fixed equation in time discretization section

Time Discretization

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In the absence of reactions, we assume that the fluid density is unchanged.

-  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

.. _Chap:Fluids

.. _Chap:Fluids:

Here we describe the fluid variables, the governing equations, and the time discretization

of the fluid evolution.

.. _Chap:GettingStarted

.. _Chap:GettingStarted:

Getting Started

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.. toctree::

   :maxdepth: 1

MFiX-Exa Introduction

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MFiX-Exa is a new massively parallel code for computing multiphase

flow in which solid particles interact with the gas surrounding them.

.. _Chap:Particles

.. _Chap:Particles:

Particles

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