Changes between Version 28 and Version 29 of FluxLimitedDiffusion
- Timestamp:
- 03/20/13 12:30:02 (12 years ago)
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FluxLimitedDiffusion
v28 v29 43 43 * If the radiation is optically thick, then 44 44 [[latex(\mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0))]] 45 which implies that ||[[latex(\kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]})]]45 which implies that [[latex(\kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]})]] 46 46 * In the optically thin regime, [[latex(|\mathbf{F}_0(\nu_0)| \rightarrow cE_0(\nu_0))]] so we would have [[latex(\kappa_{0F}=\kappa_{0E})]] however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that [[latex(\kappa_{0F}=\kappa_{0R})]] 47 47 … … 53 53 where [[latex(\lambda)]] is the flux-limiter 54 54 55 || [[latex(\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) )]] || 56 || [[latex(R=\frac{|\nabla E_0|}{\kappa_{0R}E_0})]] || 55 [[latex(\lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right ) )]] 56 [[latex(R=\frac{|\nabla E_0|}{\kappa_{0R}E_0})]] 57 57 58 58 which corresponds to a pressure tensor 59 59 60 || [[latex(\mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0])]] || 61 || [[latex(R_2=\lambda+\lambda^2R^2)]] || 60 [[latex(\mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0])]] 61 [[latex(R_2=\lambda+\lambda^2R^2)]] 62 62 63 63 … … 66 66 67 67 68 || [[latex(G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) )]] || 69 || [[latex(\mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2})]] || 68 [[latex(G^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right ) )]] 69 [[latex(\mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2})]] 70 70 71 71 = Numerics of Flux Limited Diffusion = … … 73 73 If we plug the expressions for the radiation 4-momentum back into the gas equations and keep terms necessary to maintain accuracy we get: 74 74 75 || [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]] || 76 || [[latex(\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]] || 77 || [[latex(\frac{\partial E}{\partial t} \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E} )]] || 75 [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]] 76 [[latex(\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]] 77 [[latex(\frac{\partial E}{\partial t} \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E} )]] 78 78 79 79 … … 109 109 For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions. In this case we can solve the radiation energy equations: 110 110 111 || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE))]] || 112 || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE))]] || 111 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE))]] 112 [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE))]] 113 113 114 114 where [[latex(\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E)]] … … 128 128 Then the system of equations becomes 129 129 130 || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] || 131 || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] || 130 [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] 131 [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] 132 132 133 133 … … 135 135 Which we can discretize for (1D) as 136 136 137 || [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i + \phi^n_i e^{*}_i + \theta^n_i) )]] || 138 || [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i - \phi^n_i e^{*}_i - \theta^n_i )]] || 137 [[latex(E^{n+1}_i-E^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}-E^{*}_{i} \right ) - \alpha^n_{i-1/2} \left ( E^{*}_{i}-E^{*}_{i-1} \right ) \right ] - \epsilon^n_i E^{*}_i + \phi^n_i e^{*}_i + \theta^n_i) )]] 138 [[latex(e^{n+1}_i-e^{n}_i = \epsilon^n_i E^{*}_i - \phi^n_i e^{*}_i - \theta^n_i )]] 139 139 140 140 where the diffusion coefficient is given by … … 178 178 In any event in 1D we have the following matrix coefficients 179 179 180 || [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+\bar{\psi}\phi^n_i e^n_i + \theta^n_i)]] || 181 || [[latex(\left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n )]] || 180 [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} - \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+\bar{\psi}\phi^n_i e^n_i + \theta^n_i)]] 181 [[latex(\left ( 1 +\psi \phi^n_i \right ) e^{n+1}_i - \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n )]] 182 182 183 183 184 184 If we ignore the change in the Planck function due to heating during the implicit solve, it is equivalent to replacing the term with [[latex(\psi \phi^n_i e^{n+1}_i)]] with [[latex(\psi \phi^n_i e^n_i)]] in which case the equations simplify to 185 185 186 || [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+\phi^n_i e^n_i + \theta^n_i)]] || 187 || [[latex(e^{n+1}_i = e^n_i + \epsilon^n_i \left [ \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) - \frac{4 \pi}{c} B \left ( T^n_i \right ) \right ] )]] || 186 [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i+\phi^n_i e^n_i + \theta^n_i)]] 187 [[latex(e^{n+1}_i = e^n_i + \epsilon^n_i \left [ \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) - \frac{4 \pi}{c} B \left ( T^n_i \right ) \right ] )]] 188 188 189 189 In this case the first equation decouples and can be solved on it's own, and then the solution plugged back into the second equation to solve for the new energy.
