Add page about the construction of slopes

Creating the MAC velocities

~~~~~~~~~~~~~~~~~~~~~~~~~~~

To create the normal velocities on faces, we first extrapolate from the cell centers on each side and

upwind the face value to define  :math:`U^{pred}`

To create the normal velocities on faces, we first extrapolate from the cell centers on each side using the

slopes as computed earlier, and upwind the face value to define  :math:`U^{pred}`

.  To compute the x-velocity on the x-faces of regular (ie not cut) cells, we call

   :maxdepth: 1

   FluidTimeStep

   Slopes

   BuildingMacVelocities

Computing slopes

~~~~~~~~~~~~~~~~~~~~~~~~~~~

Slopes are computed one direction at a time for each scalar and each

velocity component.

We use the second order Monotonized Central (MC)

limiter (van Leer, 1977). The scheme is described below for the u-velocity.

The limiter computes the slope at cell "i" by combining the left, central

and right u-variation "du":

.. code:: shell

     du_l = u(i) - u(i-1)               = left variation

     du_c = 0.5 * ( u(i+1) - u(i-1) )   = central (umlimited) variation

     du_r = u(i+1) - u(i)               = right variation

Finally, the u-variation at cell "i" is given by :

.. code:: shell

     du(i) = sign(du_c) * min(2|du_l|, |du_c|, 2|du_r|)) if du_l*du_r > 0

     du(i) = 0                                           otherwise

The above procedure is applied direction by direction.

BOUNDARY CONDITIONS

When periodic or Neumann's BCs are imposed, the scheme can be applied

without any change since the ghost cells at the boundary are filled

by either periodicity or by extrapolation.

For Dirichlet's BCs in the transversal direction, the scheme can again

be applied as is since the velocity is known at the first ghost cell

out of the domain.

However, for Dirichlet's BCs in the longitudinal direction, the velocity

is not known outside the domain since the BC is applied directly at the first

valid node which lies on the boundary itself. Therefore, the scheme must be

arranged as follows to use ONLY values from inside the domain.

For a left boundary (i=0), the u-variations are:

.. code:: shell

     du_l = 0                             Dont use values on the left

     du_c = -1.5*u(0) + 2*u(1) -0.5*u(2)  2nd order right-biased

     du_r = u(1) - u(0)                   Right variation