def test_exact_neumann_dirichlet():
L = 10
N = 128
Ns = 200 + 1 #20000 + 1
algo = 1
scheme = 1
W, u0, x = init_chebyshev_fredrikson(N, L)
u0[0] = 0.
print 'ETDRK4 N = ', N, ' Ns = ', Ns-1
#q3, x3 = cheb_mde_neumann_dirichlet_etdrk4(W, u0, L, Ns, algo, scheme)
lbc = BC('Neumann')
rbc = BC('Dirichlet')
etdrk4_solver = ETDRK4(L, N, Ns, h=None, lbc=lbc, rbc=rbc)
q3, x3 = etdrk4_solver.solve(W, u0)
Q3 = 0.5 * L * cheb_quadrature_clencurt(q3)
if scheme == 0:
#data_name = 'benchmark/NBC-DBC/exact/ETDRK4_Cox_N'
data_name = 'ETDRK4_Cox_N'
data_name = data_name + str(N) + '_Ns' + str(Ns-1)
else:
#data_name = 'benchmark/NBC-DBC/exact/ETDRK4_Krogstad_N'
data_name = 'ETDRK4_Krogstad_N'
data_name = data_name + str(N) + '_Ns' + str(Ns-1)
savemat(data_name,{
'N':N, 'Ns':Ns-1, 'W':W, 'u0':u0, 'Lz':L,
'x':x, 'q':q3, 'Q':Q3})
plt.plot(x3, q3, 'r')
plt.axis([0, 10, 0, 1.15])
plt.xlabel('$z$')
plt.ylabel('$q(z)$')
plt.savefig(data_name, bbox_inches='tight')
plt.show()
def build_grid(self):
config = self.config
Lx = config.grid.Lx
self.Lx = Lx
La = config.uc.a
self.La = La
Ms = config.grid.Ms
if os.path.exists(config.grid.field_data):
print "Reading from ", config.grid.field_data
mat = loadmat(config.grid.field_data)
self.w = np.zeros(Lx, dtype=np.complex128)
self.w = mat['w'].reshape(Lx)
else:
self.w = np.zeros(Lx, dtype=np.complex128)
self.w.real = np.random.rand(Lx)
self.w.imag = np.random.rand(Lx)
self.phi = np.zeros(Lx, dtype=np.complex128)
self.q = np.zeros((Ms, Lx), dtype=np.complex128)
self.q[0].real = 1.
ds = 1. / (Ms - 1)
self.ds = ds
lbc = BC(self.lbc, self.lbc_vc)
rbc = BC(self.rbc, self.rbc_vc)
self.q_solver = ETDRK4(La, Lx - 1, Ms, h=ds, lbc=lbc, rbc=rbc)
self.lam = config.grid.lam[0]
def build_grid(self):
config = self.config
Lx = config.grid.Lx
self.Lx = Lx
La = config.uc.a
self.La = La
Ms = config.grid.Ms
if os.path.exists(config.grid.field_data):
mat = loadmat(config.grid.field_data)
self.w = mat['w']
else:
#self.w = np.zeros([Lx])
self.w = np.random.rand(Lx)
self.q = np.zeros((Ms, Lx))
self.q[0, :] = 1.
ds = 1. / (Ms - 1)
self.ds = ds
lbc = BC(self.lbc, self.lbc_vc)
rbc = BC(self.rbc, self.rbc_vc)
self.q_solver = ETDRK4(La, Lx - 1, Ms, h=ds, lbc=lbc, rbc=rbc)
self.lam = config.grid.lam[0]
def build_grid(self):
config = self.config
Lx = config.grid.Lx
self.Lx = Lx
La = config.uc.a
self.La = La
Ms = config.grid.Ms
MsA = config.grid.vMs[0]
MsB = config.grid.vMs[1]
MsC = config.grid.vMs[2]
if os.path.exists(config.grid.field_data):
mat = loadmat(config.grid.field_data)
self.wA = mat['wA']
self.wB = mat['wB']
self.wC = mat['wC']
else:
self.wA = np.random.rand(Lx)
self.wB = np.random.rand(Lx)
self.wC = np.random.rand(Lx)
self.qA = np.zeros((MsA, Lx))
self.qA[0, :] = 1.
