Changes between Version 64 and Version 65 of SelfGravityDevel


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Timestamp:
10/22/19 12:26:37 (5 years ago)
Author:
Jonathan
Comment:

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  • SelfGravityDevel

    v64 v65  
    88[[latex($\frac{d \rho \mathbf{v}}{dt} = \mathbf{f}_g = -\rho \nabla \phi$)]]
    99
    10 and energy equation
     10and the energy equation now includes the gravitational self-energy on the left
    1111
    1212[[latex($\frac{d E}{dt} = \mathbf{v} \cdot \mathbf{f}_g = -\rho \mathbf{v} \cdot \nabla \phi$)]]
     
    2424AstroBEAR uses hypre's linear solver to solve Poisson's equation for the potential.  However, there are two methods for including the source terms for momentum and energy.  The non-conservative approach simply includes the terms during a source update, while the conservative approach recasts the RHS of the momentum and energy equations as total divergences - so they can be differenced conservatively.
    2525
    26 
    27 
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    29 
     26Substituting the Poisson equation into the momentum equation we have
     27
     28[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right ) \right ]$)]]
     29
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     33
     34For the energy equation, we can only find a conservative approach - if we consider the evolution of the total combined energy (hydro + gravitational). 
     35
     36[[latex($\frac{d \left(E + \frac{1}{2} \left (\rho - \rho_0 \right) \phi \right)}{dt} = -\nabla \cdot \mathbf{F_g} $)]]
     37
     38
     39Combining this with the our previous energy equation we arrive at
     40
     41[[latex($\nabla \cdot \mathbf{F_g} = -\frac{1}{2} \frac{d \left (\rho \phi \right)}{dt} + \frac{1}{2} \rho_0 \frac{d \phi}{dt}+ \rho \mathbf{v} \cdot \nabla \phi$)]]
     42
     43Expanding the time derivative and using the continuity equation, we arrive at
     44
     45
     46[[latex($\nabla \cdot \mathbf{F_g} = \frac{1}{2} \phi \nabla \cdot \left ( \rho \mathbf{v} \right) - \frac{1}{2} \left (\rho - \rho_0 \right) \dot{\phi}  + \rho \mathbf{v} \cdot \nabla \phi$)]]
     47
     48
     49Now from time differencing the poisson equation, we arrive at
     50
     51[[latex($\nabla^2 \dot{\phi} = 4 \pi G \dot{\rho} = - 4 \pi G \nabla \cdot (\rho \mathbf{v})$)]]
     52
     53and multiplying both sides by [[latex($\frac{\phi}{8 \pi G}$)]] gives
     54
     55[[latex($\frac{1}{2} \nabla \cdot (\rho \mathbf{v}) \phi = -\frac{1}{8 \pi G} \phi \nabla^2 \dot{\phi}$)]]
     56
     57
     58Adding the LHS and subtracting the RHS and substituting
     59
     60[[latex($\rho - \rho_0 = \frac{1}{4 \pi G} \nabla^2 \phi$)]]
     61
     62into the divergence equation gives us
     63
     64[[latex($\nabla \cdot \mathbf{F_g} = \nabla \cdot   \left (\phi \rho \mathbf{v} \right) + \frac{1}{8 \pi G} \left (\phi \nabla^2 \dot{\phi} - \dot{\phi}\nabla^2\phi \right) $)]]
     65
     66Now we can use the relationship
     67
     68[[latex($\nabla \cdot \left [ \phi \nabla \dot{\phi} - \dot{\phi} \nabla \phi \right ]= \phi \nabla^2 \dot{\phi} - \dot{\phi} \nabla^2 \phi$)]]
     69
     70to arrive at
     71
     72[[latex($\mathbf{F_g} = \phi \rho \mathbf{v} + \frac{1}{8 \pi G} \left (\phi \nabla \dot{\phi} - \dot{\phi}\nabla \phi \right) $)]]
     73
     74
     75[[CollapsibleStart(Notes)]]
    3076The electric field generated by a point charge[[latex($Q$)]]is
    3177
     
    231277  * Solve for gas potential after accretion?
    232278  *
     279
     280[[CollapsibleEnd]]