Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 65M10, 41A21

Retrieve articles in all journals with MSC: 65M10, 41A21

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society