self.qAc = np.zeros((MsA, Lx))
self.qB = np.zeros((MsB, Lx))
self.qBc = np.zeros((MsB, Lx))
self.qBc[0, :] = 1.
self.qC = np.zeros((MsC, Lx))
self.qC[0, :] = 1.
self.qCc = np.zeros((MsC, Lx))
ds = 1. / (Ms - 1)
self.ds = ds
lbcA = BC(self.lbc, self.lbc_vc)
rbcA = BC(self.rbc, self.rbc_vc)
self.kaA = lbcA.beta
self.kbA = rbcA.beta
fA = self.fA
fB = self.fB
lbcB = copy.deepcopy(lbcA)
lbcB.beta = -lbcA.beta * fA / fB # See Fredrickson Book p.194
rbcB = copy.deepcopy(rbcA)
rbcB.beta = -rbcA.beta * fA / fB
lbcC = copy.deepcopy(lbcA)
rbcC = copy.deepcopy(rbcA)
self.qA_solver = ETDRK4(La, Lx - 1, MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qAc_solver = ETDRK4(La, Lx - 1, MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qB_solver = ETDRK4(La, Lx - 1, MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.qBc_solver = ETDRK4(La, Lx - 1, MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.qC_solver = ETDRK4(La, Lx - 1, MsC, h=ds, lbc=lbcC, rbc=rbcC)
#self.qCc_solver = ETDRK4(La, Lx-1, MsC-1, h=ds, lbc=lbcC, rbc=rbcC)
self.qCc_solver = ETDRK4(La, Lx - 1, MsC, h=ds, lbc=lbcC, rbc=rbcC)
self.lamA = config.grid.lam[0]
self.lamB = config.grid.lam[1]
self.lamC = config.grid.lam[2]
self.lamY = config.grid.lam[3]
#self.yita = self.lamY # for compressible model
self.yita = np.zeros(Lx) # for incompressible model
def build_grid(self):
config = self.config
Nt = config.grid.Lx
Nr = config.grid.Ly # Nr is the number of grid points, EVEN only.
Nr2 = Nr / 2
self.Nt = Nt
self.Nr = Nr
self.Nr2 = Nr2
R = config.uc.a
self.R = R
Ms = config.grid.Ms
MsA = config.grid.vMs[0]
MsB = config.grid.vMs[1]
# Initiate fields from file or random numbers
if os.path.exists(config.grid.field_data):
mat = loadmat(config.grid.field_data)
self.wA = mat['wA']
self.wB = mat['wB']
else:
self.wA = np.random.rand(Nt,Nr2) - 0.5
self.wB = np.random.rand(Nt,Nr2) - 0.5
self.qA = np.zeros((MsA, Nt, Nr2))
self.qA[0,:,:] = 1.
self.qAc = np.zeros((MsA, Nt, Nr2))
self.qB = np.zeros((MsB, Nt, Nr2))
self.qBc = np.zeros((MsB, Nt, Nr2))
self.qBc[0,:,:] = 1.
