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

By Franz L. Alt (Ed.)

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Although these eigenfunctions were used above, it will be seen in Section 11 that exhibiting thcm is not necessary for the genera1 method used. As mentioned earlier, Juncosa and Young [2,3,4] and Wasow [12] have studied this convergence problem using the explicit solutions of both the difference and the differential equations. The Crank-Nicolson method may be easily formulated for problems of a more general nature. 13) the difference equation becomes + a A z 2 ( W , n + l +W,n) = $(xi, t n + : , i ( W i .

The solution of tridiagonal linear equations is arithmetically simple ; if the algebraic equations are to be of this form, then only one of the space derivatives can be evaluated at the advanced time level. 1) for the boundary value problem. 1) subject t o vanishing boundary values, and the eigenfunctions would be vijn = pn sin P=+I -1 ~ p x sin i rqyj. 3) Now, if r is large, the magnitude of t,he stability ratio can be made large by t'aking q = M - 1 and p = 1 ; hence, unlimited stability does not result from (10,l).

As nAt I T. Thus, as Ax and At tend to zero, the error tends to zero in the grid LZnorm. Of course, this is not the usual integral L2topology; however, it can be shown [27] using interpolation to define the solution over the rectangle (0 5 x 5 1,0 5 t 5 T) that the integral Lz norm of the error is also O ( ( A X ) ~ ( A t ) z ) . Note that we have preserved the local accuracy in the global error. The above argument is a special case of this method of the author [27]. It will be shown later that the optimal relation between A,?

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