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

By Franz L. Alt (Ed.)

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Example text

4) and let At, = tn+l - t,. n+l = 0. 6)advances the solut,ion, both temperature and boundary, one time step. The proof showed convergence up to t = 1; again, a minor change lcads to convergence for all time. Finally, another modification shows that the proof holds when the differential equation is generalized to a nonlinear one. A different approach would be to use a fixed time step and to allow the free boundary to intersect the line t = tn a t a point that is not necessarily a regular grid point.

8) 20 JIM DOUGLAS, JR. n+l = 0. 9) is a linear homogeneous difference equation with constant coefficients subject to homogeneous boundary conditions. 12) 0. 15) that the eigenfunctions are p = 1, . . 13) and that the corresponding eigenvalues are 1 pp = 1 - 2r sin2rpAx/2. 14) The natural topology for discussing eigenfunction expansions for differential equations is the Lz topology, and such is also the case here. Define the inner product of two vectors defined on the points xi, i = 0, . , M , as .

NUMERICAL METHODS FOR PARABOLIC EQUATIONS 15 If the coefficient matrix does not change from time step to time step, needs to be computed only once. 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.

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