Commit dd0deee3 authored by Ann Almgren's avatar Ann Almgren
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more formatting

parent 1577eeb5

docs/source/FluidTimeDiscretization.rst

+13 −10
Original line number Diff line number Diff line
@@ -10,21 +10,22 @@ Thus here we focus on the discretization of the momentum equation 


In the predictor



#. Define :math:`U^{MAC}`, the face-centered (staggered) MAC velocity which is used for advection.

-  Define :math:`U^{MAC}`, the face-centered (staggered) MAC velocity which is used for advection.



#. Define an approximation to the new-time state, :math:`(\varepsilon_g \rho_g U)^{\ast}` by setting 

-  Define an approximation to the new-time state, :math:`(\varepsilon_g \rho_g U)^{\ast}` by setting 



.. math:: (\varepsilon_g \rho_g U)^{\ast} = (\varepsilon_g \rho_g U)^n +  

           \Delta t ( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} + 

           \nabla \cdot \tau^n + \sum_{part} \beta_p (V_p - {U_g}^{\ast}) + \rho_g g )

           \Delta t ( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2}



#. Project :math:`U^{\ast}` by solving

.. math:: | \nabla \cdot \tau^n + \sum_{part} \beta_p (V_p - {U_g}^{\ast}) + \rho_g g )



-  Project :math:`U^{\ast}` by solving



.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g  U)^{\ast}



then defining



.. math:: (\varepsilon_g  U)^{n+1} = (\varepsilon_g  U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi

.. math:: U^{\ast \ast} = U^{\ast} - \frac{1}{\rho_g} \nabla \phi



and
@@ -33,20 +34,22 @@ and 


In the corrector



#. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{\ast \ast \ast}` by setting  

-  Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{\ast \ast \ast}` by setting  



.. math::  (\varepsilon_g \rho_g U)^{\ast \ast \ast} = (\varepsilon_g \rho_g U)^n + 

           \Delta t ( (-1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n -(1/2) \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast} 

          + \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{\ast \ast} + 

          + \varepsilon_g \nabla {p_g}^{n+1/2,\ast} 



.. math::  + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{\ast \ast} + 

            \sum_{part} \beta_p (V_p - {U_g}^{\ast \ast}) + \rho_g g )



#. Project :math:`U^{\ast \ast \ast}` by solving

-  Project :math:`U^{\ast \ast \ast}` by solving



.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g  U)^{\ast \ast \ast}



   then defining



.. math:: (\varepsilon_g  U)^{n+1} = (\varepsilon_g  U)^{\ast \ast \ast} - \frac{\varepsilon_g}{\rho_g} \nabla \phi

.. math:: U^{n+1} = U^{\ast \ast \ast} - \frac{1}{\rho_g} \nabla \phi



and