def solve(n):
# we have n+1 grid points
h = (xmax-xmin)/n
x = linspace(xmin,xmax,n+1)
a = afun(x)
# Right hand side
b = 2.0 * h**2 * qfun(x)
# First and last equations are bc
b[0] = 0.0
b[-1] = 0.0;
A,B,C = zeros(n+1), zeros(n+1), zeros(n+1)
A[0],C[0] = 1.0, 0.0
for i in range(1,n):
B[i] = -(a[i-1]+a[i])
A[i] = (a[i-1]+2*a[i]+a[i+1])
C[i] = -(a[i]+a[i+1])
B[-1],A[-1] = 0.0, 1.0
u = tdma(A,B,C,b)
xe = linspace(xmin,xmax,200)
ue = uexact(xe)
plt.plot(xe,ue,'-',x,u,'o--')
plt.title('FD using '+str(n+1)+' points')
plt.xlabel('x'); plt.ylabel('u')
plt.legend(('Exact','FD'))
print "Close plot window to continue"
plt.show()
return h, abs(u-uexact(x)).max()
def solve(n):
# we have n+1 grid points
h = (xmax - xmin) / n
x = linspace(xmin, xmax, n + 1)
a = afun(x)
# Right hand side
b = 2.0 * h**2 * qfun(x)
# First and last equations are bc
b[0] = 0.0
b[-1] = 0.0
A, B, C = zeros(n + 1), zeros(n + 1), zeros(n + 1)
A[0], C[0] = 1.0, 0.0
for i in range(1, n):
B[i] = -(a[i - 1] + a[i])
A[i] = (a[i - 1] + 2 * a[i] + a[i + 1])
C[i] = -(a[i] + a[i + 1])
B[-1], A[-1] = 0.0, 1.0
u = tdma(A, B, C, b)
xe = linspace(xmin, xmax, 200)
ue = uexact(xe)
plt.plot(xe, ue, '-', x, u, 'o--')
plt.title('FD using ' + str(n + 1) + ' points')
plt.xlabel('x')
plt.ylabel('u')
plt.legend(('Exact', 'FD'))
print("Close plot window to continue")
plt.show()
return h, abs(u - uexact(x)).max()
def solve(Pe, xf):
N = len(xf) - 1 # number of cells
xc = 0.5 * (xf[0:-1] + xf[1:]) # cell centers
hc = xf[1:] - xf[0:-1] # cell lenghts
hf = empty_like(xf) # h_{j+1/2}
hf[0] = hc[0]
hf[1:-1] = 0.5 * (hc[0:-1] + hc[1:])
hf[-1] = hc[-1]
# Mesh Peclet number
P = hf * Pe
# Three diagonals
A, B, C = zeros(N), zeros(N), zeros(N)
A[0], C[0] = 0.5 + 2.0 / P[0] + 1.0 / P[1], 0.5 - 1.0 / P[1]
for i in range(1, N - 1):
B[i], A[i], C[i] = -(0.5 + 1.0 / P[i]), (
1.0 / P[i] + 1.0 / P[i + 1]), 0.5 - 1.0 / P[i + 1]
B[-1], A[-1] = -(0.5 + 1.0 / P[-2]), -0.5 + 2.0 / P[-1] + 1.0 / P[-2]
a, b = 0.0, 1.0 # Dirichlet condition
# Right hand side
F = zeros(N)
F[0] = (1.0 + 2.0 / P[0]) * a
F[-1] = (-1.0 + 2.0 / P[-1]) * b
# Solve
u = tdma(A, B, C, F)
return xc, u
def error(n):
xmin, xmax = 0.0, 2.0*pi
h = (xmax - xmin)/(n - 1)
x = linspace(xmin,xmax,n)
u = zeros(n)
# BC for first and last points
u[0] = uexact(x[0])
u[-1] = uexact(x[-1])
b = h**2 * f(x[1:-1])
b[0] += u[0]
b[-1] += u[-1]
u[1:-1] = tdma(2*ones(n-2),-ones(n-2),-ones(n-2),b)
return h, abs(uexact(x)-u).max()
def error(n):
xmin, xmax = 0.0, 2.0*pi
h = (xmax - xmin)/(n - 1)
x = linspace(xmin,xmax,n)
u = zeros(n)
# BC for first and last points
u[0] = uexact(x[0])
u[-1] = uexact(x[-1])
b = h**2 * f(x[1:-1])
b[0] += u[0]
b[-1] += u[-1]
u[1:-1] = tdma(2*ones(n-2),-ones(n-2),-ones(n-2),b)
return h, abs(uexact(x)-u).max()
for i in range(len(x)):
if x[i] < 0.5:
ue[i] = a + (cos(pi*x[i])-1)/pi
else:
ue[i] = a - 1.0/pi + 2.0*cos(pi*x[i])/pi
return ue
h = (xmax-xmin)/n
x = linspace(xmin+0.5*h, xmax-0.5*h, n)
q = qfun(x)
q[0] += a/h
A,B,C = zeros(n),zeros(n),zeros(n)
A[0], C[0] = 0.5/h, 0.5/h
for i in range(1,n-1):
B[i], C[i] = -0.5/h, 0.5/h
B[-1], A[-1] = -0.5/h, 0.5/h
u = tdma(A,B,C,q)
xe = linspace(xmin,xmax,100)
ue = uexact(xe)
plt.plot(x,u,'o--',xe,ue,'-')
plt.xlabel('x')
plt.ylabel('u')
plt.legend(('Numerical','Exact'))
