def ainslie_full(nj, ni, h, k, u0=8.5, ct=ct_v90(u0), i0=8.0):
star = time()
nj += 1
ni += 1
Dmi = ct - 0.05 - (16.0 * ct - 0.5) * i0 / 1000.0
u = [[0.0 for _ in range(ni)]]
v = [[0.0 for _ in range(ni)] for _ in range(nj)]
for g in range(ni):
u[0][g] = u0 * (1.0 - Dmi * exp(- 3.56 * float(g * h) ** 2.0 / b(Dmi, ct) ** 2.0))
for j in range(1, nj):
# start = time()
A = []
B = []
C = []
R = []
i = 0
A.append(- k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct))
B.append(2.0 * (h ** 2.0 * u[j-1][i] + k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
C.append(- k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct))
R.append(k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (2.0 * u[j-1][i+1] - 2.0 * u[j-1][i]) + 2.0 * h ** 2.0 * u[j-1][i] ** 2.0)
for i in range(1, ni):
# Uncomment if v is not neglected. Radial velocity.
if j == 1:
v[j][i] = (i * h) / ((i * h) + h) * (v[j-1][i-1] - h / k * (u[j-1][i] - u[0][i]))
elif j > 1:
v[j][i] = (i * h) / ((i * h) + h) * (v[j-1][i-1] - h / k * (u[j-1][i] - u[j-2][i]))
A.append(k * (h * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) - (i * h) * h * v[j][i] - 2.0 * (i * h) * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
B.append(4.0 * (i * h) * (h ** 2.0 * u[j-1][i] + k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
C.append(k * ((i * h) * h * v[j][i] - 2.0 * (i * h) * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) - h * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
if i < ni - 1:
R.append(h * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (u[j-1][i+1] - u[j-1][i-1]) + 2.0 * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (i * h) * (u[j-1][i+1] - 2.0 * u[j-1][i] + u[j-1][i-1]) - (i * h) * h * k * v[j-1][i] * (u[j-1][i+1] - u[j-1][i-1]) + 4.0 * (i * h) * h ** 2.0 * u[j-1][i] ** 2.0)
elif i == ni - 1:
R.append(h * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (u0 - u[j-1][i-1]) + 2.0 * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (i * h) * (u0 - 2.0 * u[j-1][i] + u[j-1][i-1]) - (i * h) * h * k * v[j-1][i] * (u0 - u[j-1][i-1]) + 4.0 * (i * h) * h ** 2.0 * u[j-1][i] ** 2.0)
# print time() - start
C[0] += A[0]
del A[0]
R[-1] -= C[-1] * u0
del C[-1]
# start3 = time()
u.append(thomas(A, B, C, R))
# print time() - start3
print time() - star,
print 's'
print u[-1][0]
return u
def ainslie(ct, u0, distance_parallel, distance_perpendicular, I0):
# centreline = open('centreline.dat', 'w')
# velocity = open('velocity.dat', 'w')
h = 0.01
L = distance_parallel
n = int(L / h) + 1
Uc1 = [0.0 for _ in range(n)]
d1 = [0.0 for _ in range(n)]
Ct = ct # Thrust coefficient
U0 = u0
# dr = 0.1
Y = distance_perpendicular
# m = int(Y / dr)
Dmi = Ct - 0.05 - (16.0 * Ct - 0.5) * I0 / 10.0
# print Dmi, "Dmi"
Uc1[0] = U0 * (1.0 - Dmi) # Boundary condition at x = 2.0
# print U0, U0 * (1.0 - Dmi)
# print Uc1[0], "Uci[0]"
d1[0] = Dmi
# print d1[0], "d1[0]"
for i in range(
1, n
): # For all positions in the wake centreline direction. Recursive. Whole grid
Uc1[i] = Uc1[i - 1] + (h * 16.0 *
E(i * h, Uc1[i - 1], d1[i - 1], U0, I0, Ct) *
(Uc1[i - 1]**3.0 - U0 * Uc1[i - 1]**2.0 -
Uc1[i - 1] * U0**2.0 + U0**3.0) /
(Uc1[i - 1] * Ct * U0**2.0))
