Changes between Version 195 and Version 196 of FluxLimitedDiffusion
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 05/30/13 17:20:39 (11 years ago)
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FluxLimitedDiffusion
v195 v196 5 5 [[CollapsibleStart(Spectral Intensity)]] 6 6 == Spectral Intensity == 7 Typically when we discuss the radiation field we use the spectral intensity [[latex( I \left ( \nu, \mathbf{x}, \Omega \right ) )]] which is a function of frequency, position, and direction. This is very similar to the phase space density used in deriving the fluid equations [[latex(f \left ( \mathbf{x}, \mathbf{v} \right ))]] except that8 * light always travels at \(c\), so the velocity dependence is just a direction dependence.7 Typically when we discuss the radiation field we use the spectral intensity [[latex($I \left ( \nu, \mathbf{x}, \Omega \right ) $)]] which is a function of frequency, position, and direction. This is very similar to the phase space density used in deriving the fluid equations [[latex($f \left ( \mathbf{x}, \mathbf{v} \right ) $)]] except that 8 * light always travels at [[latex($c$)]], so the velocity dependence is just a direction dependence. 9 9 * Furthermore, photons can have different frequencies, so there is an extra dimension to the phase space. 10 10 * Instead of storing the phase space density of photons, the spectral intensity is the phase space density of energy flux... 11 11 12 Going between photon number and energy just involves a factor of \( h \nu \) and going from energy density to energy flux density just involves a factor of \(c\)so we have13 14 $$ I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $$12 Going between photon number and energy just involves a factor of [[latex($h\nu$)]] and going from energy density to energy flux density just involves a factor of [[latex($c$)]] so we have 13 14 [[latex($I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $)]] 15 15 16 16 This can also be seen by considering the differential energy 17 17 18 [[latex( dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]]18 [[latex($dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV$)]] 19 19 20 20 where the number of photons traveling normal to the surface dA that cross the surface dA in time dt is just the number of photons in the volume dV = dA c dt (assuming the photons are headed normal to dA)... 21 21 22 22 so we also have 23 [[latex( dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]]23 [[latex($dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt$)]] 24 24 25 25 which gives 26 26 27 [[latex( I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ))]]27 [[latex($I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) $)]] 28 28 29 29 … … 33 33 == Deriving the Transport Equation == 34 34 35 If we consider the Boltzmann transport equation for photons of a specific frequency \(f_\nu\)we have36 37 [[latex( \frac{\partial}{\partial t} f_\nu + \mathbf{v} \cdot \nabla f + \mathbf{F} \cdot \nabla_p f = \left ( \frac{\partial f}{\partial t} \right )_{coll})]]35 If we consider the Boltzmann transport equation for photons of a specific frequency [[latex($f_\nu$)]] we have 36 37 [[latex($\frac{\partial}{\partial t} f_\nu + \mathbf{v} \cdot \nabla f + \mathbf{F} \cdot \nabla_p f = \left (\frac{\partial f}{\partial t} \right )_{coll} $)]] 38 38 39 39 Now photons don't experience body forces, always travel at the speed of light, and in general the "collision term" consists of photon emission and absorptions... so we have 40 40 41 [[latex( \frac{\partial}{\partial t} f_\nu + c \mathbf{n} \cdot \nabla f_\nu = A_{\nu}  \chi_\nu f_\nu c)]]42 43 where \(A_{\nu} \) is the emission rate of photons of frequency \(\nu\) and the mean free path length is given by \(\chi_\nu = \sigma_\nu n\) where \(\sigma_\nu\) is the particle scattering cross section and \(n\)is the number density of particles.44 45 Now if we multiply through by \( h\nu\)we have46 [[latex( \frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot \nabla I_\nu = \eta_\nu  \sigma_\nu I_\nu)]]47 48 where \(\eta_\nu = h\nu A_{\nu}\)is the radiative power41 [[latex($\frac{\partial}{\partial t} f_\nu + c \mathbf{n} \cdot \nabla f_\nu = A_{\nu}  \chi_\nu f_\nu c $)]] 42 43 where [[latex($A_{\nu} $)]] is the emission rate of photons of frequency [[latex($\nu$)]] and the mean free path length is given by [[latex($\chi_\nu = \sigma_\nu n$)]] where [[latex($\sigma_\nu$)]] is the particle scattering cross section and [[latex($n$)]] is the number density of particles. 44 45 Now if we multiply through by [[latex($ h\nu$)]] we have 46 [[latex($\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot \nabla I_\nu = \eta_\nu  \sigma_\nu I_\nu $)]] 47 48 where [[latex($\eta_\nu = h\nu A_{\nu}$)]] is the radiative power 49 49 50 50 If we solve the transport equation along a characteristic 51 51 52 [[latex( \left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ])]]52 [[latex($\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] $)]] 53 53 54 54 we have 55 55 56 [[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(s)  \chi_\nu(s) I_\nu (s) )]] 57 58 where \(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) \) and then we can divide through by \(\chi_\nu(s)\) we get 59 60 [[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)}  I_\nu(s) = S_\nu(s)  I_\nu(s))]] 61 62 Now if we define \(d\tau_\nu = \chi_\nu(s) ds \) which gives 63 64 [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]] 65 and 66 [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] 56 [[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(s)  \chi_\nu(s) I_\nu (s) $)]] 57 58 where [[latex($f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) $)]] and then we can divide through by [[latex($\chi_\nu(s)$)]] we get 59 60 [[latex($\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)}  I_\nu(s) = S_\nu(s)  I_\nu(s)$)]] 61 62 Now if we define [[latex($d\tau_\nu = \chi_\nu(s) ds $)]] which gives 63 64 [[latex($\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds'$)]] 65 and 66 67 [[latex($s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu$)]] 67 68 68 69 we can write the transport equation in the simplest form 69 70 70 [[latex( \frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu)  I_\nu(\tau_\nu))]]71 [[latex($\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu)  I_\nu(\tau_\nu)$)]] 71 72 72 73 although the RHS is now more difficult to evaluate as 73 74 74 [[latex( f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ))]]75 76 Also if we include scattering then the source function can depend on the mean radiative flux \(\frac{cE}{4 \pi}\)and the transport equation becomes an integrodifferential equation that must be solved iteratively...75 [[latex($ f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ) $)]] 76 77 Also if we include scattering then the source function can depend on the mean radiative flux [[latex($\frac{cE}{4 \pi}$)]] and the transport equation becomes an integrodifferential equation that must be solved iteratively... 