By Franz L. Alt (Ed.)
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19) holds. The exponent p will usually be nonpositive. If s p > 0, the convergence in the desired norm has been established. The above derivation  is a natural correction of the incomplete argument used by the author in 1958 , which was a n attcmpted extension of earlier results of Lax and Richtmycr [46,47] and of the author . The multilevel difference equation can bc treated by a similar argument P71. 23) 4 2 , t ) , b(z, t ) 2 m > 0. (a($, t ) = b(z, t ) - * .. 25 It follows [30 (properly interpreted), 451 from the Courant Minimax Principle that 51.
The choice of y and the value of the constant should be consistent with stability requirements, if any. 4) where the multiplier on the right-hand side has been ignored. If the allowable error is E and constants are again ignored, (At)" + (Ax)fl = (Ax)*Y + ( A Z ) ~5 e. If a y < 0, then (Ax)". 5) and Ax tend to zero, and = cl/ay. Thus, C = e-(l+T)/w, For p and < p. 10) should be minimized as a function of y. 1I) and that emin = 1 a! 13) Thus, any increase in the global accuracy asymptotically reduces the total calculation if the order of t,he number of calculations per time step is not increased.
Stability for polygonal regions with sides parallel to the axes follows from the known distribution of eigenvalues of the Laplace difference operator ; consequently, the convergence result holds for the boundary value problem in such a region if u € C4. In the curvilinear case the local error becomes O(Ax (At)z) for either of the two methods of handling the boundary. It can be shown on the basis of the minimax principle  that stability holds when irregular boundary points are introduced, and the stability for the method of transferring data follows from the remarks above.