Changes between Version 198 and Version 199 of FluxLimitedDiffusion


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Timestamp:
05/31/13 15:34:21 (12 years ago)
Author:
Baowei Liu
Comment:

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  • FluxLimitedDiffusion

    v198 v199  
    549549
    550550which along with the other terms gives
    551 [[latex($
    552 \begin{eqnarray}
    553 E^{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 ] \\
    554  & - & \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 ] \\
    555  & - & \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 ] \\
    556 \end{eqnarray}
    557 $)]]
     551[[latex($ \begin{aligned}E^{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 ] \\ - & \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 ] \\- & \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 ] \\\end{aligned}$)]]
    558552
    559553where the diffusion coefficient is given by
    … …  
    615609Which we can arrange into the following form
    616610
    617 [[latex($
    618 \begin{eqnarray}
    619  & \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   \\
    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}  \\
    621 - & \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}  \\
    622 = & \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  \\
    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}  \\
    624 + & \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}  \\
    625 + & \frac{\theta_i}{1+\psi \phi_i}   \\
    626 \end{eqnarray}
    627 $)]]
     611[[latex($\begin{aligned}& \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\\-&\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}  \\- & \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}  \\- & \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}  \\= & \left ( 1 -\bar{\psi}\left(\alpha_{i+1/2} + \alpha_{i-1/2} + \zeta_{i+1/2} v^n_{x,i+1/2} - \zeta_{i1/2}v^n_{x,i-1/2}+\frac{\epsilon_i\xi_i}{1+\psi\phi_i}\right)\right)E^{n}_i\\\end{aligned}$)]]
     612
    628613
    629614=== 2D etc... ===