def riemann():
# grid info
xmin = 0.0
xmax = 1.0
nzones = 1
ng = 0
gr = gpu.grid(nzones, xmin=xmin, xmax=xmax)
#------------------------------------------------------------------------
# plot a domain without ghostcells
gpu.drawGrid(gr)
gpu.labelCenter(gr, 0, r"$i$")
gpu.labelCellCenter(gr, 0, r"$q_i$")
gpu.markCellLeftState(gr, 0, r"$q_{i-1/2,R}^{n+1/2}$", color="r")
gpu.markCellRightState(gr, 0, r"$q_{i+1/2,L}^{n+1/2}$", color="r")
pylab.arrow(gr.xc[0]-0.05*gr.dx, 0.5, -0.13*gr.dx, 0,
shape='full', head_width=0.075, head_length=0.05,
lw=1, width=0.01,
edgecolor="none", facecolor="r",
length_includes_head=True, zorder=100)
pylab.arrow(gr.xc[0]+0.05*gr.dx, 0.5, 0.13*gr.dx, 0,
shape='full', head_width=0.075, head_length=0.05,
lw=1, width=0.01,
edgecolor="none", facecolor="r",
length_includes_head=True, zorder=100)
pylab.xlim(gr.xl[0]-0.25*gr.dx,gr.xr[2*ng+nzones-1]+0.25*gr.dx)
# pylab.ylim(-0.25, 0.75)
pylab.axis("off")
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(4.0,2.5)
pylab.savefig("states.png")
pylab.savefig("states.eps")
def simplegrid():
nzones = 7
gr = gpu.grid(nzones, xmin=0, xmax=1)
gpu.drawGrid(gr, edgeTicks=0)
# label a few cell-centers
gpu.labelCenter(gr, nzones/2, r"$i$")
gpu.labelCenter(gr, nzones/2-1, r"$i-1$")
gpu.labelCenter(gr, nzones/2+1, r"$i+1$")
# label a few edges
gpu.labelEdge(gr, nzones/2, r"$i-1/2$")
gpu.labelEdge(gr, nzones/2+1, r"$i+1/2$")
# draw an average quantity
gpu.drawCellAvg(gr, nzones/2, 0.4, color="r")
gpu.labelCellAvg(gr, nzones/2, 0.4, r"$\,\langle a \rangle_i$", color="r")
pylab.axis([gr.xmin-1.5*gr.dx,gr.xmax+1.5*gr.dx, -0.25, 1.5])
pylab.axis("off")
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(10.0,2.5)
pylab.savefig("simplegrid2.png")
pylab.savefig("simplegrid2.eps")
def simplegrid():
# grid info
nzones = 7
ng = 1
gr = gpu.grid(nzones, ng, xmin=0.0, xmax=1.0)
#------------------------------------------------------------------------
gpu.drawGrid(gr, emphasizeEnd=1, edgeTicks=0)
# label a few
gpu.labelCenter(gr, ng + nzones / 2, r"$i$")
gpu.labelCenter(gr, ng + nzones / 2 - 1, r"$i-1$")
gpu.labelCenter(gr, ng + nzones / 2 + 1, r"$i+1$")
pylab.xlim(gr.xl[0] - 0.5 * gr.dx,
gr.xr[2 * ng + nzones - 1] + 0.5 * gr.dx)
# label dx
pylab.plot([gr.xl[ng + nzones / 2 - 2], gr.xl[ng + nzones / 2 - 2]],
[-0.35, -0.25],
color="k")
pylab.plot([gr.xr[ng + nzones / 2 - 2], gr.xr[ng + nzones / 2 - 2]],
[-0.35, -0.25],
color="k")
pylab.plot([gr.xl[ng + nzones / 2 - 2], gr.xr[ng + nzones / 2 - 2]],
[-0.3, -0.3],
color="k")
pylab.text(gr.xc[ng + nzones / 2 - 2],
-0.5,
r"$\Delta x$",
horizontalalignment="center")
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
pylab.ylim(-0.5, 1.5)
pylab.axis("off")
f = pylab.gcf()
f.set_size_inches(10.0, 2.5)
pylab.savefig("simplegrid.png")
pylab.savefig("simplegrid.eps")
def simplegrid():
# grid info
nzones = 7
ng = 1
gr = gpu.grid(nzones, ng, xmin=0.0, xmax=1.0)
#------------------------------------------------------------------------
gpu.drawGrid(gr, emphasizeEnd=1, edgeTicks=0)
# label a few
gpu.labelCenter(gr, ng+nzones/2, r"$i$")
gpu.labelCenter(gr, ng+nzones/2-1, r"$i-1$")
gpu.labelCenter(gr, ng+nzones/2+1, r"$i+1$")
pylab.xlim(gr.xl[0]-0.5*gr.dx,gr.xr[2*ng+nzones-1]+0.5*gr.dx)
# label dx
pylab.plot([gr.xl[ng+nzones/2-2], gr.xl[ng+nzones/2-2]],
