Changes between Version 2 and Version 3 of FluxLimitedDiffusion

Timestamp:

03/13/13 22:12:02 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -64 +64 @@
 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(G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) )]]   ||
+||   [[latex(\mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2})]]   ||
+
+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 )  )]]  ||
 
+