Changes between Version 176 and Version 177 of FluxLimitedDiffusion


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Timestamp:
04/03/13 16:05:27 (12 years ago)
Author:
Jonathan
Comment:

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  • FluxLimitedDiffusion

    v176 v177  
    479479and then discretized as
    480480
    481   [[latex(e^{n+1}_i-e^{n}_i = \Delta t  \left ( f \left ( e^n_i \right ) + g \left (E^*,\nabla E^* \right ) \right ) - \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i -  \psi \phi e^{n+1}_i )]]
     481  [[latex(e^{n+1}_i-e^{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 )]]
    482482
    483483where
    … …  
    490490Then if we take the semi-discretized equation for E
    491491
    492   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E - \nabla \cdot \left ( \frac{3-R_2}{2} \mathbf{v} E \right ) - f \left ( e^n_i \right ) - g \left ( E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i + \psi \phi e^{n+1}_i \right ) )]] 
     492  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c \lambda}{\kappa_{0R}} \nabla E - \nabla \cdot \left ( \frac{3-R_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  e^n_i - \psi \phi e^{n+1}_i \right ) )]] 
    493493
    494494and then plugin the solution for e^n+1^,,i,, we get
    495495
    496   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - f \left (e^n_i \right ) - g \left (E, \nabla E \right ) - \frac{1}{\Delta t} \left ( \left ( \left ( \bar{\psi} - 1  \right ) \phi \right ) e^n_i  + \psi \phi e^n_i \right ) - \frac{\psi \phi }{1-\psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) )]] 
     496  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_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 e^n_i - \psi \phi e^n_i - \frac{\psi \phi }{1+\psi \phi} \left ( f \left ( e^n_i \right ) + g \left (E,\nabla E \right ) \right ) \right ) )]] 
    497497
    498498which simplifies to
    499499
    500   [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1-\psi \phi} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) )]] 
    501 
    502 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^
     500  [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E -\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right ) - \frac{1}{1+\psi \phi} \left ( f \left ( e^n_i \right ) + g \left ( E,\nabla E \right ) \right ) )]] 
     501
     502Now we have 1 equation with 1 variable that we can solve implicitly using hypre, and then we can use \(E^{n+1}\) and \(E^\) to construct \(E^*\) which we can plug into the equation for \(e^{n+1}\)
    503503
    504504