Changes between Version 159 and Version 160 of FluxLimitedDiffusion

Timestamp:

04/02/13 01:01:39 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -36 +36 @@
 we have
 
-[[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu - \chi_\nu I_\nu )]]
-
-and then we can divide through by [[latex(\chi_\nu)]] we get
-
-[[latex(\frac{dI_\nu}{d\tau_\nu} = \frac{\eta_\nu}{\chi_\nu} - I_\nu = S_\nu - I_\nu)]]
-
-where [[latex(d\tau_\nu = \kappa_\nu ds )]] is the optical depth.  The right hand side should be evaluated along the characteristic so we have
-
-[[latex(\frac{dI_\nu}{d\tau_\nu} = \frac{\eta_\nu}{\chi_\nu} - I_\nu = S_\nu \left ( \mathbf{x}_0+c\mathbf{n} t \right )  - I_\nu \left ( \mathbf{x}_0+c\mathbf{n} t \right ) )]]
-
-There are various limits that are important to consider.
+[[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s)  )]]
+
+where [[latex(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) )]]
+
+and then we can divide through by [[latex(\chi_\nu(s))]] we get
+
+[[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s))]]
+
+Now if we define
+
+[[latex(d\tau_\nu = \chi_\nu(s) ds )]] which gives [[latex(\tau_nu(s) = \int\limits_0^s \chi_\nu(s') ds')]]
+
+[[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau^'_\nu )]] 
+
+There are a few important dimensionless numbers to consider:
 
 || [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]] || [[latex(\beta = \frac{u}{c})]] ||