Changes between Version 186 and Version 187 of FluxLimitedDiffusion


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Timestamp:
04/05/13 11:04:40 (12 years ago)
Author:
Jonathan
Comment:

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  • FluxLimitedDiffusion

    v186 v187  
    195195[[CollapsibleEnd()]]
    196196
    197 [[CollapsibleStart(Explcit Update 1)]]
    198 == Eplicit Update 1 ==
     197[[CollapsibleStart(Explicit Update 1)]]
     198== Explicit Update 1 ==
    199199The extra terms in the explicit update due to radiation energy are as follows:
    200200
    … …  
    249249Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function \(B(T)\)
    250250
    251 If we ignore the changes in the Temperature due to heating during the implicit step which would feed back into the source function.  We can improve this by writing
     251However we can expand the Plank Function about \(e_0\)
    252252
    253253[[latex(B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]]
    … …  
    272272
    273273=== Implicit Discretization 1 ===
    274 Which we can discretize for (1D) as
    275 
    276    [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i  + \phi^n_i e^{*}_i  + \theta^n_i) )]]   
    277    [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i  - \phi^n_i e^{*}_i  - \theta^n_i )]]   
     274
     275We can now discretize the equations
     276
     277   [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon E^{*}_i  + \phi e^{*}_i  + \theta_i - \phi_i) )]]   
     278   [[latex(e^{n+1}_i-e^{n}_i = \epsilon_i E^{*}_i  - \phi_i e^{*}_i  - \left(\theta_i-\phi_i \right ) )]]   
    278279
    279280where the diffusion coefficient is given by
    … …  
    286287and
    287288
    288 [[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]]
     289[[latex(\epsilon_i=c\Delta t \kappa_{0P,i})]]
    289290
    290291represents the number of absorption/emissions during the time step
    … …  
    293294and
    294295
    295 [[latex(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
    296 
    297 [[latex(\theta = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( 1 - 4\Gamma \frac{e^n_i}{T^n_i} \right ) )]]
     296[[latex(\phi_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]]
     297
     298[[latex(\theta_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) )]]
    298299
    299300
    300301and we can think of the radiative flux as
    301302
    302 [[latex(\frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) )]]
     303[[latex(\frac{\Delta t}{\Delta x}\mathbf{F}_{i+1/2} = \alpha_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) )]]
    303304
    304305=== Time Discretization ===
    … …  
    318319In any event in 1D we have the following matrix coefficients
    319320
    320    [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} +\bar{\psi}\phi^n_i e^n_i + \theta^n_i)]]   
    321    [[latex(\left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n )]]   
     321   [[latex(\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} +\bar{\psi}\phi_i e^n_i + \theta_i)]]   
     322   [[latex(\left ( 1 +\psi \phi_i \right ) e^{n+1}_i - \left ( \psi \epsilon_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi}\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i )]]   
    322323
    323324
    324325Now since the second equation has no spatial dependence, we can solve it for
    325    [[latex(\color{purple}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \}} )]]   
     326   [[latex(\color{purple}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi_i}\left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i-\theta_i \right \}} )]]   
    326327
    327328and plug the result into the first equation to get a matrix equation involving only one variable.
    328329
    329    [[latex(\color{purple}{\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \frac{\epsilon^n_i}{ 1 +\psi \phi^n_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2}  +\frac{\epsilon^n_i }{ 1 +\psi \phi^n_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n})]]   
     330   [[latex(\color{purple}{\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \frac{\epsilon_i}{ 1 +\psi \phi_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2}  +\frac{\epsilon_i }{ 1 +\psi \phi_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} + \frac{ \phi_i}{ 1 +\psi \phi_i}  e^n_i+ \frac{1}{ 1 +\psi \phi_i}\theta_i})]]   
    330331
    331332
    … …  
    333334If we ignore the change in the Planck function due to heating during the implicit solve, it is equivalent to setting \(\psi \phi = 0\)  This gives the following equations:
    334335
    335    [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+ \left ( \bar{\psi} \alpha^n_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha^n_{i-1/2} \right ) E^{n}_{i-1} +\phi^n_i e^n_i + \theta^n_i)]]   
    336    [[latex(e^{n+1}_i = e^n_i + \epsilon^n_i  \left [ \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) - \frac{4 \pi}{c} B \left ( T^n_i \right )  \right ] )]]   
     336   [[latex(\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i-1/2} + \epsilon_i \right ) \right ] E^n_i+ \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i-1/2} \right ) E^{n}_{i-1} +\phi_i e^n_i + \theta_i)]]   
     337   [[latex(e^{n+1}_i = e^n_i + \epsilon_i  \left [ \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) - \frac{4 \pi}{c} B \left ( T^n_i \right )  \right ] )]]   
    337338
    338339In this case the first equation decouples and can be solved on it's own, and then the solution plugged back into the second equation to solve for the new energy. 
    … …  
    440441[[CollapsibleEnd()]]
    441442
    442 [[CollapsibleStart(Explcit Update 2)]]
    443 == Eplicit Update 2 ==
     443[[CollapsibleStart(Explocit Update 2)]]
     444== Explicit Update 2 ==
    444445The extra terms in the explicit update due to radiation energy are as follows:
    445446