468 | | E^{n+1}_i-E^{n}_i = & \left [ \alpha^n_{i+1/2} \left ( \psi E^{n}_{i+1} + \bar{\psi} E^{n+1}_{i+1}- \psi E^{n+1}_{i} - \psi E^n_{i} \right ) - \alpha^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] \\ |
469 | | + & \left [ \zeta^n_{i+1/2} \left ( \psi E^{n}_{i+1} + \bar{\psi} E^{n+1}_{i+1} + \psi E^{n+1}_{i} + \psi E^n_{i} \right ) - \zeta^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i + \psi E^{n+1}_{i-1} + \bar{\psi}E^{n}_{i-1} \right ) \right ] \\ |
470 | | - & \frac{1}{1-\psi \phi} \left [ \epsilon \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) - \omega v^n_x \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) + \xi \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\ |
| 468 | E^{n+1}_i-E^{n}_i = & \left [ \alpha^n_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1}- \psi E^{n+1}_{i} - \bar{\psi} E^n_{i} \right ) - \alpha^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^{n}_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] \\ |
| 469 | + & \left [ \zeta^n_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1} + \psi E^{n+1}_{i} + \bar{\psi} E^n_{i} \right ) - \zeta^n_{i-1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i + \psi E^{n+1}_{i-1} + \bar{\psi}E^{n}_{i-1} \right ) \right ] \\ |
| 470 | - & \frac{1}{1-\psi \phi} \left [ \theta + \epsilon \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) - \omega v^n_x \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) + \xi \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\ |
| 479 | |
| 480 | [[latex(\zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3-R_{2,i+1/2}}{2} v^n_x,{i+1/2} )]] |
| 481 | |
| 482 | and |
| 483 | |
| 484 | [[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]] |
| 485 | |
| 486 | and |
| 487 | |
| 488 | [[latex(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]] |
| 489 | |
| 490 | and |
| 491 | |
| 492 | [[latex(\theta = f(e^n_i) = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) )]] |
| 493 | |
| 494 | and |
| 495 | |
| 496 | [[latex(\omega_{i} = \frac{\lambda_i \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]] |
| 497 | |
| 498 | |
| 499 | and |
| 500 | |
| 501 | where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} )]] |
| 502 | |
| 503 | and |
| 504 | |
480 | | |
481 | | and |
482 | | |
483 | | [[latex(\epsilon^n_i=c\Delta t \kappa^n_{0P,i})]] |
484 | | |
485 | | represents the number of absorption/emissions during the time step |
486 | | |
487 | | |
488 | | and |
489 | | |
490 | | [[latex(\phi = \epsilon^n_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) )]] |
491 | | |
492 | | [[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 ) )]] |
493 | | |
494 | | and we can think of the radiative flux as |
495 | | |
496 | | [[latex(\frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{i+1/2} \left ( E^{*}_{i+1} - E^{*}_i \right ) )]] |
497 | | |
498 | | and |
499 | | |
500 | | where [[latex(\omega_{i} = \frac{\lambda_i \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]] |
501 | | |
502 | | where |
503 | | |
504 | | [[latex(\frac{\omega}{\alpha} = \frac{v}{c} \kappa_{0R,i} \Delta x)]] |
| 507 | and |
| 508 | |
| 509 | [[latex(\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2} - \frac{1}{R_{i+1/2}} \right ) )]] |
| 510 | |
| 511 | and |
| 512 | |
| 513 | [[latex(R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]] |
| 514 | |
| 515 | and |
| 516 | |
| 517 | |
| 518 | [[latex(R_{i} = \frac{\left | E^n_{i+1}-E^n_{i-1} \right | }{2 \kappa_{0R,i} E^n_{i}})]] |
| 519 | |
| 520 | and |
| 521 | |
| 522 | [[latex(\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i} - \frac{1}{R_{i}} \right ) )]] |
| 523 | |
| 524 | and |
| 525 | |
| 526 | [[latex(R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2)]] |
| 527 | |
| 528 | |