77 78 78 79 There are also a few important dimensionless numbers to consider: 79 80 80  [[latex( \tau = l \kappa=\frac{l}{\lambda_p})]]  [[latex(\beta = \frac{u}{c})]] 81  [[latex( \tau << 1)]]  streaming limit 82  [[latex( \tau >> 1 \mbox{, } \beta \tau << 1)]]  static diffusion limit 83  [[latex( \tau >> 1 \mbox{, } \beta \tau >> 1)]]  dynamic diffusion limit 81  [[latex($\tau = l \kappa=\frac{l}{\lambda_p}$)]]  [[latex($\beta = \frac{u}{c}$)]]  82  [[latex($\tau << 1 $)]]  streaming limit  83  [[latex($\tau >> 1 \mbox{, } \beta \tau << 1$)]]  static diffusion limit  84  [[latex($\tau >> 1 \mbox{, } \beta \tau >> 1$)]]  dynamic diffusion limit  84 85 85 86 [[CollapsibleEnd()]] … … 90 91 Some of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007] 91 92 92  [[latex( \frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0)]] 93  [[latex( \frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = \nabla P+\mathbf{G})]] 94  [[latex( \frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^0)]] 95  [[latex( \frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = c G_0)]] 96  [[latex( \frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = \mathbf{G})]] 93  [[latex($\frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0$)]]  94  [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = \nabla P+\mathbf{G}$)]]  95  [[latex($\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^0$)]]  96  [[latex($\frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = c G_0 $)]]  97  [[latex($\frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = \mathbf{G}$)]]  97 98 98 99 where the moments of the specific intensity are defined as 99 100  Radiation Energy moments  Corresponding fluid moments  100  [[latex( cE =\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu))]]  [[latex(\rho=\int d \mathbf{v} f(\mathbf{v}))]] 101  [[latex( \mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu))]]  [[latex(\rho \mathbf{v} =\int d \mathbf{v} \mathbf{v} f(\mathbf{v}))]] 102  [[latex( c\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu))]]  [[latex(\mathbf{P} =\int d \mathbf{v} \mathbf{vv} f(\mathbf{v}))]] 101  [[latex($cE =\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu)$)]]  [[latex($\rho=\int d \mathbf{v} f(\mathbf{v})$)]]  102  [[latex($\mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu)$)]]  [[latex($\rho \mathbf{v} =\int d \mathbf{v} \mathbf{v} f(\mathbf{v})$)]]  103  [[latex($c\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu)$)]]  [[latex($\mathbf{P} =\int d \mathbf{v} \mathbf{vv} f(\mathbf{v})$)]]  103 104 104 105 and the radiation 4force density is given by 105 106 106  [[latex( cG^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)\eta(\mathbf{n},v) \right ])]] 107  [[latex( c\mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)\eta(\mathbf{n},v) \right ] \mathbf{n})]] 108 109 If we had a closure relation for the radiation pressure then we could solve this system. For gas particles, collisions tend to produce a Boltzmann Distribution which is isotropic and gives a pressure tensor that is a multiple of the identity tensor. Photons do not "collide" with each other and they all have the same velocity 'c' but in various directions. If the field were isotropic than \(P^{ij}=\delta^{ij} 1/3 E\) but in general \(P^{ij}=f^{ij} E\)where 'f' is the Eddington Tensor.107  [[latex($cG^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)\eta(\mathbf{n},v) \right ]$)]]  108  [[latex($c\mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)\eta(\mathbf{n},v) \right ] \mathbf{n}$)]]  109 110 If we had a closure relation for the radiation pressure then we could solve this system. For gas particles, collisions tend to produce a Boltzmann Distribution which is isotropic and gives a pressure tensor that is a multiple of the identity tensor. Photons do not "collide" with each other and they all have the same velocity 'c' but in various directions. If the field were isotropic than [[latex($P^{ij}=\delta^{ij} 1/3 E$)]] but in general [[latex($P^{ij}=f^{ij} E$)]] where 'f' is the Eddington Tensor. 110 111 111 112 [[CollapsibleEnd()]] … … 113 114 == Simplifying assumptions == 114 115 * If the flux spectrum of the radiation is directionindependent then we can write the radiation fourforce density in terms of the moments of the radiation field 115  [[latex( G^0=\gamma [ \gamma^2 \kappa_{0E} + (1\gamma^2)\kappa_{0F}]E\gamma \kappa_{0P}a_RT_0^4\gamma(\mathbf{v} \cdot \mathbf{F}/c^2)[\kappa_{0F}2\gamma^2(\kappa_{0F}\kappa_{0E})]  \gamma^3(\kappa_{0F}\kappa+{0E})(\mathbf{vv}) : \mathbf{P}/c^2)]] 116  [[latex( \mathbf{G} = \gamma \kappa_{0F}(\mathbf{F}/c)\gamma \kappa_{0P}a_RT_0^4 (\mathbf{v}/c)  \left [ \gamma^3( \kappa_{0F}  \kappa_{0E})( \mathbf{v}/c)E + \gamma \kappa_{0F}(\mathbf{v}/c) \cdot \mathbf{P} \right ] + \gamma^3 ( \kappa_{0F}  \kappa_{0E}) \left [ 2 \mathbf{v} \cdot \mathbf{F}/c^3  ( \mathbf{vv}) : \mathbf{P}/c^3 \right ] \mathbf{v})]] 116  [[latex($G^0=\gamma [ \gamma^2 \kappa_{0E} + (1\gamma^2)\kappa_{0F}]E\gamma \kappa_{0P}a_RT_0^4\gamma(\mathbf{v} \cdot \mathbf{F}/c^2)[\kappa_{0F}2\gamma^2(\kappa_{0F}\kappa_{0E})]  \gamma^3(\kappa_{0F}\kappa+{0E})(\mathbf{vv}) : \mathbf{P}/c^2$)]]  117  [[latex($\mathbf{G} = \gamma \kappa_{0F}(\mathbf{F}/c)\gamma \kappa_{0P}a_RT_0^4 (\mathbf{v}/c)  \left [ \gamma^3( \kappa_{0F}  \kappa_{0E})( \mathbf{v}/c)E + \gamma \kappa_{0F}(\mathbf{v}/c) \cdot \mathbf{P} \right ] + \gamma^3 ( \kappa_{0F}  \kappa_{0E}) \left [ 2 \mathbf{v} \cdot \mathbf{F}/c^3  ( \mathbf{vv}) : \mathbf{P}/c^3 \right ] \mathbf{v} $)]]  117 118 118 119 where 119 120 120  [[latex( \kappa_{0P})]]  comovingframe Planck function weighted opacity 121  [[latex( \kappa_{0E})]]  comovingframe radiation energy weighted opacity 122  [[latex( \kappa_{0F})]]  comovingframe radiation flux weighted opacity 123 124 * If the radiation has a blackbody spectrum then \(\kappa_{0E}=\kappa_{0P}\)121  [[latex($\kappa_{0P}$)]]  comovingframe Planck function weighted opacity  122  [[latex($\kappa_{0E}$)]]  comovingframe radiation energy weighted opacity  123  [[latex($\kappa_{0F}$)]]  comovingframe radiation flux weighted opacity  124 125 * If the radiation has a blackbody spectrum then [[latex($\kappa_{0E}=\kappa_{0P}$)]] 125 126 * If the radiation is optically thick, then 126 [[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))]]127 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]})]]128 * In the optically thin regime, \(\mathbf{F}_0(\nu_0) \rightarrow cE_0(\nu_0)\) so we would have \(\kappa_{0F}=\kappa_{0E}\) however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that \(\kappa_{0F}=\kappa_{0R}\)127 [[latex($\mathbf{F_0}(\nu_0) \propto \nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto [\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0)$)]] 128 which implies that [[latex($\kappa_{0F}^{1}=\kappa_{0R}^{1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]}$)]] 129 * 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($\kappa_{0F}=\kappa_{0R}$)]] 129 130 [[CollapsibleEnd()]] 130 131 … … 133 134 134 135 The flux limited diffusion approximation drops the radiation momentum equation in favor of 135 \(\mathbf{F}_0=\frac{c\lambda}{\kappa_{0R}}\nabla E_0\) where \(\lambda\)is the fluxlimiter and is given by136 137 [[latex( \lambda = \frac{1}{R} \left ( \coth R  \frac{1}{R} \right ))]]138 and 139 [[latex( R=\frac{\nabla E_0}{\kappa_{0R}E_0})]]136 [[latex($\mathbf{F}_0=\frac{c\lambda}{\kappa_{0R}}\nabla E_0$)]] where [[latex($\lambda$)]] is the fluxlimiter and is given by 137 138 [[latex($\lambda = \frac{1}{R} \left ( \coth R  \frac{1}{R} \right ) $)]] 139 and 140 [[latex($R=\frac{\nabla E_0}{\kappa_{0R}E_0}$)]] 140 141 141 142 which corresponds to a pressure tensor 142 [[latex( \mathbf{P}_0=\frac{E_0}{2}[(1R_2)\mathbf{I}+(3R_21)\mathbf{n}_0\mathbf{n}_0])]]143 [[latex($\mathbf{P}_0=\frac{E_0}{2}[(1R_2)\mathbf{I}+(3R_21)\mathbf{n}_0\mathbf{n}_0]$)]] 143 144 where 144 [[latex( R_2=\lambda+\lambda^2R^2)]]145 146 Here are \(\lambda(R)\) and \(R_2(R)\)plotted147 [[Image(fld.png, width=400 )]]145 [[latex($R_2=\lambda+\lambda^2R^2$)]] 146 147 Here are [[latex($\lambda(R)$)]] and [[latex($R_2(R)$)]] plotted 148 [[Image(fld.png, width=400$)]] 148 149 149 150 150 151 If we Lorentz boost the comoving terms into the lab frame and keep terms necessary to maintain accuracy we get: 151 152 152 [[latex( G^0=\kappa_{0P} \left ( E\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}  1 \right ) \mathbf{v} \cdot \nabla E  \frac{\kappa_{0P}}{c^2} E \left [ \frac{3R_2}{2}v^2 + \frac{3R_21}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E  \frac{4 \pi B}{c} \right ))]]153 [[latex( \mathbf{G} = \lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E  \frac{4 \pi B}{c} \right )  \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}}  1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2})]]153 [[latex($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{3R_2}{2}v^2 + \frac{3R_21}{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 ) $)]] 154 [[latex($\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}$)]] 154 155 155 156 [[CollapsibleEnd()]] … … 159 160 If we plug the expressions for the radiation 4momentum back into the gas equations and keep terms necessary to maintain accuracy we get: 160 161 161 [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = \nabla P\color{green}{\lambda \nabla E})]] 162 [[latex(\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \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{blue}{\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]] 163 [[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} )]] 164 165 166 For static diffusion, the terms in blue with \(\frac{v^2}{c}\) can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). 162 [[latex($\frac{\partial }{\partial t} \left(\rho\mathbf{v}\right)+\nabla\cdot\left(\rho\mathbf{vv}\right)=\nabla P\color{green}{\lambda \nabla E}$)]] 163 164 [[latex($\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] =\color{red{\kappa_{0P}(4 \pi BcE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right)\mathbf{v}\cdot \nabla E} \color{blue}{\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E}$)]] 165 [[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} $)]] 166 167 168 For static diffusion, the terms in blue with [[latex($\frac{v^2}{c}$)]] can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). 167 169 168 170 [[CollapsibleStart(Operator Splitting 1)]] … … 172 174 In AstroBEAR this would look like: 173 175 * Initialization 174 * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}^I\)176 * Prolongate, [[latex($\rho$)]], [[latex($\rho\mathbf{v}$)]], [[latex($e$)]], [[latex($E$)]], and [[latex($\dot{E}^I$)]] 175 177 * Step 1 176 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\)and do physical BC's177 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)178 * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}^I_{2\mbox{mbc}}\)179 * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}^I_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)180 * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)181 * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}^I_{\mbox{mbc}+1}\)178 * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's 179 * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}^I_1$)]] 180 * Update [[latex($E_{2\mbox{mbc}}$)]] using [[latex($\dot{E}^I_{2\mbox{mbc}}$)]] 181 * Update [[latex($e_{2\mbox{mbc}}$)]] using [[latex($E_{2\mbox{mbc}}$)]], [[latex($\dot{E}^I_{2\mbox{mbc}}$)]], and [[latex($e_{2\mbox{mbc}}$)]] 182 * Update [[latex($\dot{E}^I_0$)]] using pre IR and post IR [[latex($E_0$)]] 183 * Ghost [[latex($e_{2\mbox{mbc}}$)]], [[latex($E_{\mbox{mbc}+1}$)]], [[latex($\dot{E}^I_{\mbox{mbc}+1}$)]] 182 184 * Do first EH,,mbc,, 183 * Do ER,,mbc,,  Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\). The \(\nabla \cdot \mathbf{v}E\)term needs time centered face centered velocities which can be stored during the hydro update.184 * Store \(\dot{E}\)in child arrays to be prolongated185 * Do ER,,mbc,,  Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]]. The [[latex($\nabla \cdot \mathbf{v}E$)]] term needs time centered face centered velocities which can be stored during the hydro update. 186 * Store [[latex($\dot{E}$)]] in child arrays to be prolongated 185 187 * Step 2 186 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\)and do physical BC's187 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}^I_1\)188 * Update \(\dot{E}^I_0\) using pre IR and post IR \(E_0\)189 * Update \(E_{1}\) using \(\dot{E}^I_{1}\)190 * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}^I_{1}\)188 * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's 189 * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}^I_1$)]] 190 * Update [[latex($\dot{E}^I_0$)]] using pre IR and post IR [[latex($E_0$)]] 191 * Update [[latex($E_{1}$)]] using [[latex($\dot{E}^I_{1}$)]] 192 * Ghost [[latex($e_{mbc}$)]], [[latex($E_{1}$)]], [[latex($\dot{E}^I_{1}$)]] 191 193 * Do second EH,,0,, 192 * Do ER,,mbc,,  Terms with \(\nabla E\)can be done without ghosting since EH did not193 * Do ER,,0,,  Terms with grad E can be done without ghosting since EH did not change \(E\). The \(\nabla \cdot \mathbf{v}E\)term needs time centered face centered velocities which can be stored during the hydro update.194 * Do ER,,mbc,,  Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not 195 * Do ER,,0,,  Terms with grad E can be done without ghosting since EH did not change [[latex($E$)]]. The [[latex($\nabla \cdot \mathbf{v}E$)]] term needs time centered face centered velocities which can be stored during the hydro update. 194 196 195 197 [[CollapsibleEnd()]] … … 199 201 The extra terms in the explicit update due to radiation energy are as follows: 200 202 201 [[latex( \frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =\lambda \nabla E)]]202 [[latex( \frac{\partial e}{\partial t} = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E)]]203 [[latex( \frac{\partial E}{\partial t} = \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right ))]]203 [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =\lambda \nabla E$)]] 204 [[latex($\frac{\partial e}{\partial t} = \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E $)]] 205 [[latex($\frac{\partial E}{\partial t} = \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right ) $)]] 204 206 205 207 These can be discretized as follows: 206 208 207 [[latex( p^{n+1}_i=p^n_i  \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( E^n_{i+1}E^n_{i1} \right ))]]208 209 [[latex( e^{n+1}_i=e^n_i + \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_i \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}1 \right ) \left ( v^n_i \left ( E^n_{i+1}E^n_{i1} \right ) \right ))]]210 211 [[latex( E^{n+1}_i=E^n_i  \frac{\Delta t}{\Delta x} \left ( \frac{\lambda_i}{2} \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}1 \right ) \left ( v^n_i \left ( E^n_{i+1}E^n_{i1} \right ) \right ) + \left ( F_{i+1/2}F_{i1/2} \right ) \right ))]]209 [[latex($p^{n+1}_i=p^n_i  \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( E^n_{i+1}E^n_{i1} \right ) $)]] 210 211 [[latex($e^{n+1}_i=e^n_i + \frac{1}{2}\frac{\Delta t}{\Delta x} \lambda_i \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}1 \right ) \left ( v^n_i \left ( E^n_{i+1}E^n_{i1} \right ) \right ) $)]] 212 213 [[latex($E^{n+1}_i=E^n_i  \frac{\Delta t}{\Delta x} \left ( \frac{\lambda_i}{2} \left ( 2 \frac{\kappa^n_{i,0P}}{\kappa^n_{i,0R}}1 \right ) \left ( v^n_i \left ( E^n_{i+1}E^n_{i1} \right ) \right ) + \left ( F_{i+1/2}F_{i1/2} \right ) \right )$)]] 212 214 213 215 where 214 [[latex( F_{i+1/2}=\frac{3R_{2,i+1/2}}{8} \left(v_{i}^n+v_{i+1}^n \right ) \left ( E^n_i+E^n_{i+1} \right ))]]216 [[latex($F_{i+1/2}=\frac{3R_{2,i+1/2}}{8} \left(v_{i}^n+v_{i+1}^n \right ) \left ( E^n_i+E^n_{i+1} \right ) $)]] 215 217 216 218 where 217 219 218 [[latex( R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]]219 and 220 [[latex( R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]221 222 and 223 [[latex( \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ))]]224 225 and 226 [[latex( \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ))]]227 228 and 229 230 [[latex( \lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ))]]231 232 and 233 [[latex( R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}})]]220 [[latex($R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2$)]] 221 and 222 [[latex($R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]] 223 224 and 225 [[latex($\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ) $)]] 226 227 and 228 [[latex($\kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ) $)]] 229 230 and 231 232 [[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ) $)]] 233 234 and 235 [[latex($R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}}$)]] 234 236 235 237 [[CollapsibleEnd()]] … … 238 240 == Implicit Update 1 == 239 241 240 For now we will assume that \(\kappa_{0P}\) and \(\kappa_{0R}\)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:241 242 [[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))]]243 [[latex( \frac{\partial e}{\partial t} =  \kappa_{0P} (4 \pi B(T)cE))]]244 245 where [[latex( \mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E)]]242 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: 243 244 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)cE)$)]] 245 [[latex($\frac{\partial e}{\partial t} =  \kappa_{0P} (4 \pi B(T)cE)$)]] 246 247 where [[latex($\mathbf{F} = \frac{c\lambda}{\kappa_{0R}} \nabla E$)]] 246 248 247 249 === Expanding about e,,0,, === 248 250 249 Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still nonlinear due to the dependence on Temperature of the Planck Function \(B(T)\)250 251 However we can expand the Plank Function about \(e_0\)252 253 [[latex( B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right  _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right ))]]251 Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still nonlinear due to the dependence on Temperature of the Planck Function [[latex($B(T)$)]] 252 253 However we can expand the Plank Function about [[latex($e_0$)]] 254 255 [[latex($B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right  _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right ) $)]] 254 256 255 257 where 256 258 257 [[latex( \Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma1)}{n k_B})]]259 [[latex($\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma1)}{n k_B}$)]] 258 260 259 261 Then the system of equations becomes 260 262 261 [[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{ee_0}{T_0} \right )cE \right ])]]262 [[latex( \frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ])]]263 264 which will be accurate as long as \(\left  4\Gamma \frac{ee_0}{T_0} \right  < \xi << 1\) or \(\Delta e = \left  ee_0 \right  < \xi \frac{T_0}{4 \Gamma}\)263 [[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{ee_0}{T_0} \right )cE \right ] $)]] 264 [[latex($\frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] $)]] 265 266 which will be accurate as long as [[latex($\left  4\Gamma \frac{ee_0}{T_0} \right  < \xi << 1$)]] or [[latex($\Delta e = \left  ee_0 \right  < \xi \frac{T_0}{4 \Gamma}$)]] 265 267 266 268 We can calculate the time scale for this to be true using the evolution equation for the energy density 267 269 268 [[latex( \Delta e = \Delta t \kappa_{0P} \left  4 \pi B_0 cE \right  < \xi \frac{T_0}{4 \Gamma})]]269 270 which gives [[latex( \Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left  4 \pi B_0  cE \right  })]]270 [[latex($\Delta e = \Delta t \kappa_{0P} \left  4 \pi B_0 cE \right  < \xi \frac{T_0}{4 \Gamma}$)]] 271 272 which gives [[latex($\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left  4 \pi B_0  cE \right  }$)]] 271 273 272 274 … … 275 277 We can now discretize the equations 276 278 277 [[latex( E^{n+1}_iE^{n}_i = \left [ \alpha_{i+1/2} \left ( E^{*}_{i+1}E^{*}_{i} \right )  \alpha_{i1/2} \left ( E^{*}_{i}E^{*}_{i1} \right ) \right ]  \epsilon E^{*}_i + \phi e^{*}_i + \theta_i  \phi_i e^n_i))]]278 [[latex( e^{n+1}_ie^{n}_i = \epsilon_i E^{*}_i  \phi_i e^{*}_i  \left(\theta_i\phi_i e^n_i \right ))]]279 [[latex($E^{n+1}_iE^{n}_i = \left [ \alpha_{i+1/2} \left ( E^{*}_{i+1}E^{*}_{i} \right )  \alpha_{i1/2} \left ( E^{*}_{i}E^{*}_{i1} \right ) \right ]  \epsilon E^{*}_i + \phi e^{*}_i + \theta_i  \phi_i e^n_i) $)]] 280 [[latex($e^{n+1}_ie^{n}_i = \epsilon_i E^{*}_i  \phi_i e^{*}_i  \left(\theta_i\phi_i e^n_i \right ) $)]] 279 281 280 282 where the diffusion coefficient is given by 281 283 282 [[latex( \alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} \mbox{ where } \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ))]]284 [[latex($\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} \mbox{ where } \kappa_{0R,i+1/2} = \frac{\kappa^n_{0R,i}+\kappa^n_{0R,i+1}}{2} \mbox{ and } \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ) $)]] 283 285 and where 284 [[latex( R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]285 286 287 and 288 289 [[latex( \epsilon_i=c\Delta t \kappa_{0P,i})]]286 [[latex($R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]] 287 288 289 and 290 291 [[latex($\epsilon_i=c\Delta t \kappa_{0P,i}$)]] 290 292 291 293 represents the number of absorption/emissions during the time step … … 294 296 and 295 297 296 [[latex( \phi_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ))]]297 298 [[latex( \theta_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ))]]298 [[latex($\phi_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) $)]] 299 300 [[latex($\theta_i = \epsilon_i \frac{4 \pi}{c} B \left ( T^n_i \right ) $)]] 299 301 300 302 301 303 and we can think of the radiative flux as 302 304 303 [[latex( \frac{\Delta t}{\Delta x}\mathbf{F}_{i+1/2} = \alpha_{i+1/2} \left ( E^{*}_{i+1}  E^{*}_i \right ))]]305 [[latex($\frac{\Delta t}{\Delta x}\mathbf{F}_{i+1/2} = \alpha_{i+1/2} \left ( E^{*}_{i+1}  E^{*}_i \right ) $)]] 304 306 305 307 === Time Discretization === 306 308 307 309 Now all the terms on the right hand side that are linear in E or e have been written as E^*^ or e^*^ because there are different ways to approximate E^*^ (e^*^). For Backward Euler we have 308 [[latex( E^*_i = E^{n+1}_i)]]310 [[latex($E^*_i = E^{n+1}_i$)]] 309 311 and for Crank Nicholson we have 310 [[latex( E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ))]]312 [[latex($E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) $)]] 311 313 or we can parameterize the solution 312 [[latex( E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i)]]313 where \(\bar{\psi} = 1\psi\)314 315 Backward Euler has \(\psi=1\) and Crank Nicholson has \(\psi=1/2\)316 317 Forward Euler has \(\psi=0\)314 [[latex($E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i$)]] 315 where [[latex($\bar{\psi} = 1\psi$)]] 316 317 Backward Euler has [[latex($\psi=1$)]] and Crank Nicholson has [[latex($\psi=1/2$)]] 318 319 Forward Euler has [[latex($\psi=0$)]] 318 320 319 321 In any event in 1D we have the following matrix coefficients 320 