[-0.35,-0.25], color="k")
pylab.plot([gr.xr[ng+nzones/2-2], gr.xr[ng+nzones/2-2]],
[-0.35,-0.25], color="k")
pylab.plot([gr.xl[ng+nzones/2-2], gr.xr[ng+nzones/2-2]],
[-0.3,-0.3], color="k")
pylab.text(gr.xc[ng+nzones/2-2], -0.5, r"$\Delta x$",
horizontalalignment="center")
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
pylab.ylim(-0.5, 1.5)
pylab.axis("off")
f = pylab.gcf()
f.set_size_inches(10.0,2.5)
pylab.savefig("simplegrid.png")
pylab.savefig("simplegrid.eps")
#-----------------------------------------------------------------------------
nzones = 8
# data that lives on the grid
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones)
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, nzones / 2, r"$i$")
gpu.labelCenter(gr, nzones / 2 - 1, r"$i-1$")
gpu.labelCenter(gr, nzones / 2 + 1, r"$i+1$")
gpu.labelCenter(gr, nzones / 2 - 2, r"$i-2$")
gpu.labelCenter(gr, nzones / 2 + 2, r"$i+2$")
n = 0
while (n < nzones):
gpu.drawCellAvg(gr, n, a[n], color="r")
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -0.25, 1.2])
pylab.axis("off")
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
#-----------------------------------------------------------------------------
nzones = 8
# data that lives on the grid
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
a = numpy.array([0.3, 1.0, 1.0, 0.8, 0.2, 0.15, 0.5, 0.55])
gr = gpu.grid(nzones)
pylab.clf()
gpu.drawGrid(gr, centerOnly=1)
gpu.labelCenter(gr, nzones/2-1, r"$i$")
gpu.labelCenter(gr, nzones/2, r"$i+1$")
# compute the parabolic coefficients
ap, am, a6 = gpu.ppm(a, nolimit=1)
lap, lam, la6 = gpu.ppm(a)
n = gr.nx/2-1
while (n <= gr.nx/2):
gpu.drawParabola(gr, n, lap[n], lam[n], la6[n], color="r")
n += 1
nn = gr.nx/2-1
sigma = 0.6
gpu.ppmTraceLeft(gr, nn, lap[nn], lam[nn], la6[nn], sigma, color="0.75")
nzones = 7
ng = 2
# data that lives on the grid
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
a = numpy.array([1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones, ng)
pylab.clf()
gpu.drawGrid(gr, drawGhost=1, emphasizeEnd=1)
gpu.labelCenter(gr, ng + nzones / 2, r"$i$")
gpu.labelCenter(gr, ng + nzones / 2 - 1, r"$i-1$")
gpu.labelCenter(gr, ng + nzones / 2 + 1, r"$i+1$")
gpu.labelCenter(gr, gr.ilo, r"$\mathrm{lo}$")
gpu.labelCenter(gr, gr.ilo - 1, r"$\mathrm{lo-1}$")
gpu.labelCenter(gr, gr.ilo - 2, r"$\mathrm{lo-2}$")
gpu.labelCenter(gr, gr.ihi, r"$\mathrm{hi}$")
gpu.labelCenter(gr, gr.ihi + 1, r"$\mathrm{hi+1}$")
gpu.labelCenter(gr, gr.ihi + 2, r"$\mathrm{hi+2}$")
gpu.labelEdge(gr, ng + nzones / 2, r"$i-1/2$")
gpu.labelEdge(gr, ng + nzones / 2 + 1, r"$i+1/2$")
# draw the data
# -----------------------------------------------------------------------------
nf = 5
nc = 3
grf = gpu.grid(nf, voff=2.0, fd=1)
grc = gpu.grid(nc, fd=1)
pylab.clf()
gpu.drawGrid(grf)
gpu.drawGrid(grc)
gpu.labelCenter(grf, nf / 2 - 2, r"$i-2$")
gpu.labelCenter(grf, nf / 2 - 1, r"$i-1$")
gpu.labelCenter(grf, nf / 2, r"$i$")
gpu.labelCenter(grf, nf / 2 + 1, r"$i+1$")
gpu.labelCenter(grf, nf / 2 + 2, r"$i+2$")
gpu.labelCenter(grc, nc / 2 - 1, r"$j-1$")
gpu.labelCenter(grc, nc / 2, r"$j$")
gpu.labelCenter(grc, nc / 2 + 1, r"$j+1$")
gpu.labelCellCenter(grf, nf / 2 - 2, r"$\phi_{i-2}^h$")
gpu.labelCellCenter(grf, nf / 2 - 1, r"$\phi_{i-1}^h$")
