Changes between Version 174 and Version 175 of FluxLimitedDiffusion


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Timestamp:
04/03/13 14:10:40 (12 years ago)
Author:
Jonathan
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  • FluxLimitedDiffusion

    v174 v175  
    66== Spectral Intensity ==
    77Typically when we discuss the radiation field we use the spectral intensity [[latex(I \left ( \nu, \mathbf{x}, \Omega \right ) )]] which is a function of frequency, position, and direction.  This is very similar to the phase space density used in deriving the fluid equations [[latex(f \left ( \mathbf{x}, \mathbf{v} \right ) )]] except that
    8  * light always travels at 'c', so the velocity dependence is just a direction dependence. 
     8 * light always travels at \(c\), so the velocity dependence is just a direction dependence. 
    99 * Furthermore, photons can have different frequencies, so there is an extra dimension to the phase space. 
    1010 * Instead of storing the phase space density of photons, the spectral intensity is the phase space density of energy flux... 
    … …  
    4343where \(A_{\nu} \) is the emission rate of photons of frequency \(\nu\) and the mean free path length is given by \(\chi_\nu = \sigma_\nu n\) where \(\sigma_\nu\) is the particle scattering cross section and \(n\) is the number density of particles.
    4444
    45 Now if we multiply through by \( h\nu\)
     45Now if we multiply through by \( h\nu\) we have
     46 [[latex(\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot  \nabla I_\nu  = \eta_\nu - \sigma_\nu I_\nu  )]]
     47
     48where \(\eta_\nu = h\nu A_{\nu}\) is the radiative power
     49
     50If we solve the transport equation along a characteristic
     51
     52 [[latex(\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] )]]
    4653
    4754we have
    4855
    49  [[latex(\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot  \nabla I_\nu  = \eta_\nu - \sigma_\nu I_\nu  )]]
    50 
    51 where
    52  \(\eta_\nu = h\nu A_{\nu}\) is the radiative power
    53 
    54 If we solve the transport equation along a characteristic
    55 
    56  [[latex(\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] )]]
    57 
    58 we have
    59 
    6056 [[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s)  )]]
    6157
    62 where \(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) \)
    63 
    64 and then we can divide through by \(\chi_\nu(s)\) we get
     58where \(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) \) and then we can divide through by \(\chi_\nu(s)\) we get
    6559
    6660 [[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s))]]
    6761
    68 Now if we define
    69 
    70  \(d\tau_\nu = \chi_\nu(s) ds \)
    71 which gives
     62Now if we define \(d\tau_\nu = \chi_\nu(s) ds \) which gives
    7263
    7364 [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]]
    … …  
    142133
    143134The flux limited diffusion approximation drops the radiation momentum equation in favor of
    144 \(\mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_0\)
    145 
    146 where \(\lambda\) is the flux-limiter
     135\(\mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_0\) where \(\lambda\) is the flux-limiter and is given by
    147136
    148137  [[latex(\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) )]] 
    … …  
    156145
    157146If we Lorentz boost the comoving terms into the lab frame and keep terms necessary to maintain accuracy we get:
    158 
    159 
    160147
    161148   [[latex(G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) )]]   
    … …  
    173160
    174161
    175 For static diffusion, the terms in blue with v^2^/c can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature).
     162For static diffusion, the terms in blue with \(\frac{v^2}{c}\) can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature).
    176163
    177164[[CollapsibleStart(Operator Splitting 1)]]
    … …  
    181168In AstroBEAR this would look like:
    182169* Initialization
    183  * Prolongate, d, p, e, E, Edot
     170 * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), \(\dot{E}^I\), and \(\dot{E}^E\)
    184171* Step 1
    185  * Overlap d, p, e, E and do physical BC's
    186  * Do IR which updates e,,0,,, and E,,0,, using d,,1,,, e,,1,,, E,,1,,
    187  * Update E,,2*mbc,, using Edot,,2*mbc,,
    188  * Update e,,2*mbc,, using E,,2*mbc,,, Edot,,2*mbc,,, and e,,2*mbc,,
    189  * Update Edot,,0,, using pre IR and post IR E,,0,,
    190  * Ghost e,,2*mbc,,, E,,mbc+1,,, Edot,,mbc+1,,
     172 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
     173 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)
     174 * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}^I_{2\mbox{mbc}}\)
     175 * Update \(e_{2\mbox{mbc}} using \(E_{2\mbox{mbc}}\), \(\dot{E}^I_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)
     176 * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)
     177 * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}^I_{\mbox{mbc}+1}\)
    191178 * Do first EH,,mbc,,
    192  * Do ER,,mbc,, --- Terms with grad E can be done without ghosting since EH did not change E.  The del dot vE term needs time centered face centered velocities which can be stored during the hydro update.
    193  * Store Edot in child arrays to be prolongated
     179 * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).  The \(\nabla \cdot \mathbf{v}E\) term needs time centered face centered velocities which can be stored during the hydro update.
     180 * Store \(\dot{E}\) in child arrays to be prolongated
    194181* Step 2
    195  * Overlap d, p, e, E, and do physical BCs
    196  * Do IR which updates e,,0,,, and E,,0,, using d,,1,,, e,,1,,, E,,1,,
    197  * Update Edot,,0,, using pre IR and post IR E,,0,,
    198  * Update E,,1,, using Edot,,1,,
    199  * Ghost e,,mbc,,, E,,1,,, Edot,,1,,
     182 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\) and do physical BC's
     183 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)
     184 * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)
     185 * Update \(E_{1}\) using \(\dot{E}^I_{1}\)
     186 * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}^I_{1}\)
    200187 * Do second EH,,0,,
    201  * Do ER,,0,, --- Terms with grad E can be done without ghosting since EH did not change E.  The del dot vE term needs time centered face centered velocities which can be stored during the hydro update.
     188 * Do ER,,mbc,, --- Terms with \(\nabla E\) can be done without ghosting since EH did not
     189
     190 * Do ER,,0,, --- Terms with grad E can be done without ghosting since EH did not change \(E\).  The \(\nabla \cdot \mathbf{v}E\) term needs time centered face centered velocities which can be stored during the hydro update.
    202191
    203192