322 321 [[latex( \left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i1/2} + \epsilon_i \right ) \right ] E^{n+1}_i  \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1}  \left ( \psi \alpha_{i1/2} \right ) E^{n+1}_{i1}  \left ( \psi \phi_i \right ) e^{n+1}_i=\left [ 1  \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i1/2} + \epsilon_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i1/2} \right ) E^{n}_{i1}  \psi\phi_i e^n_i + \theta_i)]]322 323 [[latex( \left ( 1 +\psi \phi_i \right ) e^{n+1}_i  \left ( \psi \epsilon_i \right )E^{n+1}_i =\left ( 1 +\psi\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i)]]323 [[latex($\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i1/2} + \epsilon_i \right ) \right ] E^{n+1}_i  \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1}  \left ( \psi \alpha_{i1/2} \right ) E^{n+1}_{i1}  \left ( \psi \phi_i \right ) e^{n+1}_i=\left [ 1  \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i1/2} + \epsilon_i \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i1/2} \right ) E^{n}_{i1}  \psi\phi_i e^n_i + \theta_i$)]] 324 325 [[latex($\left ( 1 +\psi \phi_i \right ) e^{n+1}_i  \left ( \psi \epsilon_i \right )E^{n+1}_i =\left ( 1 +\psi\phi_i \right ) e^n_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i $)]] 324 326 325 327 326 328 Now since the second equation has no spatial dependence, we can solve it for 327 [[latex( \color{purple}{e^{n+1}_i = e^n_i + \frac{1}{ 1 +\psi \phi_i}\left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i \right \}})]]329 [[latex($\color{purple}{e^{n+1}_i = e^n_i + \frac{1}{ 1 +\psi \phi_i}\left \{ \left ( \psi \epsilon_i \right)E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i \right \}}$)]] 328 330 329 331 and plug the result into the first equation to get a matrix equation involving only one variable. 330 332 331 [[latex( \color{purple}{\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i1/2} + \frac{\epsilon_i}{ 1 +\psi \phi_i}\right ) \right ] E^{n+1}_i  \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1}  \left ( \psi \alpha_{i1/2} \right ) E^{n+1}_{i1} =\left [ 1  \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i1/2} +\frac{\epsilon_i }{ 1 +\psi \phi_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i1/2} \right ) E^{n}_{i1} + \frac{\theta_i}{ 1 +\psi \phi_i}})]]333 [[latex($\color{purple}{\left [ 1 + \psi \left( \alpha_{i+1/2} + \alpha_{i1/2} + \frac{\epsilon_i}{ 1 +\psi \phi_i}\right ) \right ] E^{n+1}_i  \left ( \psi \alpha_{i+1/2} \right ) E^{n+1}_{i+1}  \left ( \psi \alpha_{i1/2} \right ) E^{n+1}_{i1} =\left [ 1  \bar{\psi} \left( \alpha_{i+1/2} + \alpha_{i1/2} +\frac{\epsilon_i }{ 1 +\psi \phi_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \alpha_{i+1/2} \right ) E^{n}_{i+1} + \left ( \bar{\psi} \alpha_{i1/2} \right ) E^{n}_{i1} + \frac{\theta_i}{ 1 +\psi \phi_i}}$)]] 332 334 333 335 … … 343 345 For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E, and then we can solve for the new e given our guess at the new E using the same time stepping (Backward Euler, Crank Nicholson, etc...). 344 346 345 [[latex( E^{n+1}_i = E^{n}_i+\dot{E}^{I}_i \Delta t)]]346 [[latex( e^{n+1}_i = e^n_i + \frac{1}{1+\psi \phi_i} \left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i \right \})]]347 [[latex($E^{n+1}_i = E^{n}_i+\dot{E}^{I}_i \Delta t$)]] 348 [[latex($e^{n+1}_i = e^n_i + \frac{1}{1+\psi \phi_i} \left \{ \left ( \psi \epsilon_i \right )E^{n+1}_i + \left ( \bar{\psi} \epsilon_i \right ) E^n_i\theta_i \right \} $)]] 347 349 348 350 == CoarseFine Boundaries == … … 356 358 === Open (Free streaming) boundaries === 357 359 We would like the radiation to leave at the free streaming limit. So 358 [[latex( \frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( EE_g \right ) }{\Delta x})]]359 360 Clearly if we set \(E_g = 0\) and \(\lambda_g =\kappa_g \Delta x\)we should get the correct flux.360 [[latex($\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( EE_g \right ) }{\Delta x}$)]] 361 362 Clearly if we set [[latex($E_g = 0$)]] and [[latex($\lambda_g =\kappa_g \Delta x$)]] we should get the correct flux. 361 363 362 364 This corresponds to an 363 \(\alpha = c \frac{\Delta t}{\Delta x}\) 364 365 So we would just modify \(\alpha\)and zero out the matrix coefficient to the ghost zone365 [[latex($\alpha = c \frac{\Delta t}{\Delta x}$)]] 366 367 So we would just modify [[latex($\alpha$)]] and zero out the matrix coefficient to the ghost zone 366 368 367 369 === Constant Slope Boundary === 368 Here we want the flux to be constant so energy does not pile up near the boundary. If we cancel all derivative terms on both sides of the cell, this will effectively match the incoming flux with the outgoing flux. This can also be done by setting \(\alpha_g = \alpha_i = 0\)370 Here we want the flux to be constant so energy does not pile up near the boundary. If we cancel all derivative terms on both sides of the cell, this will effectively match the incoming flux with the outgoing flux. This can also be done by setting [[latex($\alpha_g = \alpha_i = 0$)]] 369 371 370 372 === Periodic Boundary === … … 376 378 377 379 === Reflecting/ZeroSlope Boundary === 378 Reflecting boundary should be fairly straightforward. This an be achieved by setting \(\alpha_g = 0\) which zeros out any flux  and has the same effect as setting \(E^{*}_g=E^{*}_i\) or \(E^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_i\)380 Reflecting boundary should be fairly straightforward. This an be achieved by setting [[latex($\alpha_g = 0$)]] which zeros out any flux  and has the same effect as setting [[latex($E^{*}_g=E^{*}_i$)]] or [[latex($E^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_i$)]] 379 381 380 382 === Constant radiative flux === 381 To have a constant radiative flux we must zero out terms involving the gradient and just add \(F_0 \frac{\Delta t}{\Delta x}\)in the source vector383 To have a constant radiative flux we must zero out terms involving the gradient and just add [[latex($F_0 \frac{\Delta t}{\Delta x}$)]] in the source vector 382 384 383 385 === Summary === 384 386 385  Numerical value  Boundary  [[latex( E^{n+1}_{i+1})]]  [[latex(E^{n+1}_i)]]  [[latex(E^{n+1}_{i1})]]  [[latex(S)]]386  0  RAD_FREE_STREAMING  [[latex( 0)]]  [[latex(\psi c \frac{\Delta t}{\Delta x})]]  [[latex(\psi\alpha_{i1/2})]]  [[latex(\bar{\psi}c \frac{\Delta t}{\Delta x} E^n_i + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right ))]] 387  1  RAD_EXTRAPOLATE_FLUX  [[latex( 0)]]  [[latex(0)]]  [[latex(0)]]  [[latex(0)]] 388  2  INTERNAL/PERIODIC  [[latex( \psi\alpha_{i+1/2})]]  [[latex(\psi\alpha_{i+1/2})]]  [[latex(\psi\alpha_{i1/2})]]  [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}E^n_i \right ) + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right ))]] 389  3  RAD_REFLECTING  [[latex( 0)]]  [[latex(0)]]  [[latex(\psi\alpha_{i1/2})]]  [[latex( \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right ))]] 390  4  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  [[latex( 0)]]  [[latex(\psi\alpha_{i+1/2})]]  [[latex(\psi\alpha_{i1/2})]]  [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}E^n_i \right )  \psi \alpha_{i+1/2} E^{n+1}_{i+1} + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right ))]] 391  5  RAD_USERDEFINED_FLUX  [[latex( 0)]]  [[latex(0)]]  [[latex(\psi\alpha_{i1/2})]]  [[latex(F_0 \frac{\Delta t}{\Delta x} + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right ))]] 387  Numerical value  Boundary  [[latex($E^{n+1}_{i+1}$)]]  [[latex($E^{n+1}_i$)]]  [[latex($E^{n+1}_{i1}$)]]  [[latex($S$)]] 388  0  RAD_FREE_STREAMING  [[latex($0$)]]  [[latex($\psi c \frac{\Delta t}{\Delta x}$)]]  [[latex($\psi\alpha_{i1/2}$)]]  [[latex($\bar{\psi}c \frac{\Delta t}{\Delta x} E^n_i + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right )$)]]  389  1  RAD_EXTRAPOLATE_FLUX  [[latex($0$)]]  [[latex($0$)]]  [[latex($0$)]]  [[latex($0$)]]  390  2  INTERNAL/PERIODIC  [[latex($\psi\alpha_{i+1/2}$)]]  [[latex($\psi\alpha_{i+1/2}$)]]  [[latex($\psi\alpha_{i1/2}$)]]  [[latex($\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}E^n_i \right ) + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right )$)]]  391  3  RAD_REFLECTING  [[latex($0$)]]  [[latex($0$)]]  [[latex($\psi\alpha_{i1/2}$)]]  [[latex($ \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right )$)]]  392  4  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  [[latex($0$)]]  [[latex($\psi\alpha_{i+1/2}$)]]  [[latex($\psi\alpha_{i1/2}$)]]  [[latex($\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}E^n_i \right )  \psi \alpha_{i+1/2} E^{n+1}_{i+1} + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right )$)]]  393  5  RAD_USERDEFINED_FLUX  [[latex($0$)]]  [[latex($0$)]]  [[latex($\psi\alpha_{i1/2}$)]]  [[latex($F_0 \frac{\Delta t}{\Delta x} + \bar{\psi}\alpha_{i1/2} \left ( E^n_{i1}E^n_i \right )$)]]  392 394 393 395 … … 401 403 402 404 403 [[latex( \frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = \nabla P\color{green}{\lambda \nabla E})]]404 [[latex( \frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{\kappa_{0P}(4 \pi BcE)} \color{magenta}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E} \color{magenta}{\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]405 [[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{magenta}{\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E} \color{magenta}{\nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )} \color{magenta}{+\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]]405 [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = \nabla P\color{green}{\lambda \nabla E}$)]] 406 [[latex($\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{\kappa_{0P}(4 \pi BcE)} \color{magenta}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E} \color{magenta}{\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E}$)]] 407 [[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{magenta}{\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E} \color{magenta}{\nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )} \color{magenta}{+\frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E} $)]] 406 408 407 409 It is possible that with included the terms in magenta in a semiimplicit method, the dynamic diffusion regime may be stable... In any event, it costs very little to add all of the terms in magenta to the implicit solve (using the old velocity). Then the momentum update can be done explicitly  though using a time centered radiation field. … … 413 415 In AstroBEAR this would look like: 414 416 * Initialization 415 * Prolongate, \(\rho\), \(\rho\mathbf{v}\), \(e\), \(E\), and \(\dot{E}\)417 * Prolongate, [[latex($\rho$)]], [[latex($\rho\mathbf{v}$)]], [[latex($e$)]], [[latex($E$)]], and [[latex($\dot{E}$)]] 416 418 * Step 1 417 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\)and do physical BC's418 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)419 * Update \(E_{2\mbox{mbc}}\) using \(\dot{E}_{2\mbox{mbc}}\)420 * Update \(e_{2\mbox{mbc}}\) using \(E_{2\mbox{mbc}}\), \(\dot{E}_{2\mbox{mbc}}\), and \(e_{2\mbox{mbc}}\)421 * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)422 * Ghost \(e_{2\mbox{mbc}}\), \(E_{\mbox{mbc}+1}\), \(\dot{E}_{\mbox{mbc}+1}\)419 * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's 420 * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}_1$)]] 421 * Update [[latex($E_{2\mbox{mbc}}$)]] using [[latex($\dot{E}_{2\mbox{mbc}}$)]] 422 * Update [[latex($e_{2\mbox{mbc}}$)]] using [[latex($E_{2\mbox{mbc}}$)]], [[latex($\dot{E}_{2\mbox{mbc}}$)]], and [[latex($e_{2\mbox{mbc}}$)]] 423 * Update [[latex($\dot{E}_0$)]] using pre IR and post IR [[latex($E_0$)]] 424 * Ghost [[latex($e_{2\mbox{mbc}}$)]], [[latex($E_{\mbox{mbc}+1}$)]], [[latex($\dot{E}_{\mbox{mbc}+1}$)]] 423 425 * Do first EH,,mbc,, 424 * Do ER,,mbc,,  Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).425 * Store \(\dot{E}\)in child arrays to be prolongated426 * Do ER,,mbc,,  Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]]. 427 * Store [[latex($\dot{E}$)]] in child arrays to be prolongated 426 428 * Step 2 427 * Overlap \(\rho\), \(\rho \mathbf{v} \) , \(e\), \(E\)and do physical BC's428 * Do IR which updates \(e_0\) and \(E_0\) using \(\rho_1\), \(e_1\), \(E_1\), and \(\dot{E}_1\)429 * Update \(\dot{E}_0\) using pre IR and post IR \(E_0\)430 * Update \(E_{1}\) using \(\dot{E}_{1}\)431 * Ghost \(e_{mbc}\), \(E_{1}\), \(\dot{E}_{1}\)429 * Overlap [[latex($\rho$)]], [[latex($\rho \mathbf{v} $)]] , [[latex($e$)]], [[latex($E$)]] and do physical BC's 430 * Do IR which updates [[latex($e_0$)]] and [[latex($E_0$)]] using [[latex($\rho_1$)]], [[latex($e_1$)]], [[latex($E_1$)]], and [[latex($\dot{E}_1$)]] 431 * Update [[latex($\dot{E}_0$)]] using pre IR and post IR [[latex($E_0$)]] 432 * Update [[latex($E_{1}$)]] using [[latex($\dot{E}_{1}$)]] 433 * Ghost [[latex($e_{mbc}$)]], [[latex($E_{1}$)]], [[latex($\dot{E}_{1}$)]] 432 434 * Do second EH,,0,, 433 * Do ER,,0,,  Terms with \(\nabla E\) can be done without ghosting since EH did not change \(E\).435 * Do ER,,0,,  Terms with [[latex($\nabla E$)]] can be done without ghosting since EH did not change [[latex($E$)]]. 434 436 435 437 … … 440 442 The extra terms in the explicit update due to radiation energy are as follows: 441 443 442 [[latex( \frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =\lambda \nabla E)]]444 [[latex($\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) =\lambda \nabla E$)]] 443 445 444 446 These can be discretized as follows: 445 447 446 [[latex( p^{n+1}_i=p^n_i  \frac{1}{4}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( \left ( E^{n}_{i+1}+E^{n+1}_{i+1} \right )  \left ( E^n_{i1}+E^{n+1}_{i1} \right ) \right ))]]447 448 [[latex( \lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ))]]449 450 and 451 [[latex( R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}})]]448 [[latex($p^{n+1}_i=p^n_i  \frac{1}{4}\frac{\Delta t}{\Delta x} \lambda_{i} \left ( \left ( E^{n}_{i+1}+E^{n+1}_{i+1} \right )  \left ( E^n_{i1}+E^{n+1}_{i1} \right ) \right ) $)]] 449 450 [[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ) $)]] 451 452 and 453 [[latex($R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}}$)]] 452 454 453 455 [[CollapsibleEnd()]] … … 456 458 == Implicit Update 2 == 457 459 458 For now we will assume that \(\kappa_{0P}\) and \(\kappa_{0R}\)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:459 460 [[latex( \frac{\partial e}{\partial t} = \kappa_{0P}(4 \pi BcE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]]461 [[latex( \frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)cE) \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E + \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right ))]]460 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: 461 462 [[latex($\frac{\partial e}{\partial t} = \kappa_{0P}(4 \pi BcE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E $)]] 463 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)cE) \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}1 \right )\mathbf{v}\cdot \nabla E + \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right ) $)]] 462 464 463 465 which we can also write as 464 466 465 [[latex( \frac{\partial e}{\partial t} = f \left ( e \right ) + g \left ( E,\nabla E \right ))]]466 [[latex( \frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  f \left ( e \right )  g \left (e,E, \nabla E \right ))]]467 [[latex($\frac{\partial e}{\partial t} = f \left ( e \right ) + g \left ( E,\nabla E \right ) $)]] 468 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  f \left ( e \right )  g \left (e,E, \nabla E \right ) $)]] 467 469 where 468 [[latex( f \left ( e \right ) = 4 \pi \kappa_{0P} B(T(e)))]]470 [[latex($ f \left ( e \right ) = 4 \pi \kappa_{0P} B(T(e)) $)]] 469 471 and 470 [[latex( g \left (e,E, \nabla E \right ) = \kappa_{0P}(cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]]472 [[latex($ g \left (e,E, \nabla E \right ) = \kappa_{0P}(cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E $)]] 471 473 472 474 Now we can linearize f about e,,0,, 473 [[latex( f \left ( e \right ) = f \left ( e_0 \right ) + \frac{\partial f}{\partial e} \left ( e  e_0 \right ))]]475 [[latex($ f \left ( e \right ) = f \left ( e_0 \right ) + \frac{\partial f}{\partial e} \left ( e  e_0 \right ) $)]] 474 476 475 477 so that the first equation can be written as 476 478 477 [[latex( \frac{\partial e}{\partial t} = f \left ( e_0 \right ) + g \left ( E,\nabla E \right ) + \frac{\partial f}{\partial e} \left ( e  e_0 \right ))]]479 [[latex($\frac{\partial e}{\partial t} = f \left ( e_0 \right ) + g \left ( E,\nabla E \right ) + \frac{\partial f}{\partial e} \left ( e  e_0 \right ) $)]] 478 480 479 481 and then discretized as 480 482 481 [[latex( e^{n+1}_ie^{n}_i = \Delta t \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) + \psi \phi e^n_i  \psi \phi e^{n+1}_i)]]483 [[latex($e^{n+1}_ie^{n}_i = \Delta t \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) + \psi \phi e^n_i  \psi \phi e^{n+1}_i $)]] 482 484 483 485 where 484 486 485 [[latex( \phi = \Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B(T(e))}{\partial e})]]487 [[latex($\phi = \Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B(T(e))}{\partial e}$)]] 486 488 487 489 which can be solved for 488 [[latex( e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 + \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ))]]490 [[latex($e^{n+1}_i = e^{n}_i + \frac{\Delta t}{1 + \psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) $)]] 489 491 490 492 or 491 493 492 [[latex( e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( \theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}E^*_{i1} \right )  \xi_i E^* \right ))]]494 [[latex($e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( \theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}E^*_{i1} \right )  \xi_i E^* \right ) $)]] 493 495 494 496 495 497 Then if we take the semidiscretized equation for E 496 498 497 [[latex( \frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E  \nabla \cdot \left ( \frac{3R_2}{2} \mathbf{v} E \right )  f \left ( e^n_i \right )  g \left ( E, \nabla E \right )  \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i  \psi \phi_i e^{n+1}_i \right ))]]499 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E  \nabla \cdot \left ( \frac{3R_2}{2} \mathbf{v} E \right )  f \left ( e^n_i \right )  g \left ( E, \nabla E \right )  \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i  \psi \phi_i e^{n+1}_i \right ) $)]] 498 500 499 501 and then plugin the solution for e^n+1^,,i,, we get 500 502 501 [[latex( \frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  f \left (e^n_i \right )  g \left (E, \nabla E \right )  \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i  \psi \phi_i e^n_i  \frac{\psi \phi_i }{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ))]]503 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  f \left (e^n_i \right )  g \left (E, \nabla E \right )  \frac{1}{\Delta t} \left ( \psi \phi_i e^n_i  \psi \phi_i e^n_i  \frac{\psi \phi_i }{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) $)]] 502 504 503 505 which simplifies to 504 506 505 [[latex( \frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  \frac{1}{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ))]]506 507 Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use \(E^{n+1}\) and \(E^n\) to construct \(E^*\) which we can plug into the equation for \(e^{n+1}\)507 [[latex($\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \nabla \cdot \left ( \frac{3R_2}{2}\mathbf{v}E \right )  \frac{1}{1+\psi \phi_i} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) $)]] 508 509 Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use [[latex($E^{n+1}$)]] and [[latex($E^n$)]] to construct [[latex($E^*$)]] which we can plug into the equation for [[latex($e^{n+1}$)]] 508 510 509 511 … … 512 514 Expanding 513 515 514 [[latex( B(T(e)) = B \left ( T_0+dT(e) \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right  _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right ))]]516 [[latex($B(T(e)) = B \left ( T_0+dT(e) \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right  _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right ) $)]] 515 517 516 518 where 517 519 518 [[latex( \Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma1)}{n k_B})]]519 520 and we can identify [[latex( \phi = \Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa_{0P} \Delta t B_0 \frac{\Gamma}{T_0})]]520 [[latex($\Gamma = \frac{\partial T}{\partial e} = \frac{(\gamma1)}{n k_B}$)]] 521 522 and we can identify [[latex($\phi = \Delta t \frac{\partial f}{\partial e} = 4 \pi \kappa_{0P} \Delta t \frac{\partial B}{\partial e} = 16 \pi \kappa_{0P} \Delta t B_0 \frac{\Gamma}{T_0}$)]] 521 523 522 524 Then the equation for e becomes 523 525 524 [[latex( \frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E)]]525 526 which will be accurate as long as \(4\Gamma \frac{\left  ee_0 \right  }{T_0} < \xi << 1\) or \(\left  \Delta e \right  = \left  ee_0 \right  < \xi \frac{T_0}{4 \Gamma}\)526 [[latex($\frac{\partial e}{\partial t} =  \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E$)]] 527 528 which will be accurate as long as [[latex($4\Gamma \frac{\left  ee_0 \right  }{T_0} < \xi << 1$)]] or [[latex($\left  \Delta e \right  = \left  ee_0 \right  < \xi \frac{T_0}{4 \Gamma}$)]] 527 529 528 530 We can calculate the time scale for this to be true using the evolution equation for the energy density 529 531 530 [[latex( \left  ee_0 \right  = \left  \Delta e \right  = \Delta t \left  \frac{\partial e}{\partial t} \right  < \xi \frac{T_0}{4 \Gamma})]]531 532 which gives [[latex( \Delta t < \xi \frac{T_0}{4 \Gamma \left  \frac{\partial e}{\partial t} \right  } = \xi \frac{T_0}{4 \Gamma \left   \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E \right  })]]532 [[latex($\left  ee_0 \right  = \left  \Delta e \right  = \Delta t \left  \frac{\partial e}{\partial t} \right  < \xi \frac{T_0}{4 \Gamma}$)]] 533 534 which gives [[latex($\Delta t < \xi \frac{T_0}{4 \Gamma \left  \frac{\partial e}{\partial t} \right  } = \xi \frac{T_0}{4 \Gamma \left   \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{ee_0}{T_0} \right )cE \right ] + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E \right  } $)]] 533 535 534 536 or in discretized form 535 537 536 [[latex( \Delta t < \xi \frac{\theta_i}{\phi_i \frac{1}{\Delta t} \left  \left ( \theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}E^*_{i1} \right )  \xi_i E^* \right ) \right })]]538 [[latex($\Delta t < \xi \frac{\theta_i}{\phi_i \frac{1}{\Delta t} \left  \left ( \theta_i + \epsilon_i E^*_i + \omega_{x,i} v^n_{x,i} \left ( E^*_{i+1}E^*_{i1} \right )  \xi_i E^* \right ) \right } $)]] 537 539 538 540 539 541 === Implicit Discretization 2 === 540 542 Now we can discretize 541 [[latex( g(E*, \nabla E*) = \kappa_{0P}cE^* + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E^* \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E^*)]]543 [[latex($g(E*, \nabla E*) = \kappa_{0P}cE^* + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}1 \right ) \mathbf{v} \cdot \nabla E^* \frac{3R_2}{2}\kappa_{0P}\frac{v^2}{c}E^*$)]] 542 544 543 545 as 544 546 545 [[latex( g = \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} \left ( \psi E^{n+1}_{i+1}  \psi E^{n+1}_{i1} + \bar{\psi} E^n_{i+1} \bar{\psi} E^n_{i1} \right )  \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ))]]547 [[latex($g = \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} \left ( \psi E^{n+1}_{i+1}  \psi E^{n+1}_{i1} + \bar{\psi} E^n_{i+1} \bar{\psi} E^n_{i1} \right )  \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) $)]] 546 548 547 549 which along with the other terms gives 548 {{{ 549 #!latex 550 [[latex($ 550 551 \begin{eqnarray} 551 552 E^{n+1}_iE^{n}_i & = & \left [ \alpha_{i+1/2} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^{n}_{i+1} \psi E^{n+1}_{i}  \bar{\psi} E^n_{i} \right )  \alpha_{i1/2} \left ( \psi E^{n+1}_{i} + \bar{\psi} E^{n}_i  \psi E^{n+1}_{i1}  \bar{\psi}E^{n}_{i1} \right ) \right ] \\ … … 553 554 &  & \frac{1}{1+\psi \phi_i} \left [  \theta_i + \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_{x,i} v^n_{x,i} \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^n_{i+1}  \psi E^{n+1}_{i1}  \bar{\psi} E^n_{i1} \right )  \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\ 554 555 \end{eqnarray} 555 }}} 556 $)]] 556 557 557 558 where the diffusion coefficient is given by 558 559 559 [[latex( \alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}})]]560 [[latex($\alpha_{i+1/2}=\frac{\Delta t}{\Delta x^2} \frac{c \lambda_{i+1/2}}{\kappa_{0R,i+1/2}} $)]] 560 561 561 562 and where 562 563 563 [[latex( \zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3R_{2,i+1/2}}{4})]]564 565 and 566 567 [[latex( \epsilon_i=c\Delta t \kappa_{0P,i})]]568 569 and 570 571 [[latex( \phi_i = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ))]]572 573 and 574 575 [[latex( \theta_i =  \Delta t f(e^n_i) = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ))]]576 577 and 578 579 [[latex( \omega_{x,i} = \frac{\lambda_{x,i} \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}\frac{1}{2} \right ))]]580 581 and 582 583 [[latex( \xi_i = \Delta t \frac{3R_{2,i}}{2}\kappa_{0P,i}\frac{v_i^2}{c})]]584 and 585 586 where [[latex( \kappa_{0R,i+1/2} = \frac{\kappa_{0R,i}+\kappa_{0R,i+1}}{2})]]587 588 and 589 590 [[latex( R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )})]]591 592 and 593 594 [[latex( \lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ))]]595 596 and 597 598 [[latex( R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2)]]599 600 and 601 602 603 [[latex( R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}})]]604 605 and 606 607 [[latex( \lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ))]]608 609 and 610 611 [[latex( R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2)]]564 [[latex($\zeta_{i+1/2}= \frac{\Delta t}{\Delta x}\frac{3R_{2,i+1/2}}{4} $)]] 565 566 and 567 568 [[latex($\epsilon_i=c\Delta t \kappa_{0P,i}$)]] 569 570 and 571 572 [[latex($\phi_i = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) \left ( \frac{4\Gamma}{T^n_i} \right ) $)]] 573 574 and 575 576 [[latex($\theta_i =  \Delta t f(e^n_i) = \Delta t 4 \pi \kappa_{0P,i} B \left ( T^n_i \right ) $)]] 577 578 and 579 580 [[latex($\omega_{x,i} = \frac{\lambda_{x,i} \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}\frac{1}{2} \right )$)]] 581 582 and 583 584 [[latex($\xi_i = \Delta t \frac{3R_{2,i}}{2}\kappa_{0P,i}\frac{v_i^2}{c} $)]] 585 and 586 587 where [[latex($ \kappa_{0R,i+1/2} = \frac{\kappa_{0R,i}+\kappa_{0R,i+1}}{2} $)]] 588 589 and 590 591 [[latex($R_{i+1/2} = \frac{\left  E^n_{i+1}E^n_{i} \right  }{2 \kappa_{0R,i+1/2} \left ( E^n_i+E^n_{i+1} \right )}$)]] 592 593 and 594 595 [[latex($\lambda_{i+1/2} = \frac{1}{R_{i+1/2}} \left ( \coth R_{i+1/2}  \frac{1}{R_{i+1/2}} \right ) $)]] 596 597 and 598 599 [[latex($R_{2,i+1/2} = \lambda_{i+1/2}+\lambda_{i+1/2}^2 R_{i+1/2}^2$)]] 600 601 and 602 603 604 [[latex($R_{i} = \frac{\left  E^n_{i+1}E^n_{i1} \right  }{2 \kappa_{0R,i} E^n_{i}}$)]] 605 606 and 607 608 [[latex($\lambda_{i} = \frac{1}{R_{i}} \left ( \coth R_{i}  \frac{1}{R_{i}} \right ) $)]] 609 610 and 611 612 [[latex($R_{2,i} = \lambda_{i}+\lambda_{i}^2 R_{i}^2$)]] 612 613 613 614 Which we can arrange into the following form 614 615 615 {{{ 616 #!latex 616 [[latex($ 617 617 \begin{eqnarray} 618 618 & \left ( 1 + \psi \left ( \alpha_{i+1/2} + \alpha_{i1/2} + \zeta_{i+1/2} v^n_{x,i+1/2}  \zeta_{i1/2} v^n_{x,i1/2} + \frac{\epsilon_i  \xi_i}{1+\psi \phi_i} \right ) \right ) E^{n+1}_i \\ … … 624 624 + & \frac{\theta_i}{1+\psi \phi_i} \\ 625 625 \end{eqnarray} 626 }}} 626 $)]] 627 627 628 628 === 2D etc... === … … 633 633 === Initial solution vector === 634 634 For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E 635 [[latex( E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t)]]635 [[latex($E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t$)]] 636 636 637 637 == CoarseFine Boundaries == … … 642 642 643 643 == Physical Boundary Conditions == 644 For each boundary type we need to specify \(E^n_g, E^{n+1}_g, \kappa_g, \mbox { and } v_g\)in terms of other quantities (ie including E^n+1^_i). The v,,g,, and kappa,,g,, terms should come from the hydro boundary conditions so we just need an equation for E^n^,,g,,644 For each boundary type we need to specify [[latex($E^n_g, E^{n+1}_g, \kappa_g, \mbox { and } v_g$)]] in terms of other quantities (ie including E^n+1^_i). The v,,g,, and kappa,,g,, terms should come from the hydro boundary conditions so we just need an equation for E^n^,,g,, 645 645 646 646 … … 648 648 We would like the radiation to leave at the free streaming limit. But we could have to modify the opacity in the ghost zone to keep the radiation energy positive 649 649 650 [[latex( \frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( EE_g \right ) }{\Delta x})]]651 652 Clearly if we set \(E_g = 0\) and \(\kappa_g << \frac{1}{\Delta x} \) we should get \(\lambda = \kappa_g \Delta x\) and a free streaming flux of \(c E \mathbf{n}\)650 [[latex($\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( EE_g \right ) }{\Delta x}$)]] 651 652 Clearly if we set [[latex($E_g = 0$)]] and [[latex($\kappa_g << \frac{1}{\Delta x} $)]] we should get [[latex($\lambda = \kappa_g \Delta x$)]] and a free streaming flux of [[latex($c E \mathbf{n}$)]] 653 653 654 654 which also corresponds to an 655 \(\alpha = c \frac{\Delta t}{\Delta x}\) 655 [[latex($\alpha = c \frac{\Delta t}{\Delta x}$)]] 656 656 657 657 === Constant Slope Boundary === 658 [[latex( E_g = 2 E_i  E_{i+1})]]658 [[latex($E_g = 2 E_i  E_{i+1}$)]] 659 659 660 660 === Periodic Boundary === … … 667 667 === Reflecting/ZeroSlope Boundary === 668 668 669 [[latex( E_g = E_i)]]669 [[latex($E_g = E_i$)]] 670 670 671 671 === Constant radiative flux === 672 672 673 [[latex( E_g = E_i  F_0 \frac{\kappa_g \Delta x}{c \lambda_g})]]674 675 [[CollapsibleEnd()]] 673 [[latex($E_g = E_i  F_0 \frac{\kappa_g \Delta x}{c \lambda_g}$)]] 674 675 [[CollapsibleEnd()]]