def cheb_D1_fchebt(v):
'''
Evaluate 1st derivative with Chebyshev polynomials via fast Fourier
transform (FFT).
Ref:
Kopriva, DA **Implementing Spectral Methods for Partial Differential
Equations**, 2009
:param:v: 1D numpy array contains the data to be differentiated.
'''
N = np.size(v) - 1
if N == 0:
return 0
# Perfrom forward Chebyshev transform to obtain the coefficients
V = cheb_fast_transform(v)
# Compute the 1st Derivative coefficients in spetral space
W = cheb_D1_coefficients(V)
# Transfrom back to spatial space for theta
w = cheb_inverse_fast_transform(W)
return w
def cheb_mde_neumann_split(W, u0, Lx, Ns):
'''
Solution of modified diffusion equation (MDE)
via Strang operator splitting as time-stepping scheme
and pseudospectral methods on Chebyshev grids.
The MDE is:
dq/dt = Dq + Wq
where D is Laplace operator. Neumann boundary condition is assumed.
'''
ds = 1. / (Ns - 1)
N = np.size(W) - 1
u = u0.copy()
u[0] = 0.
u[N] = 0.
k2 = (np.pi * np.pi) / (Lx * Lx) * np.arange(N + 1) * np.arange(N + 1)
expw = np.exp(-0.5 * ds * W)
for i in xrange(Ns - 1):
u = expw * u
ak = cheb_fast_transform(u) * np.exp(-ds * k2)
u = cheb_inverse_fast_transform(ak)
u = expw * u
ii = np.arange(N + 1)
x = 1. * ii * Lx / N
return (u, x)
def test_cheb_fast_transform():
N = 10
ii = np.arange(N + 1)
x = np.cos(np.pi * ii / N)
f = np.exp(x) * np.sin(5. * x)
F = cheb_fast_transform(f)
ff = cheb_inverse_fast_transform(F)
print f
print ff
def solve(self, w, u0, q=None):
'''
dq/dt = Dq + Wq = Dq - wq
'''
N = self.N
u = u0.reshape(u0.size) # force 1D array
f = np.zeros(u.size + 2) # u.size is N+1
ge = np.zeros(self.pep.size + 1)
go = np.zeros(self.pop.size + 1)
uc = u.astype(complex)
fc = f.astype(complex)
gec = ge.astype(complex)
goc = go.astype(complex)
expw = np.exp(-0.5 * self.h * w)
for i in xrange(self.Ns - 1):
u = expw * u
u = cheb_fast_transform(u)
f[:N + 1] = u
g = self.pg * f[0:N-1] - self.qg * f[2:N+1] \
+ self.rg * f[4:N+3]
ge[1:] = g[0:N - 1:2]
go[1:] = g[1:N - 1:2]
ue = solve_tridiag_complex_thual(self.pen, self.qen, self.ren,
self.bce, ge)
uo = solve_tridiag_complex_thual(self.pon, self.qon, self.ron,
self.bco, go)
uc[0:N + 1:2] = ue
uc[1:N + 1:2] = uo
fc[:N + 1] = uc
gc = self.pg * fc[0:N-1] - self.qg * fc[2:N+1] \
+ self.rg * fc[4:N+3]
gec[1:] = gc[0:N - 1:2]
goc[1:] = gc[1:N - 1:2]
ue = solve_tridiag_complex_thual(self.pep, self.qep, self.rep,
self.bce, gec)
uo = solve_tridiag_complex_thual(self.pop, self.qop, self.rop,
self.bco, goc)
uc[0:N + 1:2] = ue
uc[1:N + 1:2] = uo
u = uc.real
u = cheb_inverse_fast_transform(u)
u = 2. * (self.K / self.h)**2 * expw * u
if q is not None:
q[i + 1] = u
return (u, self.x)
def cheb_mde_oscheb(W, u0, Lx, Ns, bc_even, bc_odd):
'''
Solution of modified diffusion equation (MDE)
via Strang operator splitting as time-stepping scheme
and fast Chebyshev transform.
Ref:
Her, S. M.; Garcia-Cervera, C. J.; Fredrickson, G. H. macromolecules, 2012, 45, 2905.
