Read e-book online Advances in Computers, Vol. 2 PDF

By Franz L. Alt (Ed.)

ISBN-10: 0120121026

ISBN-13: 9780120121021

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I n the case of the heat equation only four arithmetic operations per grid point are necessary to evaluate Z O ~ , ~ + ~ , after pi is obtained. Note that this is one less than that required to evaluate the forward difference formula. It has been shown [13] that this method of solving the linear equations does not introduce large errors in the solution due to round-off errors in the calculation. Cuthill and Varga [14] have proposed a different algorithm which is slightly more efficient for rclf-adjoint equations with time independent coefficients; however, if the coefficients vary with time, the calculation required is greater than that required by the above algorithm.

It is also possible to increase local accuracy by using more time levels; difference equations involving several time levels will be discussed later in this section. Assume u E C6. The object for the moment is to derive a difference equation in which only the values of the solution at the six points appearing in the Crank-Nicolson equation occur and which is fourth order correct in the space coordinate and second order in time. As (Wi,n+l - w i n ) / A t is already second order correct in time at (z<, t n + + ) , it will be retained.

Only the right-hand side is affected; consequently, the same elimination method may be applied. Indeed, a considerable increase in accuracy has been obtained for a very small increase in computing. The comparative aspect of the computing requirements will be discussed later. Strang [28] has also treated the above convergence problem by different methods. His proof is based on matrix manipulations requiring explicit knowledge of the eigenfunctions of the difference operator. Although these eigenfunctions were used above, it will be seen in Section 11 that exhibiting thcm is not necessary for the genera1 method used.

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

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