Changes between Version 156 and Version 157 of FluxLimitedDiffusion
 Timestamp:
 04/01/13 19:51:15 (12 years ago)
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FluxLimitedDiffusion
v156 v157 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} )]]  31 31 32 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 33 } \right !] )]] 32 If we solve the transport equation along a characteristic 33 34 [[latex(\left \[ \mathbf{x} \left ( s \right ), t \left ( s \right ) \right \] = \left \[ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right \] )]] 34 35 35 36 we have 36 37 37 [[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 )]]38 39 and then we can divide through by [[latex(\chi )]] we get38 [[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 )]] 39 40 and then we can divide through by [[latex(\chi_\nu)]] we get 40 41 41 42 [[latex(\frac{dI_\nu}{d\tau_\nu} = \frac{\eta_\nu}{\chi_\nu}  I_\nu = S_\nu  I_\nu)]] 42 43 43 where [[latex(\tau_\nu = \int \kappa_\nu ds )]] is the optical depth 44 44 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 45 46 [[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 ) )]] 45 47 46 48 There are various limits that are important to consider.