Project

Like to use the Flux Lim Diffusion to track ionization fronts
I know this won’t work well but its better than nothing.

Jonathan what would it take to do a test of this?

Ionizing source in media with initially low optical depth but tau builds up till it hits 1.

We are basically trying to do the strongmen(Stromgren?) sphere problem.

  I think the current FLD Radiation transport (or the thermal diffusion) scheme could be adapted.  You would have to sub-cycle within the ionization/ radiation solve as the ionization fraction changes - though we do that already with the other parabolic solvers.  Usually there's some additional restriction on the hydro time step - that we would need to work out - or find in the literature...

But a first step would be to gather the necessary equations, linearize them, and then discretize them...  similar to what is done for the thermal diffusion solver.  Using the same radiation flux limiter would be straight-forward.  

https://astrobear.pas.rochester.edu/trac/wiki/ThermalConduction

Basic Equations

\frac{\partial n_{e}}{\partial t} + \nabla\cdot(n_{e} \bold{u}) = I-r

I is the rate of ionizations per unit volume and r is the rate of recombinations per unit volume

I = -\nabla\cdot J

J is the flux of ionizing photons. Inside the front where the gas is nearly completely ionized the recombination rate

r=\alpha n_{e}^{2}

The thermal energy of the gas per unit mass

E=(3/2)\kappa T(n+n_{e})/\rho

\frac{\partial E}{\partial t}=-(\bold{U}_{rel}\cdot\nabla)E-\frac{P}{\rho}(\nabla\cdot \bold{u})-\frac{\Lambda-\Gamma}{\rho}

Links and References

1. ​free-streaming limit
   

2. ​Flux limiter

3. ​Thermal Conduction

4. ​Stromgren sphere problem

5. ​Ionization front physics

6. ​Yorke 1986

7. ​Radiation pressure