# Advection using the k-Scheme¶

## CH EN 6355 - Computational Fluid Dynamics¶

Department of Chemical Engineering
University of Utah

Here, we will implement the k-scheme or kappa-schemes for advection. It is easiest to implement this scheme since for different values of k, we recover all sorts of high-order flux approximations. We will assume a positive advecting velocity for illustration purposes.

We are solving the constant speed advection equation given by $$u_t = - c u_x = - F_x;\quad F = cu$$ We will use a simple Forward Euler explicit method. Using a finite volume integration, we get $$u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x} (F_{i+\tfrac{1}{2}}^n - F_{i-\tfrac{1}{2}}^n)$$ For constant grid spacing, the k-Scheme is given by $${\phi _f} = {\phi _{\rm{C}}} + \frac{{1 - k}}{4}({\phi _{\rm{C}}} - {\phi _{\rm{U}}}) + \frac{{1 + k}}{4}({\phi _{\rm{D}}} - {\phi _{\rm{C}}})$$ which, for a positive advecting velocity, gives us $$F_{i + {\textstyle{1 \over 2}}}^n = c\phi _{i + {\textstyle{1 \over 2}}}^n = c{\phi _i} + c\frac{{1 - k}}{4}({\phi _i} - {\phi _{i - 1}}) + c\frac{{1 + k}}{4}({\phi _{i + 1}} - {\phi _i})$$

In [10]:
import numpy as np
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import matplotlib.pyplot as plt
import matplotlib.animation as animation
plt.rcParams['animation.html'] = 'html5'
from matplotlib import cm

In [11]:
def step(x,x0):
x0 = 0.6
x1 = 0.8
result = x - x0
result[x-x1<x1] = 1.0
result[x<x0] = 0.0
result[x>x1] = 0.0
return result

def gaussian(x,x0):
s = 0.08
s = s*s
result = np.exp( -(x-x0)**2/s)
return result

In [14]:
L = 1.0
n = 128 # cells
dx = L/n # n intervals
x = np.linspace(-3*dx/2, L + 3*dx/2, n+4) # include ghost cells - we will include 2 ghost cells on each side for high order schemes

# create arrays
phi = np.zeros(n+4) # cell centered quantity
f = np.zeros(n+4+1) # flux
u = np.ones(n+4+1) # velocity field - assumed to live on faces same as flux

x0 = 0.3
# u0 = np.zeros(N + 2)
# u0[1:-1] = np.sin(2*np.pi*x)
# u0 = np.zeros(N)
# phi0 = np.sin(np.pi*x)
phi0 = gaussian(x,x0) + step(x,x0)
# u0 = triangle(x,0.5,0.75,1)
# u0[0:N//2] = 1.0
plt.plot(x,phi0)

Out[14]:
[<matplotlib.lines.Line2D at 0x1199f8630>]
In [15]:
cfl =0.5
c = 1.0 # use a negative value for left traveling waves
dt = cfl*dx/abs(c)
print('dt=',dt)
print('dx=',dx)

dt= 0.00390625
dx= 0.0078125

In [17]:
# the k scheme
k = 0.5
# finite volume implementation with arrays for fluxes
t = 0
tend= L/abs(c)

sol = []
sol.append(phi0)
ims = []

fig = plt.figure(figsize=[5,3],dpi=200)
plt.rcParams["font.family"] = "serif"
plt.rcParams["font.size"] = 10
plt.rc('text', usetex=True)

# plt.grid()
plt.xlim([0.,L])
plt.ylim([-0.25,1.25])
plt.xlabel('$x$')
plt.ylabel('$\phi$')
plt.tight_layout()
# plot initial condition
plt.plot(x,phi0,'darkred',animated=True)

i = 0
while t < tend:
phin = sol[-1]

