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burgers_eqn.py
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burgers_eqn.py
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# -*- coding: utf-8 -*-
"""
Created on Wed Jul 27 15:55:37 2016
@author: clay_budin
"""
# from: https://github.com/barbagroup/CFDPython/blob/master/lessons/18_Burgers_equation.ipynb
#%matplotlib inline
import numpy
from matplotlib import pyplot
from matplotlib import animation
#from JSAnimation.IPython_display import display_animation
#Basic initial condition parameters
#defining grid size, time steps, CFL condition, etc...
nx = 81
nt = 70
sigma = 1
dx = 4.0/nx
dt = sigma*dx
#Define a quick function to set up the initial square wave condition
def u_ic():
u = numpy.ones(nx)
u[(nx-1)/2:]=0
return u
#Define two lambda functions to help with common operations encountered
#with the Euler equations
#utoE = lambda u: (u/2)**2
utoA = lambda u: u**2 # this is A*u @ (n,i)
utoE = lambda u: (u**2)/2
#utoA = lambda u: u
def laxfriedrichs(u, nt, dt, dx):
#initialize our results array with dimensions nt by nx
un = numpy.zeros((nt,len(u)))
#copy the initial u array into each row of our new array
un[:,:] = u.copy()
'''
Now, for each timestep, we're going to calculate u^n+1,
then set the value of u equal to u^n+1 so we can calculate
the next iteration. For every timestep, the entire vector
u^n is saved in a single row of our results array un.
'''
for i in range(1,nt):
E = utoE(u)
un[i,1:-1] = .5*(u[2:]+u[:-2]) - dt/(2*dx)*(E[2:]-E[:-2])
un[i,0] = 1
u = un[i].copy()
return un
u = u_ic() #make sure that u is set to our expected initial conditions
un = laxfriedrichs(u,nt,dt,dx)
print str(un.shape)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
def animate(data):
x = numpy.linspace(0,4,nx)
y = data
line.set_data(x,y)
return line,
# CFL = 1.0
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='loop')
# CFL = .5
dt = .5*dx
u = u_ic() ##Reset our initial conditions (the square wave)
un = laxfriedrichs(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
def laxwendroff(u, nt, dt, dx):
un = numpy.zeros((nt,len(u)))
un[:] = u.copy()
for i in range(1,nt):
E = utoE(u)
un[i,1:-1] = u[1:-1] - dt/(2*dx) * (E[2:]-E[:-2]) + dt**2/(4*dx**2) *\
((u[2:]+u[1:-1])*(E[2:]-E[1:-1]) -\
(u[1:-1]+u[:-2])*(E[1:-1]-E[:-2]))
un[i,0]=1
u = un[i].copy()
return un
u = u_ic()
sigma = 1
dt = sigma*dx
un = laxwendroff(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
# CFL = 1.0
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
# CFL = .5
u = u_ic()
sigma = .5
dt = sigma*dx
un = laxwendroff(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
def maccormack(u, nt, dt, dx):
un = numpy.zeros((nt,len(u)))
ustar = numpy.empty_like(u)
un[:] = u.copy()
ustar = u.copy()
for i in range(1,nt):
E = utoE(u)
ustar[:-1] = u[:-1] - dt/dx * (E[1:]-E[:-1])
Estar = utoE(ustar)
