def extra():
l_bc, r_bc = 0., 1.
lmbda = 12
N = 50
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = (.5 * (x + 1)) #**lmbda
N2 = 500
def pseudospectral_ode(y):
out = np.zeros(y.shape)
ypp = M.dot(y)
out = 4 * ypp - lmbda * np.sinh(lmbda * y)
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode, guess, method='lm', tol=1e-9)
print u.success
num_sol = BarycentricInterpolator(x, u.x)
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
plt.plot(x, guess, '*b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.axis([-1., 1., 0 - .1, 1.1])
plt.show()
plt.clf()
def exercise2():
# Solves the Poisson boundary value problem u''(x) = f(x), u(-1) = u(1) = 0
def f(x):
return np.exp(2. * x)
def anal_sol(x):
return (np.exp(2. * x) - np.sinh(2.) * x - np.cosh(2.)) / 4.
N = 3
D, x = cheb_vectorized(N)
A = np.dot(D, D)[1:N:1, 1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A, f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x, u, N)))(xx)
print "Max error is ", np.max(np.abs(uu - anal_sol(xx)))
plt.plot(x, u, '*r')
plt.plot(xx, uu, '-r')
plt.plot(xx, anal_sol(xx), '-k')
plt.axis([-1., 1., -2.5, .5])
# plt.savefig('chebyshev_points.pdf')
plt.show()
plt.clf()
def exercise2():
# Solves the Poisson boundary value problem u''(x) = f(x), u(-1) = u(1) = 0
def f(x):
return np.exp(2.*x)
def anal_sol(x):
return (np.exp(2.*x) - np.sinh(2.)*x - np.cosh(2.))/4.
N = 3
D, x = cheb_vectorized(N)
A = np.dot(D, D)[1:N:1,1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A, f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x, u, N)))(xx)
print "Max error is ", np.max(np.abs(uu - anal_sol(xx)))
plt.plot(x, u, '*r')
plt.plot(xx, uu, '-r')
plt.plot(xx, anal_sol(xx), '-k')
plt.axis([-1.,1.,-2.5,.5])
# plt.savefig('chebyshev_points.pdf')
plt.show()
plt.clf()
def extra(): l_bc, r_bc = 0., 1. lmbda = 12 N = 50 D, x = cheb_vectorized(N) M = np.dot(D, D) guess = (.5*(x+1))#**lmbda N2 = 500 def pseudospectral_ode(y): out = np.zeros(y.shape) ypp = M.dot(y) out = 4*ypp - lmbda*np.sinh(lmbda*y) out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc return out u = root(pseudospectral_ode,guess,method='lm',tol=1e-9) print u.success num_sol = BarycentricInterpolator(x,u.x) xx = np.linspace(-1, 1, N2) uu = num_sol.__call__(xx) plt.plot(x,guess,'*b') plt.plot(xx, uu, '-r') # Numerical solution via # the pseudospectral method plt.axis([-1.,1.,0-.1,1.1]) plt.show() plt.clf()
def nonlinear_minimal_area_surface_of_revolution(): l_bc, r_bc = 1., 7. N = 80 D, x = cheb_vectorized(N) M = np.dot(D, D) guess = 1. + (x--1.)*((r_bc - l_bc)/2.) def pseudospectral_ode(y): out = np.zeros(y.shape) yp, ypp = D.dot(y), M.dot(y) out = y*ypp - 1. - yp**2. out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc return out u = root(pseudospectral_ode,guess,method='lm',tol=1e-9) return x, u.x
def nonzeroDirichlet():
# Solves the Poisson boundary value problem
# u''(x) + u'(x) = f(x),
# u(-1) = a
# u(1) = b
a, b = 2, -1
def f(x):
return -(b - a) / 2. * np.ones(x.shape) + np.exp(3. * x)
def G(x):
return (b - a) / 2. * np.ones(x.shape) * x + (b + a) / 2. * np.ones(
x.shape)
def anal_sol(x):
# out = ( np.exp(-x)/( 12.*(-1 +np.exp(1))*(1 + np.exp(1))*np.exp(3)) *
# (-12.*a*np.exp(x+3.)+12.*a*np.exp(4) + 12.*b*np.exp(x+5)-12*b*np.exp(4) -
# np.exp(4*x+3) + np.exp(4*x+5) - np.exp(x+8 )+ np.exp(x)- np.exp(1) + np.exp(7)
# )
# )
N, B = np.array([[1., np.exp(1)],
[1, np.exp(-1)]]), np.array([[a - np.exp(-3) / 12.],
[b - np.exp(3) / 12.]])