#     if (i%16==0):
#         shift = int(np.ceil(c*(t-dt)/dx))
#         im = plt.plot(x[2:-2], np.roll(phin[2:-2], -shift) ,'k-o',markevery=2,markersize=3.5,markerfacecolor='deepskyblue',
#              markeredgewidth=0.25, markeredgecolor='k',linewidth=0.45, animated=True)
#         ims.append(im)

# impose periodic conditions
phin[-2] = phin[2]
phin[-1] = phin[3]
phin[0] = phin[-4]
phin[1] = phin[-3]

phi = np.zeros_like(phi0)

# predictor - take half a step and use upwind
# du/dt = -c*du/dx
if c >= 0:
ϕc = phin[1:-2] # phi upwind
else:
ϕc = phin[2:-1] # phi upwind

f[2:-2] = c*ϕc
phi[2:-2] = phin[2:-2] - dt/2.0/dx*(f[3:-2] - f[2:-3])
phi[-2] = phi[2]
phi[-1] = phi[3]
phi[0] = phi[-4]
phi[1] = phi[-3]

# du/dt = -c*du/dx
if c >= 0:
ϕc = phi[1:-2] # phi upwind
ϕu = phi[:-3]  # phi far upwind
ϕd = phi[2:-1] # phi downwind
else:
ϕc = phi[2:-1] # phi upwind
ϕu = phi[3:]  # phi far upwind
ϕd = phi[1:-2] # phi downwind

f[2:-2] = ϕc + (1-k)/4.0*(ϕc - ϕu) + (1+k)/4.0*(ϕd - ϕc)
f = c*f # multiply the flux by the velocity
phi[2:-2] = phin[2:-2] - c * dt/dx*(f[3:-2] - f[2:-3]) #+ dt/dx/dx*diffusion
t += dt
i+=1
sol.append(phi)

# plt.annotate('k = '+ str(k), xy=(0.5, 0.8), xytext=(0.015, 0.9),fontsize=8)
# plt.legend(('exact','numerical'),loc='upper left',fontsize=7)
# ani = animation.ArtistAnimation(fig, ims, interval=100, blit=True,
#                                 repeat_delay=1000)

# ani.save('k-scheme-'+str(k)+'.mp4',dpi=300,fps=24)

In [24]:
plt.plot(sol[0], label='initial condition')
plt.plot(sol[-1], label='one residence time')
plt.legend()
plt.grid()


# Create Animation in Moving Reference Frame¶

In [27]:
"""
Create Animation in Moving Reference Frame
"""
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
matplotlib.use("Agg")
fig, ax = plt.subplots(figsize=(4,3),dpi=150)
ax.grid(True)
f0 = sol[0]
line0, = ax.plot(x[2:-2], f0[2:-2] ,'r-',linewidth=0.75, animated=True)
line1, = ax.plot(x[2:-2], f0[2:-2] ,'k-o',markevery=2,markersize=3.5,markerfacecolor='deepskyblue',
markeredgewidth=0.25, markeredgecolor='k',linewidth=0.45, animated=True)

ann = ax.annotate('time ='+str(round(t,3))+' s', xy=(2, 1), xytext=(40, 200),xycoords='figure points')
ax.annotate('k ='+str(k) + ' (k-scheme)', xy=(2, 1), xytext=(40, 190),xycoords='figure points')
plt.tight_layout()

def animate_moving(i):
print('time=',i*dt)
t = i*dt
xt = x + i*1.1*c*dt
line0.set_xdata(xt[2:-2])
line1.set_xdata(xt[2:-2])
ax.axes.set_xlim(xt[0],0.0*dx + xt[-1])
f = sol[i]
ax.axes.set_ylim(1.1*min(f) - 0.1,1.1*max(f))
ann.set_text('time ='+str(round(t,4))+'s (' + str(i)+ ').')
shift =int(np.ceil(i*c*dt/dx))
line1.set_ydata(np.roll(f[2:-2], -shift))

f0 = sol[0]
line0.set_ydata(f0[2:-2])
return line0,line1

# Init only required for blitting to give a clean slate.
def init():
return line0,line1

ani = animation.FuncAnimation(fig, animate_moving, np.arange(0,len(sol),2*int(1/cfl)), init_func=init,
interval=20, blit=False)
print('done!')

done!

