Changes between Version 6 and Version 7 of FluxLimitedDiffusion
 Timestamp:
 03/18/13 13:42:56 (12 years ago)
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FluxLimitedDiffusion
v6 v7 75 75  [[latex(\frac{\partial E}{\partial t} \color{red}{  \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi BcE)} \color{green}{\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E} \color{green}{\nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E} )]]  76 76 77 For static diffusion, the terms in blue with v^2^/c can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green) using time centered radiation energy, and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). If the opacity is not temperature dependent, then the implicit solve only involves the radiation energy density. If the opacity is not a linear function of the energy, then some formof subcycling would be required.77 For static diffusion, the terms in blue with v^2^/c can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green) using time centered radiation energy, and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). If the opacity is independent of temperature and radiation energy density, then the implicit solve only involves the radiation energy density. Otherwise some sort of subcycling would be required. 78 78 79 79 For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update. In this case we can solve the radiation energy equation: … … 81 81 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi BcE))]] 82 82 83 Which we can discretize for (1D) as 84 85 [[latex(E^{n+1}_iE^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1}E^{n+1}_{i} \right )  \alpha^n_{i1/2} \left ( E^{n+1}_{i}E^{n+1}_{i1} \right ) \right ] + \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)E^{n+1}_i \right ) )]] 86 87 where 88 89 [[latex(\epsilon^n_i=\frac{c\Delta t}{ \kappa^n_{0P,i}})]] 90 91 represents the number of absorption/emissions during the time step 92 93 and the diffusion coefficient is given by 94 95 [[latex(\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{i+1/2}} \mbox{ where } \kappa_{i+1/2} = \frac{\kappa_{i}+\kappa_{i+1}}{2} \mbox{ and } \lambda_{i+1/2} = f \left ( R_{i+1/2} \right ))]] 96 97 where 98 99 [[latex(R_{i+1/2} = \frac{\left  E_{i+1}E_{i} \right  }{2 \kappa_{i+1/2} \left ( E_i+E_{i+1} \right )})]] 83 100 84 101 102 This gives matrix coefficiencts 103 104 [[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i1/2} + \epsilon^n_i \right ) E^{n+1}_i  \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1}  \left ( \alpha^n_{i1/2} \right ) E^{n+1}_{i1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]]