d1[i] = 1.0 - Uc1[i] / U0
# Code to calculate wake deficit at a specific point instead of the whole grid. Namely, the rotor's centrepoint.
# print "final"
return d1[-1] * exp(-3.56 * (Y / b(d1[-1], ct))**2.0) * (
1.0 + 7.12 * (0.07 * distance_parallel / b(d1[-1], ct)))**(-0.5)
def ainslie(ct, u0, distance_parallel, distance_perpendicular, I0):
# centreline = open('centreline.dat', 'w')
# velocity = open('velocity.dat', 'w')
h = 0.01
L = distance_parallel - 2.0
n = int(L / h) + 1
Uc1 = [0.0 for _ in range(n)]
d1 = [0.0 for _ in range(n)]
karman = 0.41 # von Karman constant
Ct = ct # Thrust coefficient
U0 = u0
# dr = 0.1
Y = distance_perpendicular
# m = int(Y / dr)
Dmi = Ct - 0.05 - (16.0 * Ct - 0.5) * I0 / 1000.0
def b(deficit): # Wake width measure
return (3.56 * Ct / (8.0 * deficit * (1.0 - 0.5 * deficit))) ** 0.5
def F(x): # Factor for near and far wake
if x >= 5.5:
return 1.0
if x < 5.5:
if x >= 4.5:
return 0.65 + ((x - 4.5) / 23.32) ** (1.0 / 3.0)
else:
return 0.65 - ((- x + 4.5) / 23.32) ** (1.0 / 3.0)
def E(x1, Uf, Ud, Dm): # Eddy viscosity term
epsilon = F(x1) * (0.015 * b(Dm) * (Uf - Ud) + (karman ** 2.0) * I0 / 100.0)
print epsilon
return epsilon
Uc1[0] = U0 * (1.0 - Dmi) # Boundary condition at x = 2.0
d1[0] = Dmi
for i in range(1, n): # For all positions in the wake centreline direction. Recursive. Whole grid
Uc1[i] = Uc1[i - 1] + (h * 16.0 * E(i * h, U0, Uc1[i - 1], d1[i - 1]) * (Uc1[i - 1] ** 3.0 - U0 * Uc1[i - 1] ** 2.0 - Uc1[i - 1] * U0 ** 2.0 + U0 ** 3.0) / (Uc1[i - 1] * Ct * U0 ** 2.0))
d1[i] = 1.0 - Uc1[i] / U0
########### Code to calculate wake deficit at a specific point instead of the whole grid. Namely, the rotor's centrepoint.
return d1[-1] * exp(- 3.56 * (Y / b(d1[-1])) ** 2.0) * (1.0 + 7.12 * (0.07 * distance_parallel / b(d1[-1]))) ** (- 0.5)
def ainslie_full(nj, ni, h, k, u0=8.5, ct=ct_v90(u0), i0=8.0):
star = time()
nj += 1
ni += 1
Dmi = ct - 0.05 - (16.0 * ct - 0.5) * i0 / 1000.0
u = [[0.0 for _ in range(ni)]]
v = [[0.0 for _ in range(ni)] for _ in range(nj)]
for g in range(ni):
u[0][g] = u0 * (1.0 -
Dmi * exp(-3.56 * float(g * h)**2.0 / b(Dmi, ct)**2.0))
for j in range(1, nj):
# start = time()
A = []
B = []
C = []
R = []
i = 0
A.append(-k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct))
B.append(2.0 * (h**2.0 * u[j - 1][i] +
k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct)))
C.append(-k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct))
R.append(k *
E(j * k, u[j - 1][i], (u0 - u[j - 1][i]) / u0, u0, i0, ct) *
(2.0 * u[j - 1][i + 1] - 2.0 * u[j - 1][i]) +
2.0 * h**2.0 * u[j - 1][i]**2.0)
for i in range(1, ni):
