def basis_laplacian_gsd(C, phi, L):
'''
Here, to avoid sigular at the boundary, we return
C * \phi * Laplaican{phi}
instead of
C * \phi^{-1} * Laplaican{phi}
'''
dphi = (2.0 / L) * cheb_D1_fchebt(phi)
d2phi = (2.0 / L) * cheb_D1_fchebt(dphi)
return C * phi * d2phi
def test_speed():
N = 32
M = 10000
ii = np.arange(N + 1)
x = np.cos(np.pi * ii / N)
f = np.exp(x) * np.sin(5. * x)
t = time()
for i in xrange(M):
D, x = cheb_D1_mat(N)
w = np.dot(D, f)
print 'Run time for mat is:', time() - t
t = time()
for i in xrange(M):
w = cheb_D1_fft(f)
print 'Run time for fft is:', time() - t
t = time()
for i in xrange(M):
w = cheb_D1_fchebt(f)
print 'Run time for fchebt is:', time() - t
t = time()
for i in xrange(M):
w = cheb_D1_dct(f)
print 'Run time for dct is:', time() - t
def basis_gradient_square_gsd(C, phi, L):
'''
Here, to avoid sigular at the boundary, we return
C * \grad{phi}**2
instead of
C * \phi^{-2} * \grad{phi}**2
'''
dphi = (2.0 / L) * cheb_D1_fchebt(phi)
return C * dphi**2
def test_cheb_D1_fchebt():
'''
Example is from p18.m of Trefethen's book, p.81.
'''
xx = np.arange(-1, 1, .01)
ff = np.exp(xx) * np.sin(5. * xx)
for N in [10, 20]:
ii = np.arange(N + 1)
x = np.cos(np.pi * ii / N)
f = np.exp(x) * np.sin(5. * x)
plt.figure()
plt.plot(x, f, '.')
plt.plot(xx, ff, 'r-')
plt.show()
err = cheb_D1_fchebt(f) - np.exp(x) * (np.sin(5. * x) +
5. * np.cos(5. * x))
plt.figure()
plt.plot(x, err, '.')
plt.plot(x, err, '-')
plt.show()
def basis_gradient_square(C, phi, L):
dphi = (2.0 / L) * cheb_D1_fchebt(phi)
return C * dphi**2
def basis_laplacian(C, phi, L):
dphi = (2.0 / L) * cheb_D1_fchebt(phi)
d2phi = (2.0 / L) * cheb_D1_fchebt(dphi)
return C * d2phi