Commit e9951cd6 authored by Ann Almgren's avatar Ann Almgren
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parent c7c0f1c9

docs/source/FluidTimeDiscretization.rst

+3 −11
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@@ -12,12 +12,9 @@ In the predictor 


#. 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)^* = (\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}^*) + \rho_g g )`

#. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{*} = (\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}^{*}) + \rho_g g )`



#. Project :math:`U^*` by solving

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

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

    then defining

         :math:(\varepsilon_g  U)^{**} = (\varepsilon_g  U)^{*} - \frac{\varepsilon_g}{\rho_g} \nabla \phi
@@ -26,12 +23,7 @@ In the predictor 


In the corrector



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

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

                                                                    + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{**}

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

#. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{***} = (\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)^{**} + \varepsilon_g \nabla {p_g}^{n+1/2,*} + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{**} + \sum_{part} \beta_p (V_p - {U_g}^{**}) + \rho_g g )`



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

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