diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index fef179d0b87af682c9fdeffac99cb71a4d3a0ae8..f079d0de82cce1f1ed3b36ac71b2a3cf1d45bb1c 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -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)^{***}`