Changes between Version 179 and Version 180 of FluxLimitedDiffusion
 Timestamp:
 04/03/13 17:09:18 (12 years ago)
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FluxLimitedDiffusion
v179 v180 521 521 Then the equation for e becomes 522 522 523 [[latex(\frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E )]]523 [[latex(\frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]] 524 524 525 525 which will be accurate as long as \(4\Gamma \frac{ee_0}{T_0} < \xi << 1\) or \(\Delta e = ee_0 < \xi \frac{T_0}{4 \Gamma}\) … … 527 527 We can calculate the time scale for this to be true using the evolution equation for the energy density 528 528 529 [[latex( \Delta e = \Delta t \kappa_{0P} \left [ 4 \pi B_0 cE \right ] < \xi \frac{T_0}{4 \Gamma})]]530 531 which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left ( 4 \pi B_0  cE \right )})]]529 [[latex(ee_0=\Delta e = \Delta t \kappa_{0P} \left [ 4 \pi B_0 cE \right ] < \xi \frac{T_0}{4 \Gamma})]] 530 531 which gives [[latex(\Delta t < \xi \frac{T_0}{4 \Gamma  \left ( \frac{\partial e}{\partial t} \right )_0  })]] 532 532 533 533