Changes between Version 4 and Version 5 of CodeExplanation/SourceTerms


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Timestamp:
01/09/12 14:58:45 (13 years ago)
Author:
Jonathan
Comment:

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  • CodeExplanation/SourceTerms

    v4 v5  
    1515* For a static potential update the momentum using the time centered density and the energy using the mass fluxes times grad phi at the cell edges
    1616* For self gravity do something slightly more complicated......
    17 [[latex($\Delta p=-\frac{\Delta t}{\Delta x} \left(F_{i+1/2}-F_{i-1/2} \right )$)]]  which is just [[latex($\dot{p} = f_{grav}$)]] where [[latex($f_{grav}=\nabla F$)]] which requires [[latex($\nabla F=\rho \nabla \phi$)]].  Since [[latex($\nabla^2 \phi = 4\pi G (\rho-\bar{\rho})$)]] we can substitute for [[latex($\rho$)]] and we have [[latex($\nabla F = (\frac{\nabla^2\phi}{4 \pi G}+\bar{\rho}) \nabla \phi$)]] which reduces to [[latex($\nabla F=\nabla \left [ \frac{1}{2} \frac{\left (\nabla \phi\right )^2}{4 \pi G} + \bar{\rho} \phi \right ]$)]] where we can identify the equivalent momentum flux as [[latex($F=\frac{1}{2} \frac{\left (\nabla \phi\right )^2}{4 \pi G} + \bar{\rho} \phi$)]]
     17[[latex($\Delta p=-\frac{\Delta t}{\Delta x} \left(F_{i+1/2}-F_{i-1/2} \right )$)]]  which is just [[latex($\dot{p} = f_{grav}$)]] where [[latex($f_{grav}=-\nabla F$)]] which requires [[latex($\nabla F=\rho \nabla \phi$)]].  Since [[latex($\nabla^2 \phi = 4\pi G (\rho-\bar{\rho})$)]] we can substitute for [[latex($\rho$)]] and we have [[latex($\nabla F = (\frac{\nabla^2\phi}{4 \pi G}+\bar{\rho}) \nabla \phi$)]] which reduces to [[latex($\nabla F=\nabla\left [\frac{1}{2} \frac{\left (\nabla \phi\right )^2}{4 \pi G} \bar{\rho} \phi \right ]$)]] where we can identify the equivalent momentum flux as [[latex($F=\frac{1}{2} \frac{\left (\nabla \phi\right )^2}{4 \pi G} + \bar{\rho} \phi$)]]
    1818* But casting it this way gives you strict momentum conservation.  If the source term were just [[latex($\nabla \phi$)]] then momentum would be conserved but since in each cell there is a factor of [[latex($\rho$)]] that currently enters, it modifies the differenced quantity from cell to cell.  In theory these all cancel but only because [[latex($\nabla^2\phi=\rho$)]]...  In the above treatment - the density used in the source term is derived from the actual potential which allows the source term to be cast as a flux difference.  This allows for the strict momentum conservation.
    1919