# this is the 4th-order McCorquodale and Colella method
import numpy as np
import pyro.compressible as comp
import pyro.mesh.array_indexer as ai
from pyro.compressible import riemann
from pyro.mesh import fourth_order, reconstruction
[docs]
def flux_cons(ivars, idir, gamma, q):
flux = q.g.scratch_array(nvar=ivars.nvar)
if idir == 1:
un = q[:, :, ivars.iu]
else:
un = q[:, :, ivars.iv]
flux[:, :, ivars.idens] = q[:, :, ivars.irho]*un
if idir == 1:
flux[:, :, ivars.ixmom] = q[:, :, ivars.irho]*q[:, :, ivars.iu]**2 + q[:, :, ivars.ip]
flux[:, :, ivars.iymom] = q[:, :, ivars.irho]*q[:, :, ivars.iv]*q[:, :, ivars.iu]
else:
flux[:, :, ivars.ixmom] = q[:, :, ivars.irho]*q[:, :, ivars.iu]*q[:, :, ivars.iv]
flux[:, :, ivars.iymom] = q[:, :, ivars.irho]*q[:, :, ivars.iv]**2 + q[:, :, ivars.ip]
flux[:, :, ivars.iener] = (q[:, :, ivars.ip]/(gamma - 1.0) +
0.5*q[:, :, ivars.irho]*(q[:, :, ivars.iu]**2 +
q[:, :, ivars.iv]**2) + q[:, :, ivars.ip])*un
if ivars.naux > 0:
flux[:, :, ivars.irhox:ivars.irhox-1+ivars.naux] = \
q[:, :, ivars.irho]*q[:, :, ivars.ix:ivars.ix-1+ivars.naux]*un
return flux
[docs]
def fluxes(myd, rp, ivars):
alpha = 0.3
beta = 0.3
myg = myd.grid
gamma = rp.get_param("eos.gamma")
# get the cell-average data
U_avg = myd.data
# convert U from cell-averages to cell-centers
U_cc = np.zeros_like(U_avg)
U_cc[:, :, ivars.idens] = myd.to_centers("density")
U_cc[:, :, ivars.ixmom] = myd.to_centers("x-momentum")
U_cc[:, :, ivars.iymom] = myd.to_centers("y-momentum")
U_cc[:, :, ivars.iener] = myd.to_centers("energy")
# compute the primitive variables of both the cell-center and averages
q_bar = comp.cons_to_prim(U_avg, gamma, ivars, myd.grid)
q_cc = comp.cons_to_prim(U_cc, gamma, ivars, myd.grid)
# compute the 4th-order approximation to the cell-average primitive state
q_avg = myg.scratch_array(nvar=ivars.nq)
for n in range(ivars.nq):
q_avg.v(n=n, buf=3)[:, :] = q_cc.v(n=n, buf=3) + myg.dx**2/24.0 * q_bar.lap(n=n, buf=3)
# flattening -- there is a single flattening coefficient (xi) for all directions
use_flattening = rp.get_param("compressible.use_flattening")
if use_flattening:
xi_x = reconstruction.flatten(myg, q_bar, 1, ivars, rp)
xi_y = reconstruction.flatten(myg, q_bar, 2, ivars, rp)
xi = reconstruction.flatten_multid(myg, q_bar, xi_x, xi_y, ivars)
else:
xi = 1.0
# for debugging
nolimit = 0
# do reconstruction on the cell-average primitive variable state, q_avg
for idir in [1, 2]:
# interpolate <W> to faces (with limiting)
q_l = myg.scratch_array(nvar=ivars.nq)
q_r = myg.scratch_array(nvar=ivars.nq)
if nolimit:
for n in range(ivars.nq):
if idir == 1:
qtmp = 7./12.*(q_avg.ip(-1, n=n, buf=1) + q_avg.v(n=n, buf=1)) - \
1./12.*(q_avg.ip(-2, n=n, buf=1) + q_avg.ip(1, n=n, buf=1))
else:
qtmp = 7./12.*(q_avg.jp(-1, n=n, buf=1) + q_avg.v(n=n, buf=1)) - \
1./12.*(q_avg.jp(-2, n=n, buf=1) + q_avg.jp(1, n=n, buf=1))
q_l.v(n=n, buf=1)[:, :] = qtmp
q_r.v(n=n, buf=1)[:, :] = qtmp
else:
for n in range(ivars.nq):
q_l[:, :, n], q_r[:, :, n] = fourth_order.states(q_avg[:, :, n], myg.ng, idir)