gpu.labelCellCenter(grf, nf / 2, r"$\phi_i^h$")
gpu.labelCellCenter(grf, nf / 2 + 1, r"$\phi_{i+1}^h$")
gpu.labelCellCenter(grf, nf / 2 + 2, r"$\phi_{i+2}^h$")
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
a = numpy.array([0.55, 0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones, ng=1, fd=1)
pylab.clf()
gpu.drawGrid(gr, drawGhost=1)
labels = ["-1", "0", "1", "", "i-1", "i", "i+1", "", "N-2", "N-1", "N"]
i = gr.ilo - gr.ng
while (i < gr.ng + gr.nx + 1):
if not labels[i] == "":
gpu.labelCenter(gr, i, r"$%s$" % (labels[i]), fontsize="medium")
i += 1
# draw the data
i = gr.ilo
while i < gr.ihi + 1:
gpu.drawFDData(gr, i, a[i - gr.ng], color="r")
i += 1
gpu.labelFD(gr, gr.ilo + 4, a[gr.ilo + 4 - gr.ng], r"$a_i$", color="r")
# label dx
pylab.plot([gr.xc[gr.ng + nzones / 2 - 1], gr.xc[gr.ng + nzones / 2 - 1]],
[-0.35, -0.25],
color="k")
pylab.plot([gr.xc[gr.ng + nzones / 2], gr.xc[gr.ng + nzones / 2]],
#-----------------------------------------------------------------------------
nzones = 8
# data that lives on the grid
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones, fd=1)
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, nzones/2, r"$i$")
gpu.labelCenter(gr, nzones/2-1, r"$i-1$")
gpu.labelCenter(gr, nzones/2+1, r"$i+1$")
gpu.labelCenter(gr, nzones/2-2, r"$i-2$")
gpu.labelCenter(gr, nzones/2+2, r"$i+2$")
# draw the data
i = 0
while i < nzones:
gpu.drawFDData(gr, i, a[i], color="r")
i += 1
gpu.labelFD(gr, nzones/2, a[nzones/2], r"$f_i$", color="r")
a = numpy.array([0.55, 0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones, ng=1, fd=1)
pylab.clf()
gpu.drawGrid(gr, drawGhost=1)
labels = ["-1", "0", "1", "", "i-1", "i", "i+1", "", "N-2", "N-1", "N"]
i = gr.ilo-gr.ng
while (i < gr.ng+gr.nx+1):
if not labels[i] == "":
gpu.labelCenter(gr, i, r"$%s$" % (labels[i]), fontsize="medium")
i += 1
# draw the data
i = gr.ilo
while i < gr.ihi+1:
gpu.drawFDData(gr, i, a[i-gr.ng], color="r")
i += 1
gpu.labelFD(gr, gr.ilo+4, a[gr.ilo+4-gr.ng], r"$a_i$", color="r")
# label dx
pylab.plot([gr.xc[gr.ng+nzones/2-1], gr.xc[gr.ng+nzones/2-1]], [-0.35,-0.25], color="k")
pylab.plot([gr.xc[gr.ng+nzones/2], gr.xc[gr.ng+nzones/2]], [-0.35,-0.25], color="k")
import numpy
import pylab
import grid_plot_util as gpu
nzones = 5
# data that lives on the grid
a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5])
gr = gpu.grid(nzones)
gpu.drawGrid(gr)
gpu.labelCenter(gr, nzones/2, r"$x_i$")
gpu.labelAvg(gr, nzones/2, a[nzones/2], r"$\langle f\rangle_i$")
gpu.labelCenter(gr, nzones/2-1, r"$x_{i-1}$")
gpu.labelCenter(gr, nzones/2+1, r"$x_{i+1}$")
gpu.labelCenter(gr, nzones/2-2, r"$x_{i-2}$")
gpu.labelCenter(gr, nzones/2+2, r"$x_{i+2}$")
gpu.labelEdge(gr, nzones/2, r"$x_{i-1/2}$")
gpu.labelEdge(gr, nzones/2+1, r"$x_{i+1/2}$")
n = 0
while (n < nzones):
gpu.drawCellAvg(gr, n, a[n])
n += 1
import math
import numpy
import pylab
import grid_plot_util as gpu
nzones = 5
# data that lives on the grid
a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5])
gr = gpu.grid(nzones)
gpu.drawGrid(gr)
gpu.labelCenter(gr, nzones / 2, r"$x_i$")
gpu.labelAvg(gr, nzones / 2, a[nzones / 2], r"$\langle f\rangle_i$")
gpu.labelCenter(gr, nzones / 2 - 1, r"$x_{i-1}$")
gpu.labelCenter(gr, nzones / 2 + 1, r"$x_{i+1}$")
gpu.labelCenter(gr, nzones / 2 - 2, r"$x_{i-2}$")
