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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The optimal accuracy of difference schemes


Authors: Arieh Iserles and Gilbert Strang
Journal: Trans. Amer. Math. Soc. 277 (1983), 779-803
MSC: Primary 65M10; Secondary 41A21
MathSciNet review: 694388
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Abstract: We consider difference approximations to the model hyperbolic equation $ {u_{t}} = {u_x}$ which compute each new value $ U(x,t + \Delta t)$ as a combination of the known values $ U(x - r\Delta x,t),\ldots,U(x + s\Delta x,\Delta t)$. For such schemes we find the optimal order of accuracy: stability is possible for small $ \Delta t/\Delta x$ if and only if $ p \leqslant \min \{ {r + s,2r + 2,2s} \}$. A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations $ P(z)/Q(z)$ to $ {z^\lambda }$ near $ z = 1$, and we find an expression for the difference $ \vert Q{\vert^2} - \vert P{\vert^2}$; this allows us to test the von Neumann condition $ \vert P/Q\vert \leqslant 1$. We also determine the number of zeros of $ Q$ in the unit circle, which decides whether the implicit part is uniformly invertible.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0694388-9
PII: S 0002-9947(1983)0694388-9
Article copyright: © Copyright 1983 American Mathematical Society