The MDE is:
dq/dt = Dq - Wq
in the interval [0, L], with homogeneous Dirichlet boundary conditions at both boundaries.
where D is Laplace operator.
Discretization:
x_j = cos(j*L_x/N), j = 0, 1, 2, ..., N
:param:W: W_j, j = 0, 1, ..., N
:param:u0: u0_j, j = 0, 1, ..., N
:param:Lx: the physical length of the interval [0, L_x]
:param:Ns: contour index, s_j, j = 0, 1, ..., N_s
'''
ds = 1. / (Ns - 1)
N = u0.size - 1
u = u0.reshape(u0.size) # force 1D array
K = (Lx / 2.)**2
lambp = K * (1. + 1j) / ds # \lambda_+
lambn = K * (1. - 1j) / ds # \lambda_-
c = np.ones(N + 1)
c[0] = 2.
c[-1] = 2. # c_0=c_N=2, c_1=c_2=...=c_{N-1}=1
b = np.ones(N + 3)
b[N - 1:] = 0. # b_{N-1}=...=b_{N+2}=0, b_0=b_1=...=b_{N-2}=1
n = np.arange(N + 1)
pe = 0.25 * c[0:N - 1:2] / n[2:N + 1:2] / (n[2:N + 1:2] - 1)
po = 0.25 * c[1:N - 1:2] / n[3:N + 1:2] / (n[3:N + 1:2] - 1)
pep = -pe * lambp
pen = -pe * lambn
pop = -po * lambp
pon = -po * lambn
qe = 0.5 * b[2:N + 1:2] / (n[2:N + 1:2]**2 - 1)
qo = 0.5 * b[3:N + 1:2] / (n[3:N + 1:2]**2 - 1)
qep = 1 + qe * lambp
qen = 1 + qe * lambn
qop = 1 + qo * lambp
qon = 1 + qo * lambn
re = 0.25 * b[4:N + 3:2] / n[2:N + 1:2] / (n[2:N + 1:2] + 1)
ro = 0.25 * b[5:N + 3:2] / n[3:N + 1:2] / (n[3:N + 1:2] + 1)
rep = -re * lambp
ren = -re * lambn
rop = -ro * lambp
ron = -ro * lambn
pg = np.zeros(pe.size + po.size)
pg[0:N - 1:2] = pe
pg[1:N - 1:2] = po
qg = np.zeros(qe.size + qo.size)
qg[0:N - 1:2] = qe
qg[1:N - 1:2] = qo
rg = np.zeros(re.size + ro.size)
rg[0:N - 1:2] = re
rg[1:N - 1:2] = ro
#dbce = np.ones(pe.size + 1)
#dbco = np.ones(po.size + 1)
dbce = bc_even
dbco = bc_odd
f = np.zeros(u.size + 2) # u.size is N+1
ge = np.zeros(pe.size + 1)
go = np.zeros(po.size + 1)
uc = u.astype(complex)
fc = f.astype(complex)
gec = ge.astype(complex)
goc = go.astype(complex)
expw = np.exp(-0.5 * ds * W)
for i in xrange(Ns - 1):
u = expw * u
u = cheb_fast_transform(u)
f[:N + 1] = u
g = pg * f[0:N - 1] - qg * f[2:N + 1] + rg * f[4:N + 3]
ge[1:] = g[0:N - 1:2]
go[1:] = g[1:N - 1:2]
ue = solve_tridiag_complex_thual(pen, qen, ren, dbce, ge)
uo = solve_tridiag_complex_thual(pon, qon, ron, dbco, go)
uc[0:N + 1:2] = ue
uc[1:N + 1:2] = uo
fc[:N + 1] = uc
gc = pg * fc[0:N - 1] - qg * fc[2:N + 1] + rg * fc[4:N + 3]
gec[1:] = gc[0:N - 1:2]
goc[1:] = gc[1:N - 1:2]
ue = solve_tridiag_complex_thual(pep, qep, rep, dbce, gec)
uo = solve_tridiag_complex_thual(pop, qop, rop, dbco, goc)
uc[0:N + 1:2] = ue
uc[1:N + 1:2] = uo
u = uc.real
u = cheb_inverse_fast_transform(u)
u = 2. * (K / ds)**2 * expw * u
return u