# Uncomment if v is not neglected. Radial velocity.
if j == 1:
v[j][i] = (i * h) / ((i * h) + h) * (v[j - 1][i - 1] - h / k *
(u[j - 1][i] - u[0][i]))
elif j > 1:
v[j][i] = (i * h) / (
(i * h) + h) * (v[j - 1][i - 1] - h / k *
(u[j - 1][i] - u[j - 2][i]))
A.append(k * (h * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) -
(i * h) * h * v[j][i] - 2.0 *
(i * h) * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct)))
B.append(4.0 * (i * h) *
(h**2.0 * u[j - 1][i] +
k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct)))
C.append(k * ((i * h) * h * v[j][i] - 2.0 *
(i * h) * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) -
h * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct)))
if i < ni - 1:
R.append(
h * k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) *
(u[j - 1][i + 1] - u[j - 1][i - 1]) +
2.0 * k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) *
(i * h) *
(u[j - 1][i + 1] - 2.0 * u[j - 1][i] + u[j - 1][i - 1]) -
(i * h) * h * k * v[j - 1][i] *
(u[j - 1][i + 1] - u[j - 1][i - 1]) + 4.0 *
(i * h) * h**2.0 * u[j - 1][i]**2.0)
elif i == ni - 1:
R.append(h * k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) *
(u0 - u[j - 1][i - 1]) +
2.0 * k * E(j * k, u[j - 1][i],
(u0 - u[j - 1][i]) / u0, u0, i0, ct) *
(i * h) * (u0 - 2.0 * u[j - 1][i] + u[j - 1][i - 1]) -
(i * h) * h * k * v[j - 1][i] *
(u0 - u[j - 1][i - 1]) + 4.0 *
(i * h) * h**2.0 * u[j - 1][i]**2.0)
# print time() - start
C[0] += A[0]
del A[0]
R[-1] -= C[-1] * u0
del C[-1]
# start3 = time()
u.append(thomas(A, B, C, R))
# print time() - start3
print time() - star,
print 's'
print u[-1][0]
return u
def ainslie_full(ct, u0, distance_parallel, distance_perpendicular, i0):
# centreline = open('centreline.dat', 'w')
# velocity = open('velocity.dat', 'w')
di = 2.0
dj = distance_parallel - 2.0
# ni = int(di * 80)
# nj = int(dj * 80)
ni = nj = 100
k = dj / float(nj)
h = di / float(ni)
nj += 1
ni += 1
Dmi = ct - 0.05 - (16.0 * ct - 0.5) * i0 / 1000.0
u = [0.0 for _ in range(ni)]
v = [0.0 for _ in range(ni)]
for g in range(ni):
u[g] = u0 * (1.0 - Dmi * exp(- 3.56 * float(g * h) ** 2.0 / b(Dmi, ct) ** 2.0))
old_u = u
u_initial = u
old_v = v
for j in range(1, nj):
# start = time()
A = []
B = []
C = []
R = []
i = 0
A.append(- k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct))
B.append(2.0 * (h ** 2.0 * old_u[i] + k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct)))
C.append(- k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct))
R.append(k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) * (2.0 * old_u[i+1] - 2.0 * old_u[i]) + 2.0 * h ** 2.0 * old_u[i] ** 2.0)
v[0] = 0.0
for i in range(1, ni):
# Uncomment if v is not neglected. Radial velocity.
if j == 1:
v[i] = (i * h) / ((i * h) + h) * (old_v[i-1] - h / k * (old_u[i] - u_initial[i]))
elif j > 1:
v[i] = (i * h) / ((i * h) + h) * (old_v[i-1] - h / k * (old_u[i] - old2_u[i]))
A.append(k * (h * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) - (i * h) * h * v[i] - 2.0 * (i * h) * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct)))
B.append(4.0 * (i * h) * (h ** 2.0 * old_u[i] + k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct)))
C.append(k * ((i * h) * h * v[i] - 2.0 * (i * h) * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) - h * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct)))
if i < ni - 1:
R.append(h * k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) * (old_u[i+1] - old_u[i-1]) + 2.0 * k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) * (i * h) * (old_u[i+1] - 2.0 * old_u[i] + old_u[i-1]) - (i * h) * h * k * old_v[i] * (old_u[i+1] - old_u[i-1]) + 4.0 * (i * h) * h ** 2.0 * old_u[i] ** 2.0)
elif i == ni - 1:
R.append(h * k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) * (u0 - old_u[i-1]) + 2.0 * k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) * (i * h) * (u0 - 2.0 * old_u[i] + old_u[i-1]) - (i * h) * h * k * old_v[i] * (u0 - old_u[i-1]) + 4.0 * (i * h) * h ** 2.0 * old_u[i] ** 2.0)
# print time() - start
C[0] += A[0]
del A[0]
R[-1] -= C[-1] * u0
del C[-1]
# start3 = time()
old2_u = old_u
old_u = thomas(A, B, C, R)
old_v = v
# print time() - start3
# print time() - star,
# print 's'
##### Code to calculate the average wake deficit in all the area of the rotor ###############
## Define function to integrate.
## p. 77 Adapting and calibration of existing wake models to meet the conditions inside offshore wind farms. For integrand squared momentum deficit.
# def G(r, theta):
# z = sqrt(Y ** 2.0 + r ** 2.0 + 2.0 * Y * r * cos(theta))
# gauss = U0 * (1.0 - d1[n - 1] * exp(- 3.56 * (z / b(d1[n - 1])) ** 2.0))
# return r * (U0 - gauss) ** 2.0
#
# A = pi * 0.5 ** 2.0 ## Unitary diameter in this program.
# U = U0 - sqrt((1.0 / A) * simpson_integrate2D(G, 0.0, 0.5, 5, 0.0, 2.0 * pi, 10))
# print old_u[int(round(distance_perpendicular * 80.0, 0))]
return 1.0 - old_u[int(distance_perpendicular * 50.0)] / u0
def E(x1, Uf, Ud, Dm): # Eddy viscosity term
epsilon = F(x1) * (0.015 * b(Dm) * (Uf - Ud) + (karman ** 2.0) * I0 / 100.0)
print epsilon
return epsilon
def ainslie_full(ct, u0, distance_parallel, distance_perpendicular, i0):
# centreline = open('centreline.dat', 'w')
# velocity = open('velocity.dat', 'w')
di = 2.0
dj = distance_parallel
# ni = int(di * 80)
# nj = int(dj * 80)
ni = nj = 100
k = dj / float(nj)
h = di / float(ni)
nj += 1
ni += 1
Dmi = ct - 0.05 - (16.0 * ct - 0.5) * i0 / 10.0
u = [0.0 for _ in range(ni)]
v = [0.0 for _ in range(ni)]
for g in range(ni):
u[g] = u0 * (1.0 -
Dmi * exp(-3.56 * float(g * h)**2.0 / b(Dmi, ct)**2.0))
old_u = u
u_initial = u
old_v = v
old2_u = 0.0
for j in range(1, nj):
# start = time()
A = []
B = []
C = []
R = []
i = 0
A.append(-k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct))
B.append(2.0 *
(h**2.0 * old_u[i] + k * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct)))
C.append(-k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct))
R.append(k * E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) *
(2.0 * old_u[i + 1] - 2.0 * old_u[i]) +
2.0 * h**2.0 * old_u[i]**2.0)
v[0] = 0.0
for i in range(1, ni):
# Uncomment if v is not neglected. Radial velocity.
if j == 1:
v[i] = (i * h) / ((i * h) + h) * (old_v[i - 1] - h / k *
(old_u[i] - u_initial[i]))
elif j > 1:
v[i] = (i * h) / ((i * h) + h) * (old_v[i - 1] - h / k *
(old_u[i] - old2_u[i]))
A.append(k * (h * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct) -
(i * h) * h * v[i] - 2.0 *
(i * h) * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct)))
B.append(
4.0 * (i * h) *
(h**2.0 * old_u[i] + k * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct)))
C.append(k * ((i * h) * h * v[i] - 2.0 *
(i * h) * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct) -
h * E(j * k, old_u[i],
(u0 - old_u[i]) / u0, u0, i0, ct)))
if i < ni - 1:
R.append(h * k *
E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) *
(old_u[i + 1] - old_u[i - 1]) + 2.0 * k *
E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) *
(i * h) *
(old_u[i + 1] - 2.0 * old_u[i] + old_u[i - 1]) -
(i * h) * h * k * old_v[i] *
(old_u[i + 1] - old_u[i - 1]) + 4.0 *
(i * h) * h**2.0 * old_u[i]**2.0)
elif i == ni - 1:
R.append(h * k *
E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) *
(u0 - old_u[i - 1]) + 2.0 * k *
E(j * k, old_u[i], (u0 - old_u[i]) / u0, u0, i0, ct) *
(i * h) * (u0 - 2.0 * old_u[i] + old_u[i - 1]) -
(i * h) * h * k * old_v[i] * (u0 - old_u[i - 1]) +
4.0 * (i * h) * h**2.0 * old_u[i]**2.0)
# print time() - start
C[0] += A[0]
del A[0]
R[-1] -= C[-1] * u0
del C[-1]
# start3 = time()
old2_u = old_u
old_u = thomas(A, B, C, R)