In [29]:
ani.save('__k-scheme_' + str(k)+'.mp4',fps=24,dpi=300)

time= 0.0
time= 0.0078125
time= 0.015625
time= 0.0234375
time= 0.03125
time= 0.0390625
time= 0.046875
time= 0.0546875
time= 0.0625
time= 0.0703125
time= 0.078125
time= 0.0859375
time= 0.09375
time= 0.1015625
time= 0.109375
time= 0.1171875
time= 0.125
time= 0.1328125
time= 0.140625
time= 0.1484375
time= 0.15625
time= 0.1640625
time= 0.171875

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
<ipython-input-29-f14b34654064> in <module>()
----> 1 ani.save('__k-scheme_' + str(k)+'.mp4',fps=24,dpi=300)

/anaconda3/lib/python3.6/site-packages/matplotlib/animation.py in save(self, filename, writer, fps, dpi, codec, bitrate, extra_args, metadata, extra_anim, savefig_kwargs)
1172                         # TODO: See if turning off blit is really necessary
1173                         anim._draw_next_frame(d, blit=False)
-> 1174                     writer.grab_frame(**savefig_kwargs)
1175
1176         # Reconnect signal for first draw if necessary

/anaconda3/lib/python3.6/site-packages/matplotlib/animation.py in grab_frame(self, **savefig_kwargs)
374             # frame format and dpi.
375             self.fig.savefig(self._frame_sink(), format=self.frame_format,
--> 376                              dpi=self.dpi, **savefig_kwargs)
377         except (RuntimeError, IOError) as e:
378             out, err = self._proc.communicate()

/anaconda3/lib/python3.6/site-packages/matplotlib/figure.py in savefig(self, fname, frameon, transparent, **kwargs)
2092             self.set_frameon(frameon)
2093
-> 2094         self.canvas.print_figure(fname, **kwargs)
2095
2096         if frameon:

/anaconda3/lib/python3.6/site-packages/matplotlib/backend_bases.py in print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, **kwargs)
2073                     orientation=orientation,
2074                     bbox_inches_restore=_bbox_inches_restore,
-> 2075                     **kwargs)
2076             finally:
2077                 if bbox_inches and restore_bbox:

/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py in print_raw(self, filename_or_obj, *args, **kwargs)
459
460     def print_raw(self, filename_or_obj, *args, **kwargs):
--> 461         FigureCanvasAgg.draw(self)
462         renderer = self.get_renderer()
463         with cbook._setattr_cm(renderer, dpi=self.figure.dpi), \

/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py in draw(self)
400         toolbar = self.toolbar
401         try:
--> 402             self.figure.draw(self.renderer)
403             # A GUI class may be need to update a window using this draw, so
404             # don't forget to call the superclass.

/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py in draw_wrapper(artist, renderer, *args, **kwargs)
48                 renderer.start_filter()
49
---> 50             return draw(artist, renderer, *args, **kwargs)
51         finally:
52             if artist.get_agg_filter() is not None:

/anaconda3/lib/python3.6/site-packages/matplotlib/figure.py in draw(self, renderer)
1647
1648             mimage._draw_list_compositing_images(
-> 1649                 renderer, self, artists, self.suppressComposite)
1650
1651             renderer.close_group('figure')

/anaconda3/lib/python3.6/site-packages/matplotlib/image.py in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
136     if not_composite or not has_images:
137         for a in artists:
--> 138             a.draw(renderer)
139     else:
140         # Composite any adjacent images together

/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py in draw_wrapper(artist, renderer, *args, **kwargs)
48                 renderer.start_filter()
49
---> 50             return draw(artist, renderer, *args, **kwargs)
51         finally:
52             if artist.get_agg_filter() is not None:

