Changes between Version 22 and Version 23 of FluxLimitedDiffusion
 Timestamp:
 03/20/13 11:37:49 (12 years ago)
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FluxLimitedDiffusion
v22 v23 136 136 137 137  [[latex(E^{n+1}_iE^{n}_i = \left [ \alpha^n_{i+1/2} \left ( E^{*}_{i+1}E^{*}_{i} \right )  \alpha^n_{i1/2} \left ( E^{*}_{i}E^{*}_{i1} \right ) \right ]  \epsilon^n_i E^{*}_i + \phi^n_i e^{*}_i + \theta^n_i) )]]  138  [[latex(e^{n+1}_ie^{n}_i = +\epsilon^n_i E^{*}_i  \phi^n_i e^{*}_i  \theta^n_i )]] 138  [[latex(e^{n+1}_ie^{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 … … 164 164 165 165 166 Now all the terms on the right hand side that are linear in E or e have been written as E^*^ because there are different ways to approximate E^*^. For Backward Euler we have166 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 167 167 [[latex(E^*_i = E^{n+1}_i)]] 168 168 and for Crank Nicholson we have 169 169 [[latex(E^*_i = \frac{1}{2} \left ( E^{n+1}_i + E^n_i \right ) )]] 170 170 or we can parameterize the solution 171 [[latex(E^*_i = \ upsilon E^{n+1}_i + \bar{\upsilon}E^n_i)]]172 where [[latex(\bar{\ upsilon} = 1\upsilon)]]173 174 Backward Euler has [[latex(\ upsilon=1)]] and Crank Nicholson has [[latex(\upsilon=1/2)]]175 176 Forward Euler has [[latex(\ upsilon=0)]]171 [[latex(E^*_i = \psi E^{n+1}_i + \bar{\psi}E^n_i)]] 172 where [[latex(\bar{\psi} = 1\psi)]] 173 174 Backward Euler has [[latex(\psi=1)]] and Crank Nicholson has [[latex(\psi=1/2)]] 175 176 Forward Euler has [[latex(\psi=0)]] 177 177 178 178 In any event in 1D we have the following matrix coefficients 179 179 180  [[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 ))]] 181  [[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 ))]] 180  [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i1/2} + \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_{i1/2} \right ) E^{n+1}_{i1}  \left ( \psi \phi^n_i \right ) e^{n+1}_i=\left [ 1  \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i1/2} + \epsilon^n_i \right ) \right ] E^n_i+\bar{\psi}\phi e^n_i + \theta^n_i)]]  181  [[latex(\left ( 1 +\psi \phi \right ) e^{n+1}_i  \left ( \psi \epsilon^n_i \right )E^{n+1}_i =\left ( 1  \bar{\psi} \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_itheta^i_n )]]  182 182 183 183 Clearly the second equation is trivial to solve after the first system of equations has been solved. So if we treat the temperature as being constant we can calculate the local heating/cooling due to radiative emission/absorption