ds = 1. / (Ms - 1)
self.ds = ds
lbcA = BC(self.lbc, self.lbc_vc)
rbcA = BC(self.rbc, self.rbc_vc)
self.kaA = lbcA.beta; self.kbA = rbcA.beta
fA = self.fA; fB = 1 - fA
lbcB = copy.deepcopy(lbcA)
lbcB.beta = -lbcA.beta * fA / fB # See Fredrickson Book p.194
rbcB = copy.deepcopy(rbcA)
rbcB.beta = -rbcA.beta * fA / fB
print 'lbcA=', lbcA.__dict__
print 'rbcA=', rbcA.__dict__
print 'lbcB=', lbcB.__dict__
print 'rbcB=', rbcB.__dict__
self.qA_solver = ETDRK4Polar(R, Nr-1, Nt,
MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qAc_solver = ETDRK4Polar(R, Nr-1, Nt,
MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qB_solver = ETDRK4Polar(R, Nr-1, Nt,
MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.qBc_solver = ETDRK4Polar(R, Nr-1, Nt,
MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.lamA = config.grid.lam[0]
self.lamB = config.grid.lam[1]
self.lamY = config.grid.lam[2]
self.yita = self.lamY # For compressible model
def build_grid(self):
config = self.config
Lx = config.grid.Lx
Ly = config.grid.Ly
self.Lx = Lx
self.Ly = Ly
La = config.uc.a
Lb = config.uc.b
self.La = La
self.Lb = Lb
Ms = config.grid.Ms
MsA = config.grid.vMs[0]
MsB = config.grid.vMs[1]
if os.path.exists(config.grid.field_data):
mat = loadmat(config.grid.field_data)
self.wA = mat['wA']
self.wB = mat['wB']
else:
self.wA = np.random.rand(Lx,Ly) - 0.5
self.wB = np.random.rand(Lx,Ly) - 0.5
self.qA = np.zeros((MsA, Lx, Ly))
self.qA[0,:,:] = 1.
self.qAc = np.zeros((MsA, Lx, Ly))
self.qB = np.zeros((MsB, Lx, Ly))
self.qBc = np.zeros((MsB, Lx, Ly))
self.qBc[0,:,:] = 1.
ds = 1. / (Ms - 1)
self.ds = ds
lbcA = BC(self.lbc, self.lbc_vc)
rbcA = BC(self.rbc, self.rbc_vc)
self.kaA = lbcA.beta; self.kbA = rbcA.beta
fA = self.fA; fB = 1 - fA
lbcB = copy.deepcopy(lbcA)
lbcB.beta = -lbcA.beta * fA / fB # See Fredrickson Book p.194
rbcB = copy.deepcopy(rbcA)
rbcB.beta = -rbcA.beta * fA / fB
print 'lbcA=', lbcA.__dict__
print 'rbcA=', rbcA.__dict__
print 'lbcB=', lbcB.__dict__
print 'rbcB=', rbcB.__dict__
self.qA_solver = ETDRK4FxCy(La, Lb, Lx, Ly-1,
MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qAc_solver = ETDRK4FxCy(La, Lb, Lx, Ly-1,
MsA, h=ds, lbc=lbcA, rbc=rbcA)
self.qB_solver = ETDRK4FxCy(La, Lb, Lx, Ly-1,
MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.qBc_solver = ETDRK4FxCy(La, Lb, Lx, Ly-1,
MsB, h=ds, lbc=lbcB, rbc=rbcB)
self.lamA = config.grid.lam[0]
self.lamB = config.grid.lam[1]
self.lamY = config.grid.lam[2]
self.yita = self.lamY # For compressible model
def __init__(self,
Lx,
Ly,
Nx,
Ny,
Ns,
h=None,
c=1.0,
lbc=BC(),
rbc=BC(),
algo=1,
scheme=1):
'''
The PDE is in 2D,
du/dt = cLu - wu
where u=u(x,y), L=d^2/dx^2 + d^2/dy^2, w=w(x,y), c is a constant.
First, perform a FFT in x direction to obtain
du(kx,y)/dt = c L u(kx,y) - {kx^2 u(kx,y) + Fx[w(x,y)u(x,y)]}
where L = D^2, with D^2 the Chebyshev 2nd order differential matrix,
and kx^2 the d^2/dx^2 in Fourier space, see detail in
the Notebook (page 2014.5.5).
The defaut left BC and right BC are DBCs.
Test: PASSED, 2014.5.8. (However, very small ds is required!)
:param:Lx: physical size of the 1D spacial grid.
:param:Lx: physical size of the 1D spacial grid.
:param:Ns: number of grid points in time.
:param:lbc: left boundary condition.
:type:lbc: class BC
:param:rbc: right boundary condition.
:type:rbc: class BC
:param:h: time step.
:param:algo: algorithm for calculation of RK4 coefficients.
:param:scheme: RK4 scheme.
