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 blog:bliu01092014 and blog:bliu01142014 . So the term \gamma_{7} won't show in the thermal diffusion equations.

1. Equations

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

where T is in cgs unit or Kelvin.

In current version of AstroBEAR, we have \frac{\partial E}{\partial T}\frac{\partial T}{\partial t}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v}}\frac{\partial}{\partial x_{cgs}}\left[T^{n}\frac{\partial T}{\partial x_{cgs}}\right]

And same as blog:bliu01092014, we define

\kappa_{1}=\frac{10\kappa_{0}K_{B}^{n}}{(\gamma-1)C_{v} } q^{*}_{cgs}=\kappa_{1}T^{*}\frac{\partial T}{\partial x_{cgs}}

And so same as blog:bliu01092014, we have

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

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^{*}\frac{\partial T}{\partial x_{cgs}} So scaleDiff=scaleFlux*lScale

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