C = solve(N, B)
return C[0] + C[1] * np.exp(-x) + np.exp(3. * x) / 12.
N = 7
D, x = cheb_vectorized(N)
A = np.dot(D, D)[1:N:1, 1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A + D[1:N:1, 1:N:1], f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x, u, N)))(xx)
print "Max error is ", np.max(np.abs(uu + G(xx) - anal_sol(xx)))
# plt.plot(x,u+G(x), '*k')
# plt.plot(xx,uu+G(xx), '-r')
plt.plot(xx, anal_sol(xx))
plt.xlabel('$x$')
plt.ylabel('$u$')
# plt.axis([-1.,1.,-2.5,.5])
# plt.savefig('nonzeroDirichlet.pdf')
plt.show()
plt.clf()
def nonzeroDirichlet():
# Solves the Poisson boundary value problem
# u''(x) + u'(x) = f(x),
# u(-1) = a
# u(1) = b
a, b = 2, -1
def f(x):
return -(b-a)/2.*np.ones(x.shape)+ np.exp(3.*x)
def G(x):
return (b-a)/2.*np.ones(x.shape) *x + (b+a)/2.*np.ones(x.shape)
def anal_sol(x):
# out = ( np.exp(-x)/( 12.*(-1 +np.exp(1))*(1 + np.exp(1))*np.exp(3)) *
# (-12.*a*np.exp(x+3.)+12.*a*np.exp(4) + 12.*b*np.exp(x+5)-12*b*np.exp(4) -
# np.exp(4*x+3) + np.exp(4*x+5) - np.exp(x+8 )+ np.exp(x)- np.exp(1) + np.exp(7)
# )
# )
N, B = np.array([[1.,np.exp(1)],[1,np.exp(-1)]]), np.array([[a-np.exp(-3)/12.], [b-np.exp(3)/12.]])
C = solve(N,B)
return C[0] + C[1]*np.exp(-x) + np.exp(3.*x)/12.
N = 7
D, x = cheb_vectorized(N)
A = np.dot(D, D)[1:N:1,1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A + D[1:N:1,1:N:1], f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x,u,N)))(xx)
print "Max error is ", np.max(np.abs(uu + G(xx) - anal_sol(xx)))
# plt.plot(x,u+G(x), '*k')
# plt.plot(xx,uu+G(xx), '-r')
plt.plot(xx, anal_sol(xx))
plt.xlabel('$x$')
plt.ylabel('$u$')
# plt.axis([-1.,1.,-2.5,.5])
# plt.savefig('nonzeroDirichlet.pdf')
plt.show()
plt.clf()
def deriv_matrix_exercise1():
def f(x):
return np.exp(x) * np.cos(6. * x)
def fp(x):
return np.exp(x) * (np.cos(6. * x) - 6. * np.sin(6. * x))
N = 10
D, x = cheb_vectorized(N)
f_exact, fp_exact = f(x), fp(x)
xx = np.linspace(-1, 1, 200)
plt.plot(xx, fp(xx), '-k') # Exact Derivative
uu = (np.poly1d(np.polyfit(x, D.dot(f(x)), N)))(xx)
plt.plot(x, D.dot(f(x)),
'*r') # Approximation to derivative at grid points
plt.plot(xx, uu, '-r') # Approximation to derivative at other values
# plt.savefig('equally_spaced_points.pdf')
plt.show()
plt.clf()
print "max error = \n", np.max(np.abs(D.dot(f(x)) - fp(x)))
def deriv_matrix_exercise1():
def f(x):
return np.exp(x)*np.cos(6.*x)
def fp(x):
return np.exp(x)*(np.cos(6.*x) - 6.*np.sin(6.*x))
N = 10
D,x = cheb_vectorized(N)
f_exact, fp_exact = f(x), fp(x)
xx = np.linspace(-1,1,200)
plt.plot(xx,fp(xx),'-k') # Exact Derivative
uu = (np.poly1d(np.polyfit(x, D.dot(f(x) ), N)))(xx)
plt.plot(x,D.dot(f(x)),'*r') # Approximation to derivative at grid points
plt.plot(xx,uu,'-r') # Approximation to derivative at other values
# plt.savefig('equally_spaced_points.pdf')
plt.show()
plt.clf()
print "max error = \n", np.max(np.abs( D.dot(f(x)) - fp(x) ))
def nonlinear_minimal_area_surface_of_revolution():
l_bc, r_bc = 1., 7.
