From 5a3b4ef22b7598d5efa2dfd67683bfa8fae6597e Mon Sep 17 00:00:00 2001
From: Johannes Blaschke <jpblaschke@lbl.gov>
Date: Mon, 22 Oct 2018 14:23:40 -0700
Subject: [PATCH] fixed equation in time discretization section

---
 docs/source/FluidTimeDiscretization.rst | 12 ++++++------
 docs/source/Fluids.rst                  |  2 +-
 docs/source/GettingStarted.rst          |  4 ++--
 docs/source/Introduction.rst            |  2 +-
 docs/source/Particles.rst               |  2 +-
 docs/source/RunningTestSuite.rst        |  2 +-
 6 files changed, 12 insertions(+), 12 deletions(-)

diff --git a/docs/source/FluidTimeDiscretization.rst b/docs/source/FluidTimeDiscretization.rst
index 931275a..c538637 100644
--- a/docs/source/FluidTimeDiscretization.rst
+++ b/docs/source/FluidTimeDiscretization.rst
@@ -1,6 +1,6 @@
 
 Time Discretization
-===============
+===================
 
 In the absence of reactions, we assume that the fluid density is unchanged.
 
@@ -38,12 +38,12 @@ 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 + 
-           \Delta t \left( (-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} 
+.. 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} + \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)
 
-.. math::  + (1/2) \nabla \cdot \tau^n + (1/2) \nabla \cdot \tau^{\ast \ast} + 
-            \sum_p \beta_p (V_p - {U_g}^{\ast \ast \ast}) + \rho_g g \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
 
 -  Project :math:`U^{\ast \ast \ast}` by solving
 
diff --git a/docs/source/Fluids.rst b/docs/source/Fluids.rst
index 84294e7..5a8a100 100644
--- a/docs/source/Fluids.rst
+++ b/docs/source/Fluids.rst
@@ -1,4 +1,4 @@
-.. _Chap:Fluids
+.. _Chap:Fluids:
 
 Here we describe the fluid variables, the governing equations, and the time discretization
 of the fluid evolution.
diff --git a/docs/source/GettingStarted.rst b/docs/source/GettingStarted.rst
index 06d5f68..ea78ff5 100644
--- a/docs/source/GettingStarted.rst
+++ b/docs/source/GettingStarted.rst
@@ -1,7 +1,7 @@
-.. _Chap:GettingStarted
+.. _Chap:GettingStarted:
 
 Getting Started
-==================
+===============
 
 .. toctree::
    :maxdepth: 1
diff --git a/docs/source/Introduction.rst b/docs/source/Introduction.rst
index 7be9080..fd7c5a6 100644
--- a/docs/source/Introduction.rst
+++ b/docs/source/Introduction.rst
@@ -1,5 +1,5 @@
 MFiX-Exa Introduction
-==================
+=====================
 
 MFiX-Exa is a new massively parallel code for computing multiphase
 flow in which solid particles interact with the gas surrounding them.
diff --git a/docs/source/Particles.rst b/docs/source/Particles.rst
index eb37c57..0fa2e7a 100644
--- a/docs/source/Particles.rst
+++ b/docs/source/Particles.rst
@@ -1,4 +1,4 @@
-.. _Chap:Particles
+.. _Chap:Particles:
 
 Particles
 =========
diff --git a/docs/source/RunningTestSuite.rst b/docs/source/RunningTestSuite.rst
index f383c2a..ea1c2e7 100644
--- a/docs/source/RunningTestSuite.rst
+++ b/docs/source/RunningTestSuite.rst
@@ -1,5 +1,5 @@
 Running the MFiX Test Suite
-=======================
+===========================
 
 MFiX-Exa comes with several tests aimed at evaluating software
 functionalities. The source files as well as the required input files
-- 
GitLab