Changes between Version 15 and Version 16 of u/erica/PoissonPlusHydro


Ignore:
Timestamp:
09/16/13 12:33:09 (11 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/PoissonPlusHydro

    v15 v16  
    8080
    8181To solve the 2nd source term ODE,
     82
     83[[latex($\frac{d}{dt}(E) = -\rho u \cdot \triangledown \phi $)]]
     84
     85replace with discrete derivatives,
     86
     87[[latex($\frac{E_i^{n+1} - E_i^{n+1/2}}{dt} = -\frac{\rho_i^{n+1/2}u_i^{n+1/2}(\phi_{i+1}^{n+1/2} - \phi_i^{n+1/2})}{dx}$)]]
     88
     89I am not exactly sure which time level to use for u (n+1 or n+1/2).
     90
     91Rearranging for the updated version of the energy, we have,
     92
     93[[latex($E_i^{n+1} = -\frac{dt}{dx}[\rho_i^{ n+1/2} u_i^{n+1/2}(\phi_{i+1}^{n+1/2}-\phi_i^{n+1/2})] + E_i^{n+1/2}$)]]