N = 80
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = 1. + (x - -1.) * ((r_bc - l_bc) / 2.)
N2 = 50
def pseudospectral_ode(y):
out = np.zeros(y.shape)
yp, ypp = D.dot(y), M.dot(y)
out = y * ypp - 1. - yp**2.
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode, guess, method='lm', tol=1e-9)
num_sol = BarycentricInterpolator(x, u.x)
# Up to this point we have found the numerical solution
# using the pseudospectral method. In the code that follows
# we check that solution with the analytic solution,
# and graph the results
def f(x):
return np.array([
x[1] * np.cosh((-1. + x[0]) / x[1]) - l_bc, x[1] * np.cosh(
(1. + x[0]) / x[1]) - r_bc
])
parameters = root(f, np.array([1., 1.]), method='lm', tol=1e-9)
A, B = parameters.x[0], parameters.x[1]
def analytic_solution(x):
out = B * np.cosh((x + A) / B)
return out
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
# print "Max error is ", np.max(np.abs(uu - analytic_solution(xx)))
plt.plot(x, guess, '-b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.plot(xx, analytic_solution(xx), '*k') # Analytic solution
plt.axis([-1., 1., l_bc - 1., r_bc + 1.])
# plt.show()
plt.clf()
theta = np.linspace(0, 2 * np.pi, N2)
X, Theta = np.meshgrid(xx, theta, indexing='ij')
print "\nxx = \n", xx
print "\nuu = \n", uu
F = uu[:, np.newaxis] + np.zeros(uu.shape)
print "\nX = \n", X
print "\nTheta = \n", Theta
print "\nF = \n", F
Y = F * np.cos(Theta)
Z = F * np.sin(Theta)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# X, Y, Z = axes3d.get_test_data(0.05)
ax.plot_wireframe(X, Y, Z, rstride=1, cstride=1)
print ax.azim, ax.elev
ax.azim = -65
ax.elev = 0
# ax.view_init(elev=-60, azim=30)
# plt.savefig('minimal_surface.pdf')
plt.show()
def nonlinear_minimal_area_surface_of_revolution():
l_bc, r_bc = 1., 7.
N = 80
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = 1. + (x--1.)*((r_bc - l_bc)/2.)
N2 = 50
def pseudospectral_ode(y):
out = np.zeros(y.shape)
yp, ypp = D.dot(y), M.dot(y)
out = y*ypp - 1. - yp**2.
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode,guess,method='lm',tol=1e-9)
num_sol = BarycentricInterpolator(x,u.x)
# Up to this point we have found the numerical solution
# using the pseudospectral method. In the code that follows
# we check that solution with the analytic solution,
# and graph the results
def f(x):
return np.array([ x[1]*np.cosh((-1.+x[0])/x[1])-l_bc,
x[1]*np.cosh((1.+x[0])/x[1])-r_bc])
parameters = root(f,np.array([1.,1.]),method='lm',tol=1e-9)
A, B = parameters.x[0], parameters.x[1]
def analytic_solution(x):
out = B*np.cosh((x + A)/B)
return out
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
# print "Max error is ", np.max(np.abs(uu - analytic_solution(xx)))
plt.plot(x,guess,'-b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.plot(xx, analytic_solution(xx), '*k') # Analytic solution
plt.axis([-1.,1.,l_bc-1.,r_bc +1.])
# plt.show()
plt.clf()
theta = np.linspace(0,2*np.pi,N2)
X,Theta = np.meshgrid(xx,theta,indexing='ij')
print "\nxx = \n", xx
print "\nuu = \n", uu
F = uu[:,np.newaxis] +np.zeros(uu.shape)
print "\nX = \n", X
print "\nTheta = \n", Theta
print "\nF = \n", F
Y = F*np.cos(Theta)
Z = F*np.sin(Theta)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# X, Y, Z = axes3d.get_test_data(0.05)
ax.plot_wireframe(X, Y, Z, rstride=1, cstride=1)
print ax.azim, ax.elev
ax.azim=-65; ax.elev = 0
# ax.view_init(elev=-60, azim=30)
# plt.savefig('minimal_surface.pdf')
plt.show()