diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst index 864257bec63150f8064332c09e6467c5145f42af..30bff60cebe18f8ebab9e4142030005e01d23ab1 100644 --- a/docs/source/FluidTimeDiscretization.rst +++ b/docs/source/FluidTimeDiscretization.rst @@ -38,12 +38,7 @@ In the corrector - Define a new 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 + \frac{\Delta t}{2} \left( - \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n - \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast}\right) + \\ &+ \frac{Delta t}{2} \left( \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast} \right) \\ &+ \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \Pi + \Gamma \right) - -where the external forcing terms are given by: - - .. math :: \Pi = \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) - .. math :: \Gamma = \rho_g g +.. math:: (\varepsilon_g \rho_g U)^{\ast \ast \ast} &= (\varepsilon_g \rho_g U)^n + \frac{\Delta t}{2} \left( - \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^n - \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g)^{\ast \ast}\right) + \\ &+ \frac{\Delta t}{2} \left( \nabla \cdot \tau^n + \nabla \cdot \tau^{\ast \ast} \right) + \Delta t \left( - \varepsilon_g \nabla {p_g}^{n+1/2,\ast} + \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast} + \rho_g g \right) - Project :math:`U^{\ast \ast \ast}` by solving