Changes between Version 12 and Version 13 of FluxLimitedDiffusion
- Timestamp:
- 03/19/13 13:19:59 (12 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
FluxLimitedDiffusion
v12 v13 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: 80 80 81 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B-cE) = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B-cE))]] 81 || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B-cE) = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B-cE))]] || 82 || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B-cE))]] || 82 83 83 84 where [[latex(\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E)]] … … 85 86 Which we can discretize for (1D) as 86 87 87 [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1}-E^{n+1}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{n+1}_{i}-E^{n+1}_{i-1} \right ) \right ] + \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]] 88 || [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1}-E^{n+1}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{n+1}_{i}-E^{n+1}_{i-1} \right ) \right ] + \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]] || 89 || [[latex(e^{n+1}_i-e^{n}_i = - \epsilon^n_i \left ( \frac{4 \pi}{c} B(T^n_i)-E^{n+1}_i \right ) )]] || 90 88 91 89 92 where 93 94 90 95 91 96 [[latex(\frac{\Delta t}{\Delta x}\mathbf{F}^n_{i+1/2} = \alpha^n_{i+1/2} \left ( E^{n+1}_{i+1} - E^{n+1}_i \right ) )]] … … 108 113 This gives matrix coefficients 109 114 110 [[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) E^{n+1}_i - \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]] 115 || [[latex(\left ( 1 + \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) E^{n+1}_i - \left ( \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1}=E^n_i+\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]] || 116 || [[latex(\left ( 1 \right ) e^{n+1}_i - \left ( \epsilon^n_i \right )E^{n+1}_i =e^n_i-\frac{4\pi \epsilon^n_i}{c}B \left (T^n_i \right ) )]] || 111 117 112 118 and for 2D the matrix coefficients would be