gpu.labelCenter(gr, nzones / 2 + 2, r"$x_{i+2}$")
gpu.labelEdge(gr, nzones / 2, r"$x_{i-1/2}$")
gpu.labelEdge(gr, nzones / 2 + 1, r"$x_{i+1/2}$")
n = 0
while (n < nzones):
gpu.drawCellAvg(gr, n, a[n])
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -0.25, 1.2])
def riemann():
# grid info
xmin = 0.0
xmax = 1.0
nzones = 1
ng = 0
gr = gpu.grid(nzones, xmin=xmin, xmax=xmax)
#------------------------------------------------------------------------
# plot a domain without ghostcells
gpu.drawGrid(gr)
gpu.labelCenter(gr, 0, r"$i$")
gpu.labelCellCenter(gr, 0, r"$q_i$")
gpu.markCellLeftState(gr, 0, r"$q_{i-1/2,R}^{n+1/2}$", color="r")
gpu.markCellRightState(gr, 0, r"$q_{i+1/2,L}^{n+1/2}$", color="r")
pylab.arrow(gr.xc[0] - 0.05 * gr.dx,
0.5,
-0.13 * gr.dx,
0,
shape='full',
head_width=0.075,
head_length=0.05,
lw=1,
width=0.01,
edgecolor="none",
facecolor="r",
length_includes_head=True,
zorder=100)
pylab.arrow(gr.xc[0] + 0.05 * gr.dx,
0.5,
0.13 * gr.dx,
0,
shape='full',
head_width=0.075,
head_length=0.05,
lw=1,
width=0.01,
edgecolor="none",
facecolor="r",
length_includes_head=True,
zorder=100)
pylab.xlim(gr.xl[0] - 0.25 * gr.dx,
gr.xr[2 * ng + nzones - 1] + 0.25 * gr.dx)
# pylab.ylim(-0.25, 0.75)
pylab.axis("off")
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
f = pylab.gcf()
f.set_size_inches(4.0, 2.5)
pylab.savefig("states.png")
pylab.savefig("states.eps")
def evolve(gr, a, C, num, nolimit=1):
#-------------------------------------------------------------------------
# first frame -- the original cell-averages
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones / 2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 2, r"$i+2$", fontsize="medium")
# draw cell averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, a[n], color="r")
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -1.25, 2.0])
pylab.axis("off")
print gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
ax = pylab.gca()
pylab.text(0.5,
0.75,
"initial state (cell averages)",
horizontalalignment="center",
fontsize=16,
color="b",
transform=ax.transAxes)
pylab.text(0.5,
0.95,
"Piecewise Linear Method for Linear Advection",
horizontalalignment="center",
fontsize=20,
color="k",
transform=ax.transAxes)
f = pylab.gcf()
f.set_size_inches(12.8, 7.2)
if (nolimit):
pylab.savefig("rea-nolimit-start_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-start_%3.3d.eps" % (num))
else:
pylab.savefig("rea-start_%3.3d.png" % (num))
pylab.savefig("rea-start_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# second frame -- reconstruction
# compute the slopes
lda = gpu.lslopes(a, nolimit=nolimit)
# draw
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones / 2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 2, r"$i+2$", fontsize="medium")
# draw cell averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, a[n], color="0.5", ls=":")
n += 1
n = gr.ilo
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="r")
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -1.25, 2.0])
pylab.axis("off")
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
pylab.text(0.5,
0.75,
"reconstructed slopes",
horizontalalignment="center",
fontsize=16,
color="b",
transform=ax.transAxes)
pylab.text(0.5,
0.95,
"Piecewise Linear Method for Linear Advection",
