import mesh.reconstruction as reconstruction
[docs]def unsplit_fluxes(my_data, rp, dt, scalar_name):
r"""
Construct the fluxes through the interfaces for the linear advection
equation:
.. math::
a_t + u a_x + v a_y = 0
We use a second-order (piecewise linear) unsplit Godunov method
(following Colella 1990).
In the pure advection case, there is no Riemann problem we need to
solve -- we just simply do upwinding. So there is only one 'state'
at each interface, and the zone the information comes from depends
on the sign of the velocity.
Our convection is that the fluxes are going to be defined on the
left edge of the computational zones::
| | | |
| | | |
-+------+------+------+------+------+------+--
| i-1 | i | i+1 |
a_l,i a_r,i a_l,i+1
a_r,i and a_l,i+1 are computed using the information in
zone i,j.
Parameters
----------
my_data : CellCenterData2d object
The data object containing the grid and advective scalar that
we are advecting.
rp : RuntimeParameters object
The runtime parameters for the simulation
dt : float
The timestep we are advancing through.
scalar_name : str
The name of the variable contained in my_data that we are
advecting
Returns
-------
out : ndarray, ndarray
The fluxes on the x- and y-interfaces
"""
myg = my_data.grid
a = my_data.get_var(scalar_name)
# get the advection velocities
u = rp.get_param("advection.u")
v = rp.get_param("advection.v")
cx = u*dt/myg.dx
cy = v*dt/myg.dy
# --------------------------------------------------------------------------
# monotonized central differences
# --------------------------------------------------------------------------
limiter = rp.get_param("advection.limiter")
ldelta_ax = reconstruction.limit(a, myg, 1, limiter)
ldelta_ay = reconstruction.limit(a, myg, 2, limiter)
a_x = myg.scratch_array()
# upwind
if u < 0:
# a_x[i,j] = a[i,j] - 0.5*(1.0 + cx)*ldelta_a[i,j]
a_x.v(buf=1)[:, :] = a.v(buf=1) - 0.5*(1.0 + cx)*ldelta_ax.v(buf=1)
else:
# a_x[i,j] = a[i-1,j] + 0.5*(1.0 - cx)*ldelta_a[i-1,j]
a_x.v(buf=1)[:, :] = a.ip(-1, buf=1) + 0.5*(1.0 - cx)*ldelta_ax.ip(-1, buf=1)
# y-direction
a_y = myg.scratch_array()
# upwind
if v < 0:
# a_y[i,j] = a[i,j] - 0.5*(1.0 + cy)*ldelta_a[i,j]
a_y.v(buf=1)[:, :] = a.v(buf=1) - 0.5*(1.0 + cy)*ldelta_ay.v(buf=1)
else:
# a_y[i,j] = a[i,j-1] + 0.5*(1.0 - cy)*ldelta_a[i,j-1]
a_y.v(buf=1)[:, :] = a.jp(-1, buf=1) + 0.5*(1.0 - cy)*ldelta_ay.jp(-1, buf=1)
# compute the transverse flux differences. The flux is just (u a)
# HOTF
F_xt = u*a_x
F_yt = v*a_y
F_x = myg.scratch_array()
F_y = myg.scratch_array()
# the zone where we grab the transverse flux derivative from
# depends on the sign of the advective velocity
if u <= 0:
mx = 0
else:
mx = -1
if v <= 0:
my = 0
else:
my = -1
dtdx2 = 0.5*dt/myg.dx
dtdy2 = 0.5*dt/myg.dy
F_x.v(buf=1)[:, :] = u*(a_x.v(buf=1) -
dtdy2*(F_yt.ip_jp(mx, 1, buf=1) -
F_yt.ip(mx, buf=1)))
F_y.v(buf=1)[:, :] = v*(a_y.v(buf=1) -
dtdx2*(F_xt.ip_jp(1, my, buf=1) -
F_xt.jp(my, buf=1)))
return F_x, F_y