9.12.4. Solution (Suggestions)

(a) and (b)

The numerical method of lines has been discussed in Problem 6.8 and applied to a semi-infinite fluid in Problem 8.17. In achieving a solution to this type of problem, it is desirable to conduct trial solutions with various number of nodes and intervals between nodes. The objective of these trial solutions for the semi-infinite slab is to have enough node intervals so that the variables at the greatest depth (large x in this case) do not change appreciably with time. The numbers of nodes can then be adjusted so that the solution is not greatly dependent upon the number of nodes as the nodes are increased over the same total interval.

The POLYMATH Simultaneous Differential Equation Solver can be used in the process of adjusting the number of nodes and their spacing. The capability to duplicate an equation is very useful in writing the finite differences for each of the nodes. Some large problems may require very large numbers of differential equations to be solved simultaneously for accurate solutions.