diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index d9583e71f4e0e27bd1d8c59a9c552e66e36ab710..6dada556305e538980414b0dd5f6b69fbfe3fac4 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -10,21 +10,22 @@ Thus here we focus on the discretization of the momentum equation In the predictor -#. Define :math:`U^{MAC}`, the face-centered (staggered) MAC velocity which is used for advection. +- 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)^{\ast}` by setting +- 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 ) + \Delta t ( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} -#. Project :math:`U^{\ast}` by solving +.. math:: | \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)^{n+1} = (\varepsilon_g U)^{***} - \frac{\varepsilon_g}{\rho_g} \nabla \phi +.. math:: U^{\ast \ast} = U^{\ast} - \frac{1}{\rho_g} \nabla \phi and @@ -33,20 +34,22 @@ and In the corrector -#. Define an approximation to the new-time state,:math:`(\varepsilon_g \rho_g U)^{\ast \ast \ast}` by setting +- 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} + + + \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 ) -#. Project :math:`U^{\ast \ast \ast}` 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)^{\ast \ast \ast} -then defining + then defining -.. math:: (\varepsilon_g U)^{n+1} = (\varepsilon_g U)^{\ast \ast \ast} - \frac{\varepsilon_g}{\rho_g} \nabla \phi +.. math:: U^{n+1} = U^{\ast \ast \ast} - \frac{1}{\rho_g} \nabla \phi and