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Coggeshall #8
Verification Problem

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
Coggeshall (1991) published a collection of analytic self-similar test problems, and "Coggeshall #8" or "Cog8" is the eighth one listed. The solution represents an adiabatic expansion plus heat conduction. The heat conduction's area weighted flux on each cell face is equal. That is, conduction moves as much energy into a cell as it removes. Thus, answers with and without conduction look much the same; which is part of the test! This article, this article, and this article, discuss analytic and numerical solutions for the Su-Olson problem.

The tools in cog8.tbz provide solutions in as a function of time and position in one-dimensional spherical coordinates and two-dimensional $r-z $coordinates.
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Curiously, and unpublished, the two-dimensional axisymmetric case has an exact solution involving hypergeometric terms such as \begin{equation} z \ r^{2 + \chi} \ {}_{2}F_{1} \left( \frac{1}{2}, - \frac{2 + \chi}{2}, \frac{3}{2}, -\frac{z^2}{ r^2} \right ) \label{eq1} \tag{1} \end{equation} that reduce to the one-dimensional spherical case expresions when $z \rightarrow 0$.
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