horizontalalignment="center",
fontsize=20,
color="k",
transform=ax.transAxes)
f = pylab.gcf()
f.set_size_inches(12.8, 7.2)
if (nolimit):
pylab.savefig("rea-nolimit-reconstruction_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-reconstruction_%3.3d.eps" % (num))
else:
pylab.savefig("rea-reconstruction_%3.3d.png" % (num))
pylab.savefig("rea-reconstruction_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# third frame -- evolve
# draw
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones / 2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 2, r"$i+2$", fontsize="medium")
# draw cell slopes
n = gr.ilo
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="0.75", ls=":")
n += 1
# evolve
n = gr.ilo
while (n <= gr.ihi):
gpu.evolveToRight(gr, n, lda, a, C, color="r")
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -1.25, 2.0])
pylab.axis("off")
print gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
pylab.text(0.5,
0.75,
"evolved with C = {}".format(C),
horizontalalignment="center",
fontsize=16,
color="b",
transform=ax.transAxes)
pylab.text(0.5,
0.95,
"Piecewise Linear Method for Linear Advection",
horizontalalignment="center",
fontsize=20,
color="k",
transform=ax.transAxes)
f = pylab.gcf()
f.set_size_inches(12.8, 7.2)
if (nolimit):
pylab.savefig("rea-nolimit-evolve_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-evolve_%3.3d.eps" % (num))
else:
pylab.savefig("rea-evolve_%3.3d.png" % (num))
pylab.savefig("rea-evolve_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# fourth frame -- re-average
# left states (we don't need the right state when u > 0)
al = numpy.zeros(2 * gr.ng + gr.nx, dtype=numpy.float64)
n = gr.ilo
while (n <= gr.ihi + 1):
al[n] = a[n - 1] + 0.5 * (1 - C) * lda[n - 1]
n += 1
# the Riemann problem just picks the left state. Do a conservative
# update
anew = numpy.zeros(2 * gr.ng + gr.nx, dtype=numpy.float64)
anew[gr.ilo:gr.ihi+1] = a[gr.ilo:gr.ihi+1] + \
C*(al[gr.ilo:gr.ihi+1] - al[gr.ilo+1:gr.ihi+2])
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones / 2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 - 2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones / 2 + 2, r"$i+2$", fontsize="medium")
# show the evolved profiles from the old time
n = gr.ilo
while (n <= gr.ihi):
gpu.evolveToRight(gr, n, lda, a, C, color="0.5", ls=":")
n += 1
# draw new averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, anew[n], color="red")
n += 1
pylab.axis([gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx, -1.25, 2.0])
pylab.axis("off")
print gr.xmin - 0.5 * gr.dx, gr.xmax + 0.5 * gr.dx
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
pylab.text(0.5,
0.75,
"averaged profile (final state)",
horizontalalignment="center",
fontsize=16,
color="b",
transform=ax.transAxes)
pylab.text(0.5,
0.95,
"Piecewise Linear Method for Linear Advection",
horizontalalignment="center",
fontsize=20,
color="k",
transform=ax.transAxes)
f = pylab.gcf()
f.set_size_inches(12.8, 7.2)
if (nolimit):
pylab.savefig("rea-nolimit-final_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-final_%3.3d.eps" % (num))
else:
pylab.savefig("rea-final_%3.3d.png" % (num))
pylab.savefig("rea-final_%3.3d.eps" % (num))
return anew
def riemann():
# grid info
xmin = 0.0
xmax = 1.0
nzones = 4