'''
self.Lx, self.Ly = Lx, Ly
self.Nx, self.Ny = Nx, Ny
self.Ns = Ns
if h is None:
self.h = 1. / (Ns - 1)
else:
self.h = h
self.c = c
self.lbc = lbc
self.rbc = rbc
self.algo = algo
self.scheme = scheme
self.update()
def build_grid(self):
config = self.config
Lx = config.grid.Lx
self.Lx = Lx
L = config.uc.a # in unit of Rg
self.L = L / self.z_hat # in unit of \hat{z}
N = Lx - 1
self.N = N
Ms = config.grid.Ms
if os.path.exists(config.grid.field_data):
mat = loadmat(config.grid.field_data)
self.w = mat['w']
self.w.shape = (self.w.size, )
else:
self.w = np.random.rand(Lx)
self.q = np.zeros((Ms, Lx))
self.q[0, :] = 1.
self.qc = np.zeros((Ms, Lx))
lbc = BC(self.lbc, self.lbc_vc)
rbc = BC(self.rbc, self.rbc_vc)
h = 1. / (Ms - 1)
c = 0.25 / self.beta
self.q_solver = ETDRK4(L,
N,
Ms,
h=h,
c=c,
lbc=lbc,
rbc=rbc,
algo=1,
scheme=1)
#self.qc_solver = ETDRK4(L, N, Ms, h=h, c=c, lbc=lbc, rbc=rbc,
# algo=1, scheme=1)
self.qc_solver = ETDRK4(L,
N,
Ms - 1,
h=h,
c=c,
lbc=lbc,
rbc=rbc,
algo=1,
scheme=1) # CKE
def test_etdrk4_complex():
'''
The test case is according to R. C. Daileda Lecture notes.
du/dt = (1/25) u_xx , x@(0,3)
with boundary conditions:
u(0,t) = 0
u_x(3,t) = -(1/2) u(3,t)
u(x,0) = 100*(1-x/3)
Conclusion:
We find that the numerical solution is much more accurate than the five
term approximation of the exact analytical solution.
'''
Nx = 64
Lx = 3
t = 1.
Ns = 101
ds = t / (Ns - 1)
ii = np.arange(Nx + 1)
x = np.cos(np.pi * ii / Nx) # yy [-1, 1]
x = 0.5 * (x + 1) * Lx # mapping to [0, Ly]
w = np.zeros(Nx + 1, dtype=np.complex128)
q = np.zeros([Ns, Nx + 1], dtype=np.complex128)
q[0] = 100 * (1 - x / 3)
# The approximation of exact solution by first 5 terms
q_exact = 47.0449*np.exp(-0.0210*t)*np.sin(0.7249*x) + \
45.1413*np.exp(-0.1113*t)*np.sin(1.6679*x) + \
21.3586*np.exp(-0.2872*t)*np.sin(2.6795*x) + \
19.3403*np.exp(-0.5505*t)*np.sin(3.7098*x) + \
12.9674*np.exp(-0.9015*t)*np.sin(4.7474*x)
lbc = BC(DIRICHLET, [0, 1, 0])
rbc = BC(ROBIN, [1., 0.5, 0])
q_solver = ETDRK4(Lx, Nx, Ns, h=ds, c=1. / 25, lbc=lbc, rbc=rbc)
q1, x = q_solver.solve(w, q[0], q)
plt.plot(x, q[0].real, label='q_0')
plt.plot(x, q1.real, label='q_solution1')
plt.plot(x, q[-1].real, label='q_solution2')
plt.plot(x, q_exact, label='q_exact')
plt.legend(loc='best')
plt.show()
def build_grid(self):
config = self.config
Lx = config.grid.Lx
self.Lx = Lx
La = config.uc.a
self.La = La
Ms = config.grid.Ms
if os.path.exists(config.grid.field_data):
print "Reading from ", config.grid.field_data
mat = loadmat(config.grid.field_data)
self.w = np.zeros(Lx, dtype=np.complex128)
#try:
# self.w = mat['iw_avg'][0,:]
#except:
# self.w.real = mat['w'].reshape(Lx)
#y = np.arange(-1, 1, 2.0/Lx)
#self.w.real = cheb_interpolation_1d(y, w.real)
#self.w.imag = cheb_interpolation_1d(y, w.imag)
self.phi = np.zeros(Lx, dtype=np.complex128)
try:
self.phi.real = mat['phi_avg'][0, :].real
except:
self.phi.real = mat['phi'][0, :]
#self.phi.real = cheb_interpolation_1d(y, phi.real)
else:
print "Abort: you must provide a density field."
