Changes between Version 194 and Version 195 of FluxLimitedDiffusion


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Timestamp:
04/05/13 14:41:25 (12 years ago)
Author:
Jonathan
Comment:

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  • FluxLimitedDiffusion

    v194 v195  
    490490or
    491491
    492 [[latex(e^{n+1}_i = e^{n}_i + \frac{1}{1 + \psi \phi_i} \left ( -\theta_i + \epsilon_i E^*_i \omega_i v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) )]]
     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^*_{i-1} \right ) - \xi_i E^* \right ) )]]
    493493
    494494
    … …  
    534534or in discretized form
    535535
    536 [[latex(\Delta t < \xi \frac{\theta_i}{\phi_i \frac{1}{\Delta t} \left | \left ( -\theta_i + \epsilon_i E^*_i + \omega_i v^n_{x,i} \left ( E^*_{i+1}-E^*_{i-1} \right ) - \xi_i E^* \right ) \right |} )]]
     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^*_{i-1} \right ) - \xi_i E^* \right ) \right |} )]]
    537537
    538538
    … …  
    543543as
    544544
    545 [[latex(g = \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_i v^n_{x,i} \left ( \psi E^{n+1}_{i+1} - \psi E^{n+1}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) )]]
     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}_{i-1} + \bar{\psi} E^n_{i+1}- \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) )]]
    546546
    547547which along with the other terms gives
    … …  
    551551E^{n+1}_i-E^{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_{i-1/2} \left ( \psi  E^{n+1}_{i} + \bar{\psi} E^{n}_i - \psi E^{n+1}_{i-1} - \bar{\psi}E^{n}_{i-1} \right ) \right ] \\
    552552 & - & \left [ \zeta_{i+1/2} v^n_{x,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 ) - \zeta_{i-1/2} v^n_{x,i-1/2}\left ( \psi E^{n+1}_{i} + \bar{\psi} E^n_i + \psi E^{n+1}_{i-1} + \bar{\psi}E^{n}_{i-1} \right ) \right ] \\
    553  & - & \frac{1}{1+\psi \phi_i}  \left [ - \theta_i  + \epsilon_i \left ( \psi E^{n+1}_i + \bar{\psi} E^n_i \right ) + \omega_i v^n_x \left ( \psi E^{n+1}_{i+1} + \bar{\psi} E^n_{i+1} - \psi E^{n+1}_{i-1} - \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\
     553 & - & \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}_{i-1} - \bar{\psi} E^n_{i-1} \right ) - \xi_i \left ( \psi E^{n+1}_i + \bar{\psi} E^{n}_i \right ) \right ] \\
    554554\end{eqnarray}
    555555}}}
    … …  
    577577and
    578578
    579 [[latex(\omega_{i} = \frac{\lambda_i \Delta t}{\Delta x} \left ( \frac{\kappa_{0P,i}}{\kappa_{0R,i}}-\frac{1}{2} \right ))]]
     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 ))]]
    580580
    581581and
    … …  
    617617\begin{eqnarray}
    618618 & \left ( 1 + \psi \left ( \alpha_{i+1/2} +  \alpha_{i-1/2} + \zeta_{i+1/2} v^n_{x,i+1/2} - \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\epsilon_i - \xi_i}{1+\psi \phi_i} \right ) \right ) E^{n+1}_i   \\
    619 - & \left ( \psi \left (  \alpha_{i+1/2} - \zeta_{i+1/2} v^n_{x,i+1/2} - \frac{\omega_i v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n+1}_{i+1}  \\
    620 - & \left ( \psi \left (  \alpha_{i-1/2} + \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\omega_i v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n+1}_{i-1}  \\
     619- & \left ( \psi \left (  \alpha_{i+1/2} - \zeta_{i+1/2} v^n_{x,i+1/2} - \frac{\omega_{x,i} v^n_{x,i}}{1+\psi \phi_i} \right ) \right ) E^{n+1}_{i+1}  \\
     620- & \left ( \psi \left (  \alpha_{i-1/2} + \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\omega_{x,i} v^n_{x,i}}{1+\psi \phi_i} \right ) \right ) E^{n+1}_{i-1}  \\
    621621= & \left ( 1 - \bar{\psi} \left ( \alpha_{i+1/2} + \alpha_{i-1/2} + \zeta_{i+1/2} v^n_{x,i+1/2} -  \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\epsilon_i - \xi_i}{1+\psi \phi_i} \right ) \right ) E^{n}_i  \\
    622 + & \left ( \bar{\psi} \left (  \alpha_{i+1/2} - \zeta_{i+1/2} v^n_{x,i+1/2} - \frac{\omega_i v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n}_{i+1}  \\
    623 + & \left ( \bar{\psi} \left (  \alpha_{i-1/2} + \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\omega_i v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n}_{i-1}  \\
     622+ & \left ( \bar{\psi} \left (  \alpha_{i+1/2} - \zeta_{i+1/2} v^n_{x,i+1/2} - \frac{\omega_{x,i} v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n}_{i+1}  \\
     623+ & \left ( \bar{\psi} \left (  \alpha_{i-1/2} + \zeta_{i-1/2} v^n_{x,i-1/2} + \frac{\omega_{x,i} v^n_x}{1+\psi \phi_i} \right ) \right ) E^{n}_{i-1}  \\
    624624+ & \frac{\theta_i}{1+\psi \phi_i}   \\
    625625\end{eqnarray}