Needed factor of eps in front of (rho g)

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

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

           \Delta t \left( \nabla \cdot \tau^n + \sum_p \beta_p (V_p - {U_g}^{\ast}) + \rho_g g \right)

           \Delta t \left( \nabla \cdot \tau^n + \sum_p \beta_p (V_p - {U_g}^{\ast}) + \rho_g \varepsilon_g g \right)

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

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

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

then defining

-  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 - \frac{\Delta t}{2} \left( \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n + \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast}\right) + \\ &+ \frac{\Delta t}{2} \left( \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast} \right) + \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \rho_g g \right)

.. math:: (\varepsilon_g \rho_g U)^{\ast \ast \ast} &= (\varepsilon_g \rho_g U)^n - \frac{\Delta t}{2} \left( \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n + \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast}\right) + \\ &+ \frac{\Delta t}{2} \left( \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast} \right) + \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \varepsilon_g \rho_g g \right)

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

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

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

then defining