• The following equations are based on blog:bliu01092014. and ​Scales in AstroBEAR

1. Equations and Parameters

Equation solved in Betti's code (SI units and Temperature in Joules)

\rho C^{\prime}_{v}\frac{\partial K_{B}T}{\partial t}=\frac{\partial}{\partial x}\left[\kappa_{0}(K_{B}T)^{n}\frac{\partial (K_{B}T)}{\partial x}\right] where K_{B} is Boltzmann constant and C^{\prime}_{v}=C_{v}/K_{B} as in ​Betti's document and C_{v} is the normal specific heat capacity. And the flux is q_{0}=\kappa_{0}(K_{B}T)^{n}\frac{\partial (K_{B}T)}{\partial x_{SI}}

To convert this to cgs units we write

10^{3}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\kappa_{0}K_{B}^{n}}{C^{\prime}_{v}}\frac{\partial}{10^{-2}\partial x_{cgs}}\left[T^{n}\frac{\partial T}{10^{-2}\partial x_{cgs}}\right]

That is

\frac{1}{\gamma-1}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]

Here the term \frac{1}{\gamma-1} was put for the convenience of scale convert in the code which will be explained in the part of computational scales.

In AstroBEAR we define \kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C^{\prime}_{v}} and q^{*}_{cgs}=\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}}

Comparing the definition of q_{0} and q^{*}_{cgs} we have q_{0}=10(\gamma-1)q^{*}_{cgs}C^{\prime}_{v}K_{B}

In ​Betti's data, C^{\prime}_{v}=7.816e26 and \gamma=5/3 so q_{0}=\frac{2}{3}*7.816*1.38*10^{4}*q^{*}_{cgs}

2. Computational Scales

In AstroBEAR code, everything is in computational scales. In this part, I summarize the relations of scales for different physics quantities. The details can be found in ​Parameters and Scales for the Ablative RT problem (and in the code).

The following table gives scales the relationship to physics variables in C.U. in the code X_{cu} = X_{cgs}/xScale

Variable	scale	relationship in cu
length	lScale
time	timeScale
velocity	velScale	v=l/t
density	rScale
temperature	TempScale
pressure	pScale	p=\rho T
energy		E=\frac{1}{\gamma-1}(p+0.5*\rho *v^2)

More details about physics relations in the code can be found in the ​visit expressions

The above heat diffusion equation (everything is in C.U.) in AstroBEAR is

\frac{1}{\gamma-1}\rho\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left[\kappa_{1}T^{n}\frac{\partial T}{\partial x}\right]

The reason the \frac{1}{\gamma-1} is there because we want the left-hand-side to be energy where \rho T gives pressure in C.U.