# apply flattening
for n in range(ivars.nq):
if idir == 1:
q_l.ip(1, n=n, buf=2)[:, :] = xi.v(buf=2)*q_l.ip(1, n=n, buf=2) + \
(1.0 - xi.v(buf=2))*q_avg.v(n=n, buf=2)
q_r.v(n=n, buf=2)[:, :] = xi.v(buf=2)*q_r.v(n=n, buf=2) + \
(1.0 - xi.v(buf=2))*q_avg.v(n=n, buf=2)
else:
q_l.jp(1, n=n, buf=2)[:, :] = xi.v(buf=2)*q_l.jp(1, n=n, buf=2) + \
(1.0 - xi.v(buf=2))*q_avg.v(n=n, buf=2)
q_r.v(n=n, buf=2)[:, :] = xi.v(buf=2)*q_r.v(n=n, buf=2) + \
(1.0 - xi.v(buf=2))*q_avg.v(n=n, buf=2)
# solve the Riemann problem to find the face-average q
_q = riemann.riemann_prim(idir, myg.ng,
ivars.irho, ivars.iu, ivars.iv, ivars.ip, ivars.ix, ivars.naux,
0, 0,
gamma, q_l, q_r)
q_int_avg = ai.ArrayIndexer(_q, grid=myg)
# calculate the face-centered q using the transverse Laplacian
q_int_fc = myg.scratch_array(nvar=ivars.nq)
if idir == 1:
for n in range(ivars.nq):
q_int_fc.v(n=n, buf=myg.ng-1)[:, :] = q_int_avg.v(n=n, buf=myg.ng-1) - \
1.0/24.0 * (q_int_avg.jp(1, n=n, buf=myg.ng-1) -
2*q_int_avg.v(n=n, buf=myg.ng-1) +
q_int_avg.jp(-1, n=n, buf=myg.ng-1))
else:
for n in range(ivars.nq):
q_int_fc.v(n=n, buf=myg.ng-1)[:, :] = q_int_avg.v(n=n, buf=myg.ng-1) - \
1.0/24.0 * (q_int_avg.ip(1, n=n, buf=myg.ng-1) -
2*q_int_avg.v(n=n, buf=myg.ng-1) +
q_int_avg.ip(-1, n=n, buf=myg.ng-1))
# compute the final fluxes using both the face-average state, q_int_avg,
# and face-centered q, q_int_fc
F_fc = flux_cons(ivars, idir, gamma, q_int_fc)
F_avg = flux_cons(ivars, idir, gamma, q_int_avg)
if idir == 1:
F_x = myg.scratch_array(nvar=ivars.nvar)
for n in range(ivars.nvar):
F_x.v(n=n, buf=1)[:, :] = F_fc.v(n=n, buf=1) + \
1.0/24.0 * (F_avg.jp(1, n=n, buf=1) -
2*F_avg.v(n=n, buf=1) +
F_avg.jp(-1, n=n, buf=1))
else:
F_y = myg.scratch_array(nvar=ivars.nvar)
for n in range(ivars.nvar):
F_y.v(n=n, buf=1)[:, :] = F_fc.v(n=n, buf=1) + \
1.0/24.0 * (F_avg.ip(1, n=n, buf=1) -
2*F_avg.v(n=n, buf=1) +
F_avg.ip(-1, n=n, buf=1))
# artificial viscosity McCorquodale & Colella Eq. 35, 36
# first find face-centered div
lam = myg.scratch_array()
if idir == 1:
lam.v(buf=1)[:, :] = (q_bar.v(buf=1, n=ivars.iu) -
q_bar.ip(-1, buf=1, n=ivars.iu))/myg.dx + \
0.25*(q_bar.jp(1, buf=1, n=ivars.iv) -
q_bar.jp(-1, buf=1, n=ivars.iv) +
q_bar.ip_jp(-1, 1, buf=1, n=ivars.iv) -
q_bar.ip_jp(-1, -1, buf=1, n=ivars.iv))/myg.dy
else:
lam.v(buf=1)[:, :] = (q_bar.v(buf=1, n=ivars.iv) -
q_bar.jp(-1, buf=1, n=ivars.iv))/myg.dy + \
0.25*(q_bar.ip(1, buf=1, n=ivars.iu) -
q_bar.ip(-1, buf=1, n=ivars.iu) +
q_bar.ip_jp(1, -1, buf=1, n=ivars.iu) -
q_bar.ip_jp(-1, -1, buf=1, n=ivars.iu))/myg.dx
test = myg.scratch_array()
test.v(buf=1)[:, :] = (myg.dx*lam.v(buf=1))**2 / \
(beta * gamma * q_bar.v(buf=1, n=ivars.ip) /
q_bar.v(buf=1, n=ivars.irho))
nu = myg.dx * lam * np.minimum(test, 1.0)
nu[lam >= 0.0] = 0.0
if idir == 1:
for n in range(ivars.nvar):
F_x.v(buf=1, n=n)[:, :] += alpha * nu.v(buf=1) * (U_avg.v(buf=1, n=n) - U_avg.ip(-1, buf=1, n=n))
else:
for n in range(ivars.nvar):
F_y.v(buf=1, n=n)[:, :] += alpha * nu.v(buf=1) * (U_avg.v(buf=1, n=n) - U_avg.jp(-1, buf=1, n=n))
return F_x, F_y
