Updated equations

In the predictor

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

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

-  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}

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

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

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

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

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

.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( \varepsilon_g  U)^{\ast}+ \varepsilon_g \nabla {p_g}^{n-1/2} \right)

then defining

and 

.. math:: {p_g}^{n+1/2, \ast} = {p_g}^{n-1/2} + \phi

.. math:: {p_g}^{n+1/2, \ast} = \phi

In the corrector

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

-  Define :math:`U^{MAC,\ast \ast}` at the "new" time using :math:`U^{\ast \ast}`

-  Define a new 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} 

           \Delta t \left( (-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} 

.. 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 )

            \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \rho_g g \right)

-  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}

.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( (\varepsilon_g  U)^{\ast \ast \ast} + \varepsilon_g \nabla {p_g}^{n+1/2,\ast} \right)

then defining

and 

.. math:: {p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi

.. math:: {p_g}^{n+1/2} = \phi