diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index f079d0de82cce1f1ed3b36ac71b2a3cf1d45bb1c..95baf6afbe22f23fbdfd09f8e916b4be1e21f1fc 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -12,22 +12,43 @@ 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)^{\ast}` by setting -#. Project :math:`U^{*}` by solving - :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^*` +.. 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 ) + +#. 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)^{**} = (\varepsilon_g U)^{*} - \frac{\varepsilon_g}{\rho_g} \nabla \phi + +.. math:: (\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi + + and + +.. math:: {p_g}^{n+1/2, \ast} = {p_g}^{n-1/2} + \phi + + :math:(\varepsilon_g U)^{\ast \ast} = (\varepsilon_g U)^{\ast} - \frac{\varepsilon_g}{\rho_g} \nabla \phi and - :math:`{p_g}^{n+1/2,*} = {p_g}^{n-1/2} + \phi` + :math:`{p_g}^{n+1/2,\ast} = {p_g}^{n-1/2} + \phi` 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)^{\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} + + \sum_{part} \beta_p (V_p - {U_g}^{\ast \ast}) + \rho_g g ) #. Project :math:`U^{***}` by solving - :math:`\nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^{***}` + +.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot (\varepsilon_g U)^{***} + then defining - :math:(\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi + +.. math:: (\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi + and - :math:`{p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi` + +.. math:: {p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi