# 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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**Example text**

Perhaps the simplest way would be to add the point (z*, yj) to the lattice and assign the given boundary dsta there; also, it is feasible to transfer the data from (z*, Yj) to either (xi, gi) or (zi+l, yj). Let us consider both of these procedures. If the point (x*,yj) is added to the lattice, it is nixessary to replace uIz a t (xi, yj) by a n uncentered second difference. Let L* - xi = Ax1 and xi - xi-1 = Ax2. Then, (72u.. tjn= ax2 Axl ‘, In) - Uijn + A Z ~[u(x*ly’AxI - Uijn - U&l,j,,! Ax2 1 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 33 uzt+ being evaluated where required by the Taylor series with remainder expansion.

3) converges with an error that is uniformly O ( ( A X ) ~ At). The method of proof is essentially Lees’s energy approach. 1), the algebraic equations are linear. The Crank-Nicolson equation may also be generalized; however, a proof + JIM DOUGLAS, JR. 1) has the self-adjoint form a ax (p(x, t ) 2) = ~ ( xt ), $ + F (x, t, u, lfg(x, t, s, u(x, 8)) ds The difference equation becomes Note that the quadrature has been changed in order to preserve the sccond order accuracy in time. It is the author’s conjecture that any convergent difference analogue of the differential equation can be adapted to the integro-differential equation; moreover, it should be possible to maintain the same order of accuracy by a proper choice of the quadrature method.

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. Now assume that the solution of the differential equation is sufficiently smooth that the best estimates of the error obtained for each difference equation hold. 14) and for thcir generalizations. Note that the higher order correct equations lead to a very large reduction in the amount of arithmetical work required to complete a problem if a small crror is prescribcd.

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

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