Changes between Version 4 and Version 5 of FluxLimitedDiffusion

Timestamp:

03/18/13 12:49:27 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -71 +71 @@
 which if we plug back into the gas equations and keep terms necessary to maintain accuracy we get:
 
-||  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P-\lambda \nabla E)]]   ||
-||  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E - \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]]  ||
-||  [[latex(\frac{\partial E}{\partial t} - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E = \kappa_{0P} (4 \pi B-cE) - \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E + \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )  )]]  ||
+||  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]]   ||
+||  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]  ||
+||  [[latex(\frac{\partial E}{\partial t}  \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}  )]]  ||
 
+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) using time centered radiation energy, and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature).  If the opacity is not temperature dependent, then the implicit solve only involves the radiation energy density.  If the opacity is not a linear function of the energy, then some form of sub-cycling would be required.
 
-For static diffusion, the terms with v^2^/c can be dropped and the system can be split into the usual hydro update, radiative source terms (using time centered radiation energy), and a coupled implicit solve for the radiation energy density and thermal energy density (ie temperature).  If the opacity is not temperature dependent, then the implicit solve only involves the radiation energy density.