old_v = v
# print time() - start3
# print time() - star,
# print 's'
# Code to calculate the average wake deficit in all the area of the rotor ###############
# Define function to integrate.
# p. 77 Adapting and calibration of existing wake models to meet the conditions inside
# -----offshore wind farms. For integrand squared momentum deficit.
# def G(r, theta):
# z = sqrt(Y ** 2.0 + r ** 2.0 + 2.0 * Y * r * cos(theta))
# gauss = U0 * (1.0 - d1[n - 1] * exp(- 3.56 * (z / b(d1[n - 1])) ** 2.0))
# return r * (U0 - gauss) ** 2.0
#
# A = pi * 0.5 ** 2.0 ## Unitary diameter in this program.
# U = U0 - sqrt((1.0 / A) * simpson_integrate2D(G, 0.0, 0.5, 5, 0.0, 2.0 * pi, 10))
# print old_u[int(round(distance_perpendicular * 80.0, 0))]
return 1.0 - old_u[int(distance_perpendicular * 50.0)] / u0
def ainslie_full(parallel, perpendicular, discretisation_parallel, discretisation_perpendicular, ct=0.79, u0=8.5, i0=8.0):
Dmi = ct - 0.05 - (16.0 * ct - 0.5) * i0 / 1000.0
if perpendicular == 0.0:
perpendicular = 0.01
h = perpendicular / discretisation_perpendicular
k = parallel / discretisation_parallel
nj = discretisation_parallel
ni = discretisation_perpendicular
u = [[0.0 for _ in range(ni)]]
v = [[0.0 for _ in range(ni)] for _ in range(nj)]
for g in range(ni):
u[0][g] = u0 * (1.0 - Dmi * exp(- 3.56 * float(g * h) ** 2.0 / b(Dmi, ct) ** 2.0))
print('\rPercentage: 0.0 %. Time remaining: s'),
for j in range(1, nj):
start = time()
A = []
B = []
C = []
R = []
for i in range(ni):
# Uncomment if v is not neglected. Radial velocity.
if i == 1:
v[j][i] = (i * h) / ((i * h) + h) * (v[j-1][i-1] - h / k * (u[j-1][i] - u[0][i]))
elif i > 1:
v[j][i] = (i * h) / ((i * h) + h) * (v[j-1][i-1] - h / k * (u[j-1][i] - u[j-2][i]))
if i == 0:
A.append(- k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct))
B.append(2.0 * (h ** 2.0 * u[j-1][i] + k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
C.append(- k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct))
R.append(k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (2.0 * u[j-1][i+1] - 2.0 * u[j-1][i]) + 2.0 * h ** 2.0 * u[j-1][i] ** 2.0)
else:
A.append(k * (h * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) - (i * h) * h * v[j][i] - 2.0 * (i * h) * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
B.append(4.0 * (i * h) * (h ** 2.0 * u[j-1][i] + k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
C.append(k * ((i * h) * h * v[j][i] - 2.0 * (i * h) * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) - h * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct)))
if i < ni - 1:
R.append(h * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (u[j-1][i+1] - u[j-1][i-1]) + 2.0 * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (i * h) * (u[j-1][i+1] - 2.0 * u[j-1][i] + u[j-1][i-1]) - (i * h) * h * k * v[j-1][i] * (u[j-1][i+1] - u[j-1][i-1]) + 4.0 * (i * h) * h ** 2.0 * u[j-1][i] ** 2.0)
elif i == ni - 1:
R.append(h * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (u[j-1][i] - u[j-1][i-1]) + 2.0 * k * E(j * k, u[j-1][i], (u0 - u[j-1][i]) / u0, u0, i0, ct) * (i * h) * (u[j-1][i] - 2.0 * u[j-1][i] + u[j-1][i-1]) - (i * h) * h * k * v[j-1][i] * (u[j-1][i] - u[j-1][i-1]) + 4.0 * (i * h) * h ** 2.0 * u[j-1][i] ** 2.0)
C[0] += A[0]
del A[0]
R[-1] -= C[-1] * u0
del C[-1]
u.append(thomas(A, B, C, R))
periter = time()
lleva = float(j) / float(nj)
falta = (periter - start) * (nj - j - 1)
# print('\rPercentage: {0:2.2f} %. Time remaining: {1:2.1f} s'.format(100.0 * lleva, falta)),
print 'Position = ' + str(nj * k) + ', ' + str(ni * h)
return u[-1][-1]