/anaconda3/lib/python3.6/site-packages/matplotlib/axes/_base.py in draw(self, renderer, inframe)
2626             renderer.stop_rasterizing()
2627
-> 2628         mimage._draw_list_compositing_images(renderer, self, artists)
2629
2630         renderer.close_group('axes')

/anaconda3/lib/python3.6/site-packages/matplotlib/image.py in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
136     if not_composite or not has_images:
137         for a in artists:
--> 138             a.draw(renderer)
139     else:
140         # Composite any adjacent images together

/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py in draw_wrapper(artist, renderer, *args, **kwargs)
48                 renderer.start_filter()
49
---> 50             return draw(artist, renderer, *args, **kwargs)
51         finally:
52             if artist.get_agg_filter() is not None:

/anaconda3/lib/python3.6/site-packages/matplotlib/text.py in draw(self, renderer)
2391         # Draw text, including FancyBboxPatch, after FancyArrowPatch.
2392         # Otherwise, a wedge arrowstyle can land partly on top of the Bbox.
-> 2393         Text.draw(self, renderer)
2394
2395     def get_window_extent(self, renderer=None):

/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py in draw_wrapper(artist, renderer, *args, **kwargs)
48                 renderer.start_filter()
49
---> 50             return draw(artist, renderer, *args, **kwargs)
51         finally:
52             if artist.get_agg_filter() is not None:

/anaconda3/lib/python3.6/site-packages/matplotlib/text.py in draw(self, renderer)
752                     textrenderer.draw_tex(gc, x, y, clean_line,
753                                           textobj._fontproperties, angle,
--> 754                                           mtext=mtext)
755                 else:
756                     textrenderer.draw_text(gc, x, y, clean_line,

/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py in draw_tex(self, gc, x, y, s, prop, angle, ismath, mtext)
231         texmanager = self.get_texmanager()
232
--> 233         Z = texmanager.get_grey(s, size, self.dpi)
234         Z = np.array(Z * 255.0, np.uint8)
235

/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py in get_grey(self, tex, fontsize, dpi)
418         alpha = self.grey_arrayd.get(key)
419         if alpha is None:
--> 420             pngfile = self.make_png(tex, fontsize, dpi)
421             X = _png.read_png(os.path.join(self.texcache, pngfile))
422             self.grey_arrayd[key] = alpha = X[:, :, -1]

/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py in make_png(self, tex, fontsize, dpi)
383             self._run_checked_subprocess(
384                 ["dvipng", "-bg", "Transparent", "-D", str(dpi),
--> 385                  "-T", "tight", "-o", pngfile, dvifile], tex)
386         return pngfile
387

/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py in _run_checked_subprocess(self, command, tex)
296             report = subprocess.check_output(command,
297                                              cwd=self.texcache,
--> 298                                              stderr=subprocess.STDOUT)
299         except subprocess.CalledProcessError as exc:
300             raise RuntimeError(

/anaconda3/lib/python3.6/subprocess.py in check_output(timeout, *popenargs, **kwargs)
354
355     return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
--> 356                **kwargs).stdout
357
358

/anaconda3/lib/python3.6/subprocess.py in run(input, timeout, check, *popenargs, **kwargs)
423     with Popen(*popenargs, **kwargs) as process:
424         try:
--> 425             stdout, stderr = process.communicate(input, timeout=timeout)
426         except TimeoutExpired:
427             process.kill()

/anaconda3/lib/python3.6/subprocess.py in communicate(self, input, timeout)
848                 self._stdin_write(input)
849             elif self.stdout:
--> 850                 stdout = self.stdout.read()
851                 self.stdout.close()
852             elif self.stderr:

KeyboardInterrupt: 
In [26]:
import urllib
import requests
from IPython.core.display import HTML
def css_styling():