Advection using the k-Scheme

CH EN 6355 - Computational Fluid Dynamics

Prof. Tony Saad (www.tsaad.net)
slides at: www.tsaad.net
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 \begin{equation} u_t = - c u_x = - F_x;\quad F = cu \end{equation} We will use a simple Forward Euler explicit method. Using a finite volume integration, we get \begin{equation} u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x} (F_{i+\tfrac{1}{2}}^n - F_{i-\tfrac{1}{2}}^n) \end{equation} For constant grid spacing, the k-Scheme is given by \begin{equation} {\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}}}) \end{equation} which, for a positive advecting velocity, gives us \begin{equation} 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}) \end{equation}

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
    # advect
    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():
    line0.set_ydata(np.ma.array(x[2:-2], mask=True))
    line1.set_ydata(np.ma.array(x[2:-2], mask=True))    
    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():
    styles = requests.get("https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css")
    return HTML(styles.text)
css_styling()
Out[26]:
CSS style adapted from https://github.com/barbagroup/CFDPython. Copyright (c) Barba group
In [ ]: