Changes between Version 1 and Version 2 of FluxLimitedDiffusion

Timestamp:

03/12/13 15:15:36 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -13 +13 @@
 ||  [[latex(\frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0)]]  ||
 ||  [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P+\mathbf{G})]]  ||
-||  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( \rho e + P \right ) \mathbf{v} \right ] = c G^0)]]  ||
+||  [[latex(\frac{\partial e}{\partial t}  + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^0)]]  ||
 ||  [[latex(\frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = -c G_0 )]]  ||
 ||  [[latex(\frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = -\mathbf{G})]]  ||
@@ -41 +41 @@
 
 * If the radiation has a blackbody spectrum then [[latex(\kappa_{0E}=\kappa_{0P})]]
+* If the radiation is optically thick, then 
+[[latex(\mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0))]] 
+ which implies that ||  [[latex(\kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]})]] 
+* In the optically thin regime, [[latex(|\mathbf{F}_0(\nu_0)| \rightarrow cE_0(\nu_0))]] so we would have [[latex(\kappa_{0F}=\kappa_{0E})]] however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that [[latex(\kappa_{0F}=\kappa_{0R})]]
 
-* What about the flux weighted opacity?
- * If the radiation is optically thick, then [[latex(\mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0))]] which gives: ||  [[latex(\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{})]] 
+== Flux limited diffusion ==
+
+The flux limited diffusion approximation drops the radiation momentum equation in favor of 
+[[latex(\mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_0)]]
+
+where [[latex(\lambda)]] is the flux-limiter
+
+||  [[latex(\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) )]]  ||
+||  [[latex(R=\frac{|\nabla E_0|}{\kappa_{0R}E_0})]]   ||
+
+which corresponds to a pressure tensor
+
+||   [[latex(\mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0])]]   ||
+||   [[latex(R_2=\lambda+\lambda^2R^2)]]   ||
+
+
+If we Lorentz boost the comoving terms into the lab frame 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 )  )]]  ||
+