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Conduction Front Analytic test

The self-similar solution of the temperature of a thermal conduction front (without hydro) is described by the following set of equations:

T = T_c (1-x^2/x_0^2)^{1/n}

where T_C is the peak temperature:

T_c = (\frac{Q^2}{\kappa t})^{1/(n+2)}(\frac{n}{2(n+2)}\xi_0^2)^{1/n}

x_0 is the interface position:

x_0 = (\kappa Q^nt)^{1/(n+2)}\xi_0

\xi_0 is a constant:

\xi_0 = [\frac{(n+2)^{1+n}2^{1-n}}{n\pi^{n/2}}]^{1/(n+2)}[\frac{\Gamma(1/2+1/n)}{\Gamma(1/n)}]^{n/(n+2)}

here \kappa is the thermal conduction constant, n is thermal conduction index. The idea of the test is that for any given Q (which determines total energy inside the grid), we can generate a set of parameters x_0, T_c to get a profile T(x,t). We can therefore first find a starting profile at T(x,t_0) by input a time t_0, run the pure conduction code with said \kappa to t_1, and compare it with profile T(x, t_1).

Also notice that the above equations only give the profile left to x_0. We have to hand pick a value for x > x_0. This usually depends on the resolution. I recommend cut off the profile left to x_0 at a point (x-x_0) = \epsilon, and use T(\epsilon) as the uniform temperature to the right.

I wrote a quick thrown together executable that will return T_c and x_0 by getting input of \kappa, Q and time, located at /home/shuleli/gadgets/cond. for example, type the following command on the cmd line and hit enter,
./cond 0.001 100.0 0.1
would return "at time 0.1, Tc is 36.2443, x0 is 1.68733".

astrobear

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Conduction Front Analytic test

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