From 21af62aa17c9148dff5ee278c64e073c2b41c7c9 Mon Sep 17 00:00:00 2001
From: Ann Almgren <asalmgren@lbl.gov>
Date: Mon, 22 Oct 2018 21:33:56 -0700
Subject: [PATCH] more formatting
---
docs/source/FluidTimeDiscretization.rst | 13 ++++++-------
1 file changed, 6 insertions(+), 7 deletions(-)
diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst
index 1802983..61cee6a 100644
--- a/docs/source/FluidTimeDiscretization.rst
+++ b/docs/source/FluidTimeDiscretization.rst
@@ -14,14 +14,13 @@ 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 \left( -\nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) - \varepsilon_g \nabla {p_g}^{n-1/2} \right.
-
-.. math:: \left. \nabla \cdot \tau^n + \sum_p \beta_p (V_p - {U_g}^{\ast}) + \rho_g g \right)
+.. math:: (\varepsilon_g \rho_g U)^{\ast} &= (\varepsilon_g \rho_g U)^n -
+ \Delta t \left( \nabla \cdot (\varepsilon_g \rho_g U^{MAC} U_g) + \varepsilon_g \nabla {p_g}^{n-1/2} \right) \\ &+
+ \Delta t \left( \nabla \cdot \tau^n + \sum_p \beta_p (V_p - {U_g}^{\ast}) + \rho_g g \right)
- Project :math:`U^{\ast}` by solving
-.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( (\varepsilon_g U)^{\ast}+ \Delta t \varepsilon_g \nabla {p_g}^{n-1/2} \right)
+.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( (\frac{1}{\Delta t \varepsilon_g U)^{\ast}+ \varepsilon_g \nabla {p_g}^{n-1/2} \right)
then defining
@@ -38,11 +37,11 @@ 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} + \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast} + \rho_g g \right)
+.. 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
-.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( (\varepsilon_g U)^{\ast \ast \ast} + \Delta t \varepsilon_g \nabla {p_g}^{n+1/2,\ast} \right)
+.. math:: \nabla \cdot \frac{\varepsilon_g}{\rho_g} \nabla \phi = \nabla \cdot \left( \frac{1}{\Delta t} (\varepsilon_g U)^{\ast \ast \ast} + \varepsilon_g \nabla {p_g}^{n+1/2,\ast} \right)
then defining
--
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