/
Diffusion.py
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Diffusion.py
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from scitools.std import *
class Diffusion:
"""
This class solves the diffusion equation in 1D by means of the following algorithms: explicit,implicit,
cranck-Nicholson -scheme, and by consider a random walker in 1D.
"""
def __init__(self,t):
self.a = 0
self.b = 1
self.Nx = 36
self.hx = (self.b - self.a)/float((self.Nx -1))
self.Nt = 50
self.t = t
self.ht = t/float(self.Nt-1)
self.D = 0.1
self.alpha = (self.D*self.ht)/(self.hx*self.hx)
def exact(self):
N = 100
x = linspace(self.a, self.b,self.Nx)
u_exact = zeros(self.Nx)
tic = time.time()
for n in range(1,N+1):
An = -2/(n*pi)
u_exact[:] = u_exact[:] + An*sin(n*pi*x[:])*exp(-self.D*((n*pi)**2)*self.t)
u_exact[:] = 1 - x[:] + u_exact[:]
t_exact = linspace(self.a,self.b, len(x))
toc = time.time()
return t_exact, u_exact
def explicit(self):
Nx,Nt,t, alpha = self.Nx, self.Nt, self.t, self.alpha
if alpha >= 0.5:
print "The explicit scheme is unstable for this spatio-temporal gridding"
sys.exit(1)
print alpha
u = zeros(Nx)
unew = zeros(Nx)
u[0] = 1 ; unew[0] = 1; u[-1] = 0; unew[-1] = 0
j = 1
while j <= (Nt-1):
unew[1:-1] = alpha*u[:-2] + (1-2*alpha)*u[1:-1] + alpha*u[2:]
u[:] = unew[:]
j +=1
return unew
def implicit(self):
Nx, alpha, Nt = self.Nx, self.alpha, self.Nt
b = (1+2*alpha); a = (-1*alpha); c = a
unew = zeros(Nx)
u = zeros(Nx)
vnew = zeros(Nx)
v = zeros(Nx)
grid = linspace(self.a, self.b,Nx)
#initial condition
u[0] = 1 ; unew[0] = 1; u[-1] = 0; unew[-1] = 0
g = lambda x:1-x
v = u - g(grid)
vnew = unew - g(grid)
for i in range(1,Nt):
new = self.tridiagonal(a,b,c,vnew,v,Nx)
v[:] = vnew[:]
unew = vnew + g(grid)
return unew
def cranck_nicolson(self):
Nx, alpha, Nt = self.Nx, self.alpha, self.Nt
b = (2+2*alpha); a = (-1*alpha); c = a
unew = zeros(Nx)
u = zeros(Nx)
temp = zeros(Nx)
v = zeros(Nx)
vnew = zeros(Nx)
grid = linspace(self.a, self.b,Nx)
#Initial condition
u[0] = 1 ; unew[0] = 1; u[-1] = 0; unew[-1] = 0
g = lambda x:1-x
v = u - g(grid)
vnew = unew - g(grid)
#Time loop
for i in range(1,Nt):
for j in range(1,len(v)-1):
temp[j] = alpha*v[j-1] + alpha*v[j+1] + (2-2*alpha)*v[j]
vnew = self.tridiagonal(a,b,c,vnew,v,Nx)
v[:] =vnew[:]
unew = vnew + g(grid)
return unew
def tridiagonal(self, a,b,c,vnew,v,Nx):
temp = zeros(Nx)
btemp = b
vnew[1] = v[1]/btemp
#forward substitution
for i in range(1,self.Nx-1):
temp[i] = c/btemp
btemp = b-a*temp[i]
vnew[i] = (v[i] - a*vnew[i-1])/btemp
#backward substitution
for i in range(self.Nx-2,0,-1):
vnew[i] = vnew[i] - temp[i+1]*vnew[i+1]
return vnew
def MC_uniform(self,walkers):
l0 = sqrt(2*self.D*self.ht)
temp_1 = int((1./l0)) + 1
position = zeros(temp_1,int)
temp = zeros(temp_1,int)
position[0] = walkers
temp[0] = walkers
for j in range(self.Nt):
for i in range(len(position)):
if temp[i] != 0:
for k in range(temp[i]):
r = random.random()
#right
if r > 0.5 and i!= 0 and i!= (len(position)-2):
position[i+1] += 1
position[i] -= 1
if r > 0.5 and i== 0:
position[i+1] += 1
if r > 0.5 and i == (len(position)-2):
position[i+1] = 0
#left
if r < 0.5 and i!= 0 and i!= 1:
position[i-1] +=1
position[i] -=1
if r< 0.5 and i==1:
position[i] -=1
temp[:]= position[:]
position_scale = (1./position[0])*position
t_uniform = linspace(self.a, self.b, len(position))
return t_uniform, position_scale
def MC_normal(self,walkers):
Nbins = 10
t = linspace(0,1,Nbins)
dudes = []
for i in range(walkers):
dudes.append(0)
total = walkers
count = 0
for j in range(self.Nt):
count = 0
total = len(dudes)
while count < total:
epsilon = random.normal(0,(1/sqrt(2)))
dudes[count] = dudes[count] + sqrt(2*self.D*self.ht)*epsilon
if dudes[count]< 0:
del dudes[count]
dudes.append(0)
if dudes[count] >1:
del dudes[count]
count += 1
hist, bin_edges = histogram(dudes)
hist = (1./max(hist)) * hist
return hist, t
def error():
Nx = 10
explicit_error = zeros(Nx)
implicit_error = zeros(Nx)
MC_uniform_error = zeros(Nx)
for i in range(1,11):
d1 = Diffusion(t=0.3)
t_exact,u_exact = d1.exact()
u_explicit = d1.explicit()
u_implicit = d1.implicit()
def compare_MC_exact():
max_error = []
sum_error = []
walkers = []
for i in range(500,20000,500):
d1 = Diffusion(t=0.2)
t_exact,u_exact = d1.exact()
t_uniform , mc_uniform = d1.MC_uniform(i)
print len(u_exact)
print len(mc_uniform)
diff = u_exact[:] - mc_uniform[:]
temp = max(abs(diff))
max_error.append(temp)
temp = sum(diff)
sum_error.append(temp)
temp = i
print temp
walkers.append(temp)
from matplotlib.pylab import plt
plt.figure(2)
plt.plot(walkers,max_error,'o-')
plt.xlabel('Number of walkers')
plt.ylabel('Maximum error')
plt.savefig('mcuniform_error1.eps')
plt.figure(3)
plt.plot(walkers,sum_error,'o-')
plt.xlabel('Number of walkers')
plt.ylabel('Accumulated error')
plt.savefig('mcuniform_error2.eps')
def execute():
d1 = Diffusion(t=1)
#t_exact,u_exact = d1.exact()
hist,Nbins = d1.MC_normal(100)
#plot(t_exact,u_exact,'*-')
#hold('on')
plot(hist)
if __name__ == '__main__':
execute()
#compare_MC_exact()