ng = 2
gr = gpu.grid(nzones, ng=ng)
# interior
atemp = numpy.array([0.8, 0.7, 0.4, 0.5])
a = numpy.zeros(2*gr.ng + gr.nx, dtype=numpy.float64)
# fill interior and ghost cells
a[gr.ilo:gr.ihi+1] = atemp[:]
a[0:gr.ilo] = a[gr.ihi-1:gr.ihi+1]
a[gr.ihi:2*gr.ng+gr.nx] = a[gr.ihi]
#------------------------------------------------------------------------
# plot a domain without ghostcells
gpu.drawGrid(gr, emphasizeEnd=1, drawGhost=1)
gpu.labelCenter(gr, gr.ng-2, r"$\mathrm{lo-2}$")
gpu.labelCenter(gr, gr.ng-1, r"$\mathrm{lo-1}$")
gpu.labelCenter(gr, gr.ng, r"$\mathrm{lo}$")
gpu.labelCenter(gr, gr.ng+1, r"$\mathrm{lo+1}$")
gpu.labelEdge(gr, gr.ng, r"$\mathrm{lo}-1/2$")
# draw cell averages
n = 0
while n < gr.ng+gr.nx:
gpu.drawCellAvg(gr, n, a[n], color="0.5", ls=":")
n += 1
# get slopes
lda = gpu.lslopes(a, nolimit=1)
n = gr.ilo-1
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="r")
n += 1
# compute the states to the left and right of lo-1/2
C = 0.7 # CFL
al = a[gr.ilo-1] + 0.5*gr.dx*(1.0 - C)*lda[gr.ilo-1]
ar = a[gr.ilo] - 0.5*gr.dx*(1.0 + C)*lda[gr.ilo]
# L
gpu.markCellRightState(gr, ng-1, r"$a_{\mathrm{lo}+1/2,L}^{n+1/2}$",
value=al, vertical="top", color="b")
# R
gpu.markCellLeftState(gr, ng, r"$a_{\mathrm{lo}+1/2,R}^{n+1/2}$",
value=ar, vertical="top", color="b")
pylab.xlim(gr.xl[0]-0.15*gr.dx,gr.xr[ng+1]+0.15*gr.dx)
pylab.ylim(-0.25, 1.1)
pylab.axis("off")
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0,2.0)
pylab.tight_layout()
pylab.savefig("riemann-bc.png")
pylab.savefig("riemann-bc.eps")
anode = numpy.append(anode, anode[ng])
acell = numpy.insert(acell, 0, acell[ncells - 1])
acell = numpy.append(acell, acell[ng])
print "anode: ", len(anode), nnodes, ng
#-----------------------------------------------------------------------------
# finite difference
gr = gpu.grid(nnodes, ng=ng, fd=1)
pylab.clf()
gpu.drawGrid(gr, emphasizeEnd=1, drawGhost=1)
gpu.labelCenter(gr, ng + nnodes / 2, r"$i$")
gpu.labelCenter(gr, ng + nnodes / 2 - 1, r"$i-1$")
gpu.labelCenter(gr, ng + nnodes / 2 + 1, r"$i+1$")
#gpu.labelCenter(gr, ng+nnodes/2-2, r"$i-2$")
#gpu.labelCenter(gr, ng+nnodes/2+2, r"$i+2$")
# draw the data
i = 0
while i < nnodes + 2 * ng:
gpu.drawFDData(gr, i, anode[i], color="r")
i += 1
gpu.labelFD(gr, ng + nnodes / 2, anode[ng + nnodes / 2], r"$f_i$", color="r")
# label dx
pylab.plot([gr.xc[gr.ng + nnodes / 2 - 1], gr.xc[gr.ng + nnodes / 2 - 1]],
def riemann():
# grid info
xmin = 0.0
xmax = 1.0
nzones = 4
ng = 2
gr = gpu.grid(nzones, ng=ng)
# interior
atemp = numpy.array([0.8, 0.7, 0.4, 0.5])
a = numpy.zeros(2 * gr.ng + gr.nx, dtype=numpy.float64)
# fill interior and ghost cells
a[gr.ilo:gr.ihi + 1] = atemp[:]
a[0:gr.ilo] = a[gr.ihi - 1:gr.ihi + 1]
a[gr.ihi:2 * gr.ng + gr.nx] = a[gr.ihi]
#------------------------------------------------------------------------
# plot a domain without ghostcells
gpu.drawGrid(gr, emphasizeEnd=1, drawGhost=1)
gpu.labelCenter(gr, gr.ng - 2, r"$\mathrm{lo-2}$")
gpu.labelCenter(gr, gr.ng - 1, r"$\mathrm{lo-1}$")
gpu.labelCenter(gr, gr.ng, r"$\mathrm{lo}$")
gpu.labelCenter(gr, gr.ng + 1, r"$\mathrm{lo+1}$")
gpu.labelEdge(gr, gr.ng, r"$\mathrm{lo}-1/2$")
# draw cell averages
n = 0
while n < gr.ng + gr.nx:
gpu.drawCellAvg(gr, n, a[n], color="0.5", ls=":")
n += 1
# get slopes
lda = gpu.lslopes(a, nolimit=1)
