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## 1-d problem with mixed boundary conditions

Consider the solution of the diffusion equation in one dimension. Suppose that
 (191)

for , subject to the mixed spatial boundary conditions
 (192)

at , and
 (193)

at . Here, , , etc., are known functions of time. Of course, must be specified at some initial time .

Equation (191) needs to be discretized in both time and space. In time, we discretize on the equally spaced grid

 (194)

where is the time-step. Adopting a simple first-order differencing scheme, Eq. (191) becomes
 (195)

In space, we discretize on the usual equally spaced grid-points specified in Eq. (114), and approximate via the second-order, central difference scheme introduced in Eq. (115). The spatial boundary conditions are discretized in a similar manner to Eqs. (134) and (135). Thus, Eq. (195) yields
 (196)

or
 (197)

for , where , and . The discretized boundary conditions take the form
 (198) (199)

where , etc. The discretization scheme outlined above is termed first-order in time and second-order in space.

Equations (197)-(199) constitute a fairly straightforward iterative scheme which can be used to evolve the in time.

Next: An example 1-d diffusion Up: The diffusion equation Previous: Introduction
Richard Fitzpatrick 2006-03-29