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

By Franz L. Alt (Ed.)

ISBN-10: 0120121026

ISBN-13: 9780120121021

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Extra resources for Advances in Computers, Vol. 2

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For convenience in the discussion let Ax = Ay. The time step may be taken t o be O(Ax) so that the time and distance truncation errors are in balance, whercas this balancing is not possible with the forward difference equation; consequently, many fewer time steps can be taken with the backward equation without affecting the order of the error. A serious algebraic difficulty arises from the use of the backward difference equation. The linear equations are no longer tridiagonal and elimination methods require an excessive amount of calculation if the mesh size is small.

They do not show that the alternating direction methods fail for such problems; indeed, the methods are very useful for the more complex problems. 11) on an arbitrary domain. At the moment his results are based on fixed r. 6) and those of the methods of Douglas and Rachford [21] for both two and three space variables have been treated. 12) v. (4x1 Y ) V d = ", If t appears in the coefficients, the evaluation should be at time t,+i to preEerve second order correctness in time. The forms of other generalizations are evident.

Moreover, these norms arc uniformly comparable with the usual LZnorm, and p = 0. 27) assuming At/Ax to be held fixed. 23) and to other difference equations, including higher order correct equations in some cases. IICkllb 44 JIM DOUGLAS, JR. We see that stability and consistency are sufficient for convergence; Lax and Richtmyer [46, 471 have shown that the converse is essentially true, at least for properly posed pure initial value problems. They have demonstrated that if the solution of the difference equation converges to the solution of the differential equation for every initial condition in a reasonable class (Lee,dense in L2), then the difference equation must be stable.