Commit 95cd65d6 authored by Ann Almgren's avatar Ann Almgren
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More tweaks to formatting

parent 5360afc9
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+1 −6
Original line number Diff line number Diff line
@@ -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