Changes between Version 156 and Version 157 of FluxLimitedDiffusion

Timestamp:

04/01/13 19:51:15 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -30 +30 @@
 ||  [[latex(\frac{\partial}{\partial t} f + \mathbf{v} \cdot  \nabla f + \mathbf{a} \cdot \nabla_v f = \left ( \frac{\partial f}{\partial t} \right )_{coll} )]]  ||
 
-If we solve the transport equation along a characteristic [[latex(\left ![ \mathbf{x}\left ( s \right ), t(s) \right \] = \left \[ \mathbf{x0} + \mathbf{n} s, \frac{s}{c
-} \right !] )]]
+If we solve the transport equation along a characteristic 
+
+[[latex(\left \[ \mathbf{x} \left ( s \right ), t \left ( s \right ) \right \] = \left \[ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right \] )]]
 
 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)]] we get
+[[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(\tau_\nu = \int \kappa_\nu ds )]] is the optical depth
-
+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.