un[i,1:] = .5 * (u[1:]+ustar[1:] - dt/dx * (Estar[1:] - Estar[:-1]))
u = un[i].copy()
return un
u = u_ic()
sigma = 1
dt = sigma*dx
un = maccormack(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
u = u_ic()
sigma = 0.5
dt = sigma*dx
un = maccormack(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
from scipy import linalg
def beamwarming(u, nt, dt, dx):
##Tridiagonal setup##
a = numpy.zeros_like(u)
b = numpy.ones_like(u)
c = numpy.zeros_like(u)
d = numpy.zeros_like(u)
un = numpy.zeros((nt,len(u)))
un[:]=u.copy()
for n in range(1,nt):
u[0] = 1
E = utoE(u)
au = utoA(u)
a[0] = -dt/(4*dx)*u[0]
a[1:] = -dt/(4*dx)*u[0:-1]
a[-1] = -dt/(4*dx)*u[-1]
#b is all ones
c[:-1] = dt/(4*dx)*u[1:]
print str(-.5*dt/dx*(E[2:]-E[0:-2])+dt/(4*dx)*(au[2:]-au[:-2]))
#print str(dt/(4*dx)*(au[2:]-au[:-2]))
d[1:-1] = u[1:-1]-.5*dt/dx*(E[2:]-E[0:-2])+dt/(4*dx)*(au[2:]-au[:-2])
###subtract a[0]*LHS B.C to 'fix' thomas algorithm
d[0] = u[0] - .5*dt/dx*(E[1]-E[0])+dt/(4*dx)*(au[1]-au[0]) - a[0]
ab = numpy.matrix([c,b,a])
u = linalg.solve_banded((1,1), ab, d)
u[0]=1
un[n] = u.copy()
return un
u = u_ic()
sigma = .5
dt = sigma*dx
nt=60
nt = 40
un = beamwarming(u,nt,dt,dx)
fig = pyplot.figure();
ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
line, = ax.plot([],[],lw=2);
anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
#display_animation(anim, default_mode='once')
'''
def dampit(u,eps,dt,dx):
d = u[2]-.5*dt/dx*(u[3]**2/2-u[1]**2/2)+dt/(4*dx)*(u[3]**2-u[1]**2)\
-eps*(u[4]-4*u[3]+6*u[2]-4*u[1]+u[0])
return d
def beamwarming_damp(u, nt, dt, dx):
##Tridiagonal setup##
a = numpy.zeros_like(u)
b = numpy.ones_like(u)
c = numpy.zeros_like(u)
d = numpy.zeros_like(u)
un = numpy.zeros((nt,len(u)))
un[:] = u.copy()
eps = .125
for n in range(1,nt):
u[0] = 1
E = utoE(u)
au = utoA(u)
a[0] = -dt/(4*dx)*u[0]
a[1:] = -dt/(4*dx)*u[0:-1]
a[-1] = -dt/(4*dx)*u[-1]
#b is all ones
c[:-1] = dt/(4*dx)*u[1:]
###Calculate the damping factor for MOST of our u_vector
d[2:-2] = u[2:-2]-.5*dt/dx*(E[3:-1]-E[1:-3])+dt/(4*dx)\
*(au[3:-1]-au[1:-3])\
-eps*(u[4:]-4*u[3:-1]+6*u[2:-2]-4*u[1:-3]+u[:-4])
###Calculate the damping factor for d[0] and d[1]
damp = numpy.concatenate((np.ones(2), u[:3]))
d[0] = dampit(damp,eps,dt,dx)
damp = numpy.concatenate((np.ones(1), u[:4]))
d[1] = dampit(damp,eps,dt,dx)
###subtract a[0]*LHS B.C to 'fix' thomas algorithm
d[0] = d[0] - u[0] * a[0]
ab = numpy.matrix([c,b,a])
u = linalg.solve_banded((1,1), ab, d)
u[0]=1
un[n] = u.copy()
return un
u = u_ic()
sigma = 0.5
dt = sigma*dx
nt = 120
un = beamwarming_damp(u,nt,dt,dx)
#fig = pyplot.figure();
#ax = pyplot.axes(xlim=(0,4),ylim=(-.5,2));
#line, = ax.plot([],[],lw=2);
#anim = animation.FuncAnimation(fig, animate, frames=un, interval=50)
display_animation(anim, default_mode='once')
from IPython.core.display import HTML
def css_styling():
styles = open("../styles/custom.css", "r").read()
return HTML(styles)
css_styling()
'''
pyplot.show()