The current version of the code has changed the temperature from computational units to Kelvin or solves the energy equation instead of temperature. This note is to check the parameter and equations changes need to do to be consistent comparing the old ways of ​thermal diffusion equation in Astrobear. So lhs of the equation was replaced by the derivative of the energy.

NOT COMPLETE !!

1. Equations

The heat diffusion equation solved in AstroBEAR is equivalent to

C_{v}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{cgs}}\left[\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}}\right]

where everything is in cgs unit or Kelvin and C_{v} is the specific heat capacity.

In order to find the relationship between \kappa_{1} and \kappa_{0} in ​Betti's document, we have to rewrite the first equation in wiki:TempEquations to make temperature as in Kelvin instead of K_{B}T which is in Joule: C_{v}\rho_{SI}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{SI}}\left[\kappa_{0}K_{B}^{n+1}T^{n}\frac{\partial T}{\partial x_{SI}}\right]

where C_{v} is in unit of Joule per kg per kelvin. and q_{0}=\kappa_{0}(K_{B}T)^{n}\frac{\partial K_{B}T}{\partial x_{SI}}

or q_{0}=10^{2}\kappa_{0}(K_{B}T)^{n+1}\frac{\partial T}{\partial x_{cgs}}

By converting everything but the Boltzmann constant in the above equation to cgs units we have

10^{3}10^{-4}C_{v}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\partial}{\partial 10^{-2}x_{cgs}}\left[\kappa_{0}K_{B}^{n+1}T^{n}\frac{\partial T}{\partial 10^{-2}x_{cgs}}\right] Here or C_{v}\rho_{cgs}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{cgs}}\left[\kappa_{0}10^{5}K_{B}^{n+1}T^{n}\frac{\partial T}{\partial x_{cgs}}\right]

where C_{v} is in unit of erg per g per kelvin and 1 erg per g per kelvin = 10^{-4} Joule per kg per kelvin

So the \kappa_{1} in the new version of the code should be.

\kappa_{1}=10^{5}\kappa_{0}K_{B}^{n+1}

and K_{B} is in SI unit?

And since we use the subroutine getdEdT to get the energy derivative so \frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{cgs}}\left[\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}}\right]

And same as blog:bliu01092014, we define

\kappa_{1}=10^{5}\kappa_{0}K_{B}^{n+1} q^{*}_{cgs}=10^{5}\kappa_{0}K_{B}^{n+1}T^{n}\frac{\partial T}{\partial x_{cgs}}

And so same as blog:bliu01092014, we have

In Betti's data, C^{\prime}_{v}=7.816e26, \kappa_{0}=3.734E69 q_{0}=5.876E18 so \kappa_{1}=10^{5}*3.734E69*(1.38EK_{B}^{n+1} q^{*}_{cgs}=10^{3}*q_{0}=5.876E21

2. Scales for converting from cgs to cu ScaleFlux is the scale which converts the heat flux from cgs units to computational units in AstroBEAR. Since

\frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{\partial Q}{\partial x} The new scaleFlux is EScale*velScale or pScale*velScale as the temperature is in Kelvin now. While in the older version where the temperature is in c.u. scale, \frac{1}{\gamma-1}\rho\frac{\partial T}{\partial t}=\frac{\partial Q}{\partial x} So scaleFlux is rScale*TempScale*velScale instead…

scaleDiff is the scale that converts the kappa1 from cgs units to computational units. Since q^{*}_{cgs}=\kappa_{1}T^{n}\frac{\partial T}{\partial x_{cgs}} So scaleDiff=scaleFlux*lScale

These scales are defined in physics_control.f90

3. subroutine getdEdT in EOS.f90

This subroutine will return the value of dE/dT which has a unit as C.U. Energy/Kelvin

4. source and stencil_fix(0) in diffusion.f90 will be modified as

    source = ((ndiff-1.0)/(ndiff+1.0))*dot_product(T(0:2*nDim),stencil_fixed(0:2*nDim)) - T(0)*GetdEdT(Info%q(i,j,k,:))
    stencil_fixed(0) = stencil_fixed(0) - GetdEdT(Info%q(i,j,k,:)) 

comparing with the old way

!    source = ((ndiff-1.0)/(ndiff+1.0))*dot_product(T(0:2*nDim),stencil_fixed(0:2*nDim)) - T(0)*gamma7*Info%q(i,j,k,1)
!    stencil_fixed(0) = stencil_fixed(0) - gamma7*Info%q(i,j,k,1)

So the lefthand side vector has unit of Kelvin and righthand side vector has unit of C.U.energy