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="0.5")
n += 1
# compute the slopes
da = gpu.lslopes(a, nolimit=1)
lda = gpu.lslopes(a)
n = 2
while (n < nzones - 2):
gpu.drawSlope(gr, n, da[n], a[n], color="r", ls=":")
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)
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")
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.savefig("rea-start.eps")
pylab.savefig("rea-start.png")
#-----------------------------------------------------------------------------
# second frame -- reconstruction
# compute the slopes
#lda = gpu.lslopes(a, nolimit=1)
lda = gpu.lslopes(a)
# draw
pylab.clf()
gpu.drawGrid(gr)
gpu.labelCenter(gr, gr.ng + nzones/2, r"$i$")
gpu.labelCenter(gr, gr.ng + nzones/2-1, r"$i-1$")
gpu.labelCenter(gr, gr.ng + nzones/2+1, r"$i+1$")
gpu.labelCenter(gr, gr.ng + nzones/2-2, r"$i-2$")
gpu.labelCenter(gr, gr.ng + nzones/2+2, r"$i+2$")
# draw cell averages
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="0.5")
n += 1
# compute the slopes
da = gpu.lslopes(a, nolimit=1)
lda = gpu.lslopes(a)
n = 2
while (n < nzones-2):
gpu.drawSlope(gr, n, da[n], a[n], color="r", ls=":")
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()
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")
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