n = gr.ilo - 1
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="r")
n += 1
# compute the states to the left and right of lo-1/2
C = 0.7 # CFL
al = a[gr.ilo - 1] + 0.5 * gr.dx * (1.0 - C) * lda[gr.ilo - 1]
ar = a[gr.ilo] - 0.5 * gr.dx * (1.0 + C) * lda[gr.ilo]
# L
gpu.markCellRightState(gr,
ng - 1,
r"$a_{\mathrm{lo}+1/2,L}^{n+1/2}$",
value=al,
vertical="top",
color="b")
# R
gpu.markCellLeftState(gr,
ng,
r"$a_{\mathrm{lo}+1/2,R}^{n+1/2}$",
value=ar,
vertical="top",
color="b")
pylab.xlim(gr.xl[0] - 0.15 * gr.dx, gr.xr[ng + 1] + 0.15 * gr.dx)
pylab.ylim(-0.25, 1.1)
pylab.axis("off")
pylab.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0, 2.0)
pylab.tight_layout()
pylab.savefig("riemann-bc.png")
pylab.savefig("riemann-bc.eps")
# plot two stacked fv grids of different (2x) resolution to show prolongation
#-----------------------------------------------------------------------------
nf = 5
nc = 3
grf = gpu.grid(nf, voff=2.0, fd=1)
grc = gpu.grid(nc, fd=1)
pylab.clf()
gpu.drawGrid(grf)
gpu.drawGrid(grc)
gpu.labelCenter(grf, nf / 2 - 2, r"$i-2$")
gpu.labelCenter(grf, nf / 2 - 1, r"$i-1$")
gpu.labelCenter(grf, nf / 2, r"$i$")
gpu.labelCenter(grf, nf / 2 + 1, r"$i+1$")
gpu.labelCenter(grf, nf / 2 + 2, r"$i+2$")
gpu.labelCenter(grc, nc / 2 - 1, r"$j-1$")
gpu.labelCenter(grc, nc / 2, r"$j$")
gpu.labelCenter(grc, nc / 2 + 1, r"$j+1$")
gpu.labelCellCenter(grf, nf / 2 - 2, r"$\phi_{i-2}^h$")
gpu.labelCellCenter(grf, nf / 2 - 1, r"$\phi_{i-1}^h$")
gpu.labelCellCenter(grf, nf / 2, r"$\phi_i^h$")
gpu.labelCellCenter(grf, nf / 2 + 1, r"$\phi_{i+1}^h$")
gpu.labelCellCenter(grf, nf / 2 + 2, r"$\phi_{i+2}^h$")
nzones = 7
ng = 2
# data that lives on the grid
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])
#a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
a = numpy.array([1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])
gr = gpu.grid(nzones, ng)
pylab.clf()
gpu.drawGrid(gr, drawGhost=1, emphasizeEnd=1)
gpu.labelCenter(gr, ng+nzones/2, r"$i$")
gpu.labelCenter(gr, ng+nzones/2-1, r"$i-1$")
gpu.labelCenter(gr, ng+nzones/2+1, r"$i+1$")
gpu.labelCenter(gr, gr.ilo, r"$\mathrm{lo}$")
gpu.labelCenter(gr, gr.ilo-1, r"$\mathrm{lo-1}$")
gpu.labelCenter(gr, gr.ilo-2, r"$\mathrm{lo-2}$")
gpu.labelCenter(gr, gr.ihi, r"$\mathrm{hi}$")
gpu.labelCenter(gr, gr.ihi+1, r"$\mathrm{hi+1}$")
gpu.labelCenter(gr, gr.ihi+2, r"$\mathrm{hi+2}$")
gpu.labelEdge(gr, ng+nzones/2, r"$i-1/2$")
gpu.labelEdge(gr, ng+nzones/2+1, r"$i+1/2$")
# draw the data
def evolve(gr, a, C, num, nolimit=1):
#-------------------------------------------------------------------------
# first frame -- the original cell-averages
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones/2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+2, r"$i+2$", fontsize="medium")
# draw cell averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, a[n], color="r")
n += 1
pylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2])
pylab.axis("off")
print gr.xmin-0.5*gr.dx, gr.xmax+0.5*gr.dx
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0,2.0)
if (nolimit):
pylab.savefig("rea-nolimit-start_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-start_%3.3d.eps" % (num))
else:
pylab.savefig("rea-start_%3.3d.png" % (num))