plt.title('Central scheme')
plt.show()
args = parser.parse_args()
N, Pe = args.N, args.Pe
a, b = 0.0, 1.0
h = (b - a) / N
P = h * Pe
A, B, C = zeros(N), zeros(N), zeros(N)
A[0], C[0] = 1.0 + 6.0 / P, 1.0 - 2.0 / P
for i in range(1, N - 1):
B[i], A[i], C[i] = -(1.0 + 2.0 / P), 4.0 / P, 1.0 - 2.0 / P
B[-1], A[-1] = -(1.0 + 2.0 / P), -1.0 + 6.0 / P
F = zeros(N)
F[0] = (2.0 + 4.0 / P) * a
F[-1] = (-2.0 + 4.0 / P) * b
u = tdma(A, B, C, F)
x = linspace(a + 0.5 * h, b - 0.5 * h, N)
xe = linspace(a, b, 100)
ue = a + (b - a) * (exp((xe - 1) * Pe) - exp(-Pe)) / (1 - exp(-Pe))
plt.plot(x, u, 'o-', xe, ue)
plt.xlabel('x')
plt.ylabel('u')
plt.legend(('Numerical', 'Exact'), loc='upper left')
T = 'N=' + str(N) + ', Pe=' + str(Pe) + ', P=' + str(P)
plt.title(T)
plt.show()
for i in range(len(x)):
if x[i] < 0.5:
ue[i] = a + (cos(pi * x[i]) - 1) / pi
else:
ue[i] = a - 1.0 / pi + 2.0 * cos(pi * x[i]) / pi
return ue
h = (xmax - xmin) / n
x = linspace(xmin + 0.5 * h, xmax - 0.5 * h, n)
q = qfun(x)
q[0] += a / h
A, B, C = zeros(n), zeros(n), zeros(n)
A[0], C[0] = 0.5 / h, 0.5 / h
for i in range(1, n - 1):
B[i], C[i] = -0.5 / h, 0.5 / h
B[-1], A[-1] = -0.5 / h, 0.5 / h
u = tdma(A, B, C, q)
xe = linspace(xmin, xmax, 100)
ue = uexact(xe)
plt.plot(x, u, 'o--', xe, ue, '-')
plt.xlabel('x')
plt.ylabel('u')
plt.legend(('Numerical', 'Exact'))
plt.title('Central scheme')
plt.show()
xmin, xmax = 0.0, 2.0 * pi
# Grid of n points
n = 20
h = (xmax - xmin) / (n - 1)
x = linspace(xmin, xmax, n)
# array for solution
u = zeros(n)
# BC for first and last points
u[0] = uexact(x[0])
u[-1] = uexact(x[-1])
b = h**2 * f(x[1:-1])
b[0] += u[0]
b[-1] += u[-1]
u[1:-1] = tdma(2 * ones(n - 2), -ones(n - 2), -ones(n - 2), b)
print("Max error = ", abs(uexact(x) - u).max())
# Exact solution on fine mesh for plotting
xe = linspace(xmin, xmax, 100)
ue = uexact(xe)
# Plot exact and numerical solution
plt.plot(xe, ue, x, u, 'o')
plt.legend(('Exact solution', 'Numerical solution'))
plt.xlabel('x')
plt.ylabel('u')
plt.show()
def f(x):
return sin(x)
# exact solution
def uexact(x):
return sin(x)
xmin, xmax = 0.0, 2.0*pi
n = 20
h = (xmax - xmin)/(n - 1)
x = linspace(xmin,xmax,n)
u = zeros(n)
# BC for first and last points
u[0] = uexact(x[0])
u[-1] = uexact(x[-1])
b = h**2 * f(x[1:-1])
b[0] += u[0]
b[-1] += u[-1]
u[1:-1] = tdma(2*ones(n-2),-ones(n-2),-ones(n-2),b)
print "Max error = ", abs(uexact(x)-u).max()
xe = linspace(xmin, xmax, 100); ue = uexact(xe)
plt.plot(xe,ue,x,u,'o-')
plt.legend(('Exact solution','Numerical solution'))
plt.xlabel('x')
plt.ylabel('u')
plt.show()
h = (b - a) / N
P = h * Pe
# Construct diagonals of matrix
A, B, C = zeros(N), zeros(N), zeros(N)
A[0], C[0] = 1.0 + 2.0 / P, -1.0 / P
for i in range(1, N - 1):
B[i], A[i], C[i] = -(1.0 + 1.0 / P), 1.0 + 2.0 / P, -1.0 / P
B[-1], A[-1] = -(1.0 + 1.0 / P), 1.0 + 3.0 / P
# construct rhs
F = zeros(N)
F[0] = (1.0 + 1.0 / P) * a
F[-1] = (2.0 / P) * b
# solve
u = tdma(A, B, C, F)
# plot the solution
x = linspace(a + 0.5 * h, b - 0.5 * h, N)
xe = linspace(a, b, 100)
ue = a + (b - a) * (exp((xe - 1) * Pe) - exp(-Pe)) / (1 - exp(-Pe))
plt.plot(x, u, "o-", xe, ue)
plt.xlabel("x")
plt.ylabel("u")
plt.legend(("Numerical", "Exact"), loc="upper left")
T = "N=" + str(N) + ", Pe=" + str(Pe) + ", P=" + str(P)
plt.title(T)
plt.show()