exit(1)
self.q = np.zeros((Ms, Lx), dtype=np.complex128)
self.q[0].real = 1.
ds = 1. / (Ms - 1)
self.ds = ds
lbc = BC(self.lbc, self.lbc_vc)
rbc = BC(self.rbc, self.rbc_vc)
self.q_solver = ETDRK4(La, Lx - 1, Ms, h=ds, lbc=lbc, rbc=rbc)
self.lam = config.grid.lam[0]
def __init__(self, Lx, N, Ns, h=None, bc=BC()):
'''
Both BCs are DBCs or NBCs. The default is DBC.
:param:Lx: physical size of the 1D spacial grid.
:param:Ns: number of grid points in time.
:param:N: number of grid points in space.
:param:h: time step.
'''
self.Lx = Lx
self.N = N
self.Ns = Ns
self.bc = bc
if h is None:
self.h = 1. / (Ns - 1)
else:
self.h = h
self.update()
def test_etdrk4polar():
'''
The PDE is
du/dt = (d^2/dr^2 + (1/r)d/dr + (1/r^2)d^2/dtheta^2) u - w u
in the domain [0,R]x[0,2*pi] for time t=0 to t=1,
with boundary conditions
u(r,theta,t) = u(r,theta+2*pi,t) # periodic in theta direction
du/dr |{r=1} + ka u(r=1) = 0, # RBC
'''
Nt = 64 # theta
R = 1.0 # r [0, R]
Nr = 63 # Nr must be odd
N2 = (Nr + 1) / 2
Ns = 101
ds = 1. / (Ns - 1)
# Periodic in x direction, Fourier
tt = np.arange(Nt + 1) * 2 * np.pi / Nt
# Non-periodic in y direction, Chebyshev
ii = np.arange(Nr + 1)
rr = np.cos(np.pi * ii / Nr) # rr [-1, 1]
rr = rr[:N2] # rr in (0, 1] with r[0] = 1
w = np.zeros([Nt, N2])
A = 1.0
q = np.zeros([Ns, Nt, N2])
r, t = np.meshgrid(rr, tt[:-1])
rp, tp = np.meshgrid(rr, tt)
q[0] = np.exp(-A * r**2) * np.sin(t)
w = np.cos(t)
#plt.contourf(r*np.cos(t), r*np.sin(t), q[0])
q0p = np.zeros([Nt + 1, N2])
q0p[:-1, :] = q[0]
q0p[-1, :] = q[0, 0, :]
print rp.shape, tp.shape, q0p.shape
scft_contourf(rp * np.cos(tp), rp * np.sin(tp), q0p)
plt.xlabel('q[0]')
plt.show()
#plt.contourf(r*np.cos(t), r*np.sin(t), w)
wp = np.zeros([Nt + 1, N2])
wp[:-1, :] = w
wp[-1, :] = w[0, :]
scft_contourf(rp * np.cos(tp), rp * np.sin(tp), wp)
plt.xlabel('w')
plt.show()
#lbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
#rbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
lbc = BC(ROBIN, [1.0, -0.1, 0.0])
rbc = BC(ROBIN, [1.0, 0.1, 0.0])
q_solver = ETDRK4Polar(R, Nr, Nt, Ns, h=ds, lbc=lbc, rbc=rbc)
q1 = q_solver.solve(w, q[0], q)
#plt.contourf(r*np.cos(t), r*np.sin(t), q1)
q1p = np.zeros([Nt + 1, N2])
q1p[:-1, :] = q1
q1p[-1, :] = q1[0, :]
scft_contourf(rp * np.cos(tp), rp * np.sin(tp), q1p)
plt.xlabel('q_solution')
plt.show()
plt.plot(rr, q1[0, :], label='$\\theta=0$')
plt.plot(rr, q1[Nt / 8, :], label='$\\