Changes between Version 6 and Version 7 of FluxLimitedDiffusion

Timestamp:

03/18/13 13:42:56 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -75 +75 @@
 ||  [[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 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 independent of temperature and radiation energy density, then the implicit solve only involves the radiation energy density.  Otherwise some sort of sub-cycling would be required.
 
 For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update.  In this case we can solve the radiation energy equation:
@@ -81 +81 @@
 [[latex(\frac{\partial E}{\partial t}  = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B-cE))]]
 
+Which we can discretize for (1D) as
+
+[[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1}-E^{n+1}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{n+1}_{i}-E^{n+1}_{i-1} \right ) \right ] + \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]]
+
+where
+
+[[latex(\epsilon^n_i=\frac{c\Delta t}{ \kappa^n_{0P,i}})]]
+
+represents the number of absorption/emissions during the time step
+
+and the diffusion coefficient is given by 
+
+[[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2}  \frac{c \lambda_{i+1/2}}{\kappa_{i+1/2}} \mbox{ where } \kappa_{i+1/2} = \frac{\kappa_{i}+\kappa_{i+1}}{2} \mbox{ and } \lambda_{i+1/2} = f \left ( R_{i+1/2} \right ))]]
+
+where
+
+[[latex(R_{i+1/2} = \frac{\left | E_{i+1}-E_{i} \right | }{2 \kappa_{i+1/2} \left ( E_i+E_{i+1} \right )})]]
 
 
+This gives matrix coefficiencts 
+
+[[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) E^{n+1}_i - \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]