diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index 95baf6afbe22f23fbdfd09f8e916b4be1e21f1fc..d9583e71f4e0e27bd1d8c59a9c552e66e36ab710 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -14,23 +14,22 @@ In the predictor #. 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 ) +.. 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:: \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 - and +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,\ast} = {p_g}^{n-1/2} + \phi` In the corrector @@ -41,14 +40,14 @@ In the corrector + \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 +#. Project :math:`U^{\ast \ast \ast}` 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)^{\ast \ast \ast} - then defining +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)^{\ast \ast \ast} - \frac{\varepsilon_g}{\rho_g} \nabla \phi - and +and .. math:: {p_g}^{n+1/2} = {p_g}^{n-1/2} + \phi