Download e-book for iPad: Advances in Computers, Vol. 2 by Franz L. Alt (Ed.)

By Franz L. Alt (Ed.)

ISBN-10: 0120121026

ISBN-13: 9780120121021

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There are examples [48] of convergent, but unstable, difference analogues of the heat equation subject to special initial conditions; however, the result of Lax and Richtmyer shows that this cannot be expected to occur generally. Even in the case of specia1:data where the solution of an unstable difference equation is convergent, round-off error would grow sufficiently rapidly to invalidate the computed results. The proof that convergence implies stability is a Banach space argument based on the principle of uniform boundedness.

34)is used, as the author has observed each time he has forgotten this simple example. It should be mentioned that the eigenfunctions used in the stability analysis change when the slope is specified. Frequently, cosines instead of sines appear. 6. An Unconditionally Unstable Difference Equation A seemingly simpler way to produce second order correctness in time than that of the Crank-Nicolson equation is to use a centered first difference for ut, while replacing uzzby a difference at the middle time.

11) obviously generalizes. The analysis proceeds in the same manner for a rectangular region and then extends to more general regions essentially like that for the Crank-Nicolson equation. 21) by differences. One such replacement leads to the difference equation [34] - 12 [At2wijn - 4A22Au2~ijn]. 22) The equation is fourth order correct in space and second order in time; a simple but uninteresting argument shows unconditional Lzstability for the boundary value problem on a rectangle. 4) does not.

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Advances in Computers, Vol. 2 by Franz L. Alt (Ed.)

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