pylab.savefig("rea-start_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# second frame -- reconstruction
# compute the slopes
lda = gpu.lslopes(a, nolimit=nolimit)
# draw
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones/2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+2, r"$i+2$", fontsize="medium")
# draw cell averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, a[n], color="0.5", ls=":")
n += 1
n = gr.ilo
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="r")
n += 1
pylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2])
pylab.axis("off")
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0,2.0)
if (nolimit):
pylab.savefig("rea-nolimit-reconstruction_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-reconstruction_%3.3d.eps" % (num))
else:
pylab.savefig("rea-reconstruction_%3.3d.png" % (num))
pylab.savefig("rea-reconstruction_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# third frame -- evolve
# draw
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones/2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+2, r"$i+2$", fontsize="medium")
# draw cell slopes
n = gr.ilo
while (n <= gr.ihi):
gpu.drawSlope(gr, n, lda[n], a[n], color="0.75", ls=":")
n += 1
# evolve
n = gr.ilo
while (n <= gr.ihi):
gpu.evolveToRight(gr, n, lda, a, C, color="r")
n += 1
pylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2])
pylab.axis("off")
print gr.xmin-0.5*gr.dx, gr.xmax+0.5*gr.dx
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0,2.0)
if (nolimit):
pylab.savefig("rea-nolimit-evolve_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-evolve_%3.3d.eps" % (num))
else:
pylab.savefig("rea-evolve_%3.3d.png" % (num))
pylab.savefig("rea-evolve_%3.3d.eps" % (num))
#-------------------------------------------------------------------------
# fourth frame -- re-average
# left states (we don't need the right state when u > 0)
al = numpy.zeros(2*gr.ng + gr.nx, dtype=numpy.float64)
n = gr.ilo
while (n <= gr.ihi+1):
al[n] = a[n-1] + 0.5*(1 - C)*lda[n-1]
n += 1
# the Riemann problem just picks the left state. Do a conservative
# update
anew = numpy.zeros(2*gr.ng + gr.nx, dtype=numpy.float64)
anew[gr.ilo:gr.ihi+1] = a[gr.ilo:gr.ihi+1] + \
C*(al[gr.ilo:gr.ihi+1] - al[gr.ilo+1:gr.ihi+2])
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones/2, r"$i$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-1, r"$i-1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+1, r"$i+1$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2-2, r"$i-2$", fontsize="medium")
gpu.labelCenter(gr, gr.ng + nzones/2+2, r"$i+2$", fontsize="medium")
# show the evolved profiles from the old time
n = gr.ilo
while (n <= gr.ihi):
gpu.evolveToRight(gr, n, lda, a, C, color="0.5", ls=":")
n += 1
# draw new averages
n = gr.ilo
while (n <= gr.ihi):
gpu.drawCellAvg(gr, n, anew[n], color="red")
n += 1
pylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2])
pylab.axis("off")
print gr.xmin-0.5*gr.dx, gr.xmax+0.5*gr.dx
pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)
f = pylab.gcf()
f.set_size_inches(8.0,2.0)
if (nolimit):
pylab.savefig("rea-nolimit-final_%3.3d.png" % (num))
pylab.savefig("rea-nolimit-final_%3.3d.eps" % (num))
else:
pylab.savefig("rea-final_%3.3d.png" % (num))
pylab.savefig("rea-final_%3.3d.eps" % (num))
return anew
