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

By Franz L. Alt (Ed.)

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1)that is analogous to a procedure iiitroduced by Lax [60] to compute shock wave propagation in first order, hyperbolic equations. He h s iiot yt:t succcedcd i n demoristratiig coiivergence for the method, but his ~iumcricalevidence is encouragiiig. 1)is replaced by a first order systcin with temperature 48 JIM DOUGLAS, JR. and internal energy as dependent variables. An explicit difference system is employed. Very little has been done for Stefan problems involving more that one space variable. Rose notes that his method is essentially independent of dimensionality, but he has yet to try such a problem.

The method was originally applied to differential equations, and many of the complications that arise in the application to difference equations do not appear in the differential case. It is the intention here to outline the procedure in a heuristic fashion by switching back and forth between difference and differential equations as the argument progresses for a very simple case. Let us consider the backward difference equation for the boundary value problem for the heat equation. 1) e c = O ( ( k > ' 4- A t ) , provided the solution of the differential equation is sufficiently smooth.

Gallie and the author [20] have shown that At may be increased following the relation 18 JIM DOUGLAS, JR. for the backward difference equation for the heat equation without a reduction in the rate of convergence, provided the derivatives appearing in the truncation error terms decay exponentially with time. The result also holds for certain analogues of the heat equation in several space variables [all, and the ideas of the proof extend to any of the other difference equations discussed in this paper for linear parabolic equations.

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