Changes between Version 83 and Version 84 of FluxLimitedDiffusion
- Timestamp:
- 03/27/13 22:37:47 (12 years ago)
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FluxLimitedDiffusion
v83 v84 324 324 [[CollapsibleEnd()]] 325 325 326 == OR==326 == Alternative Splitting Method == 327 327 328 328 [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]] … … 376 376 377 377 [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =-\lambda \nabla E)]] 378 [[latex(\frac{\partial e}{\partial t} = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]] 379 [[latex(\frac{\partial E}{\partial t} = -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) )]] 378 [[latex(\frac{\partial E}{\partial t} = -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) )]] 380 379 381 380 These can be discretized as follows: … … 383 382 [[latex(p^{n+1}_i=p^n_i - \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( E^n_{i+1}-E^n_{i-1} \right ) )]] 384 383 385 [[latex(e^{n+1}_i=e^n_i + \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_i \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) )]] 386 387 [[latex(E^{n+1}_i=E^n_i - \frac{\Delta t}{\Delta x} \left ( \frac{\lambda_i}{2} \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}-1 \right ) \left ( v^n_i \left ( E^n_{i+1}-E^n_{i-1} \right ) \right ) + \left ( F_{i+1/2}-F_{i-1/2} \right ) \right ))]] 384 [[latex(E^{n+1}_i=E^n_i + \frac{\Delta t}{\Delta x} \left ( F_{i+1/2}-F_{i-1/2} \right ) \right ))]] 388 385 389 386 where … … 393 390 394 391 [[latex(R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]] 392 395 393 and 396 394 [[latex(R_{i+1/2} = \frac{\left | E^n_{i+1}-E^n_{i} \right | }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]] … … 411 409 [[CollapsibleEnd()]] 412 410 413 [[CollapsibleStart(Implicit Update 1)]]414 == Implicit Update 1==411 [[CollapsibleStart(Implicit Update 2)]] 412 == Implicit Update 2 == 415 413 416 414 For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions. In this case we can solve the radiation energy equations: 417 415 418 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE))]] 419 [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE))]] 420 421 where [[latex(\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E)]] 416 [[latex(\frac{\partial e}{\partial t} = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]] 417 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) -\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E )]] 422 418 423 419 === Expanding about e,,0,, === … … 435 431 Then the system of equations becomes 436 432 437 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]438 [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]]433 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] - \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]] 434 [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E )]] 439 435 440 436 which will be accurate as long as [[latex(4\Gamma \frac{e-e_0}{T_0} < \xi << 1)]] or [[latex(\Delta e = e-e_0 < \xi \frac{T_0}{4 \Gamma})]] … … 450 446 Which we can discretize for (1D) as 451 447 452 [[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) )]]453 [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i - \phi^n_i e^{*}_i - \theta^n_i )]]448 [[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) - omega_{x,i} v_{x,i} \left ( E^{*}_{i+1}-E^*_{i-1} \right ) )]] 449 [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i - \phi^n_i e^{*}_i - \theta^n_i + omega_{x,i} v_{x,i} \left ( E^{*}_{i+1}-E^*_{i-1} \right ) )]] 454 450 455 451 where the diffusion coefficient is given by … … 473 469 [[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 ) )]] 474 470 471 and 472 473 where [[latex(\omega_{x,i} = \frac{\Delta t}{\Delta x} \lambda_i \left ( \frac{\kappa_{0P},i}{\kappa_{0R,i}}-\frac{1}{2} \right ) )]] 475 474 476 475 and we can think of the radiative flux as
