with b > a, (b) y is a vector containing s components and (c) y(a) is given. The theta-method is applied to solve approximately the system of ODEs on a set of prescribed grid points. If N is the number of time steps that are to be carried out, then this numerical method can be defined using the following set of relationships, y(n) = y(n-1) + h(1 - theta)f + (t(n-1), y(n-1)) + h theta f(t(n), y(n)), theta is an element of [0.5, 1 0], n = 1, 2, , N, h = (b - a)/N, t(n) = t(n-1) + h = t(0) + nh, t(0) = a, t(N) = b As a rule, the accuracy of approximations (y vertical bar n = 1, 2, N] can be improved by applying the Richardson Extrapolation under the assumption that the stability of the computational process is preserved. Therefore, it is natural to require that the combined numerical method (Richardson Extrapolation + the theta-method) is in some sense stable It is proved in this paper that the combined method is strongly A-stable when theta is an element of [2/3, 1 0] It is furthermore shown that some theorems proved in a previous paper by the same authors, Farago et al. (2009)[1], are simple corollaries of the main result obtained in the present work. The usefulness of the main result in the solution of many problems arising in different scientific and engineering areas is demonstrated by performing a series of tests with an extremely badly scaled and very stiff atmospheric chemistry scheme which is actually used in several well-known large-scale air pollution models. (C) 2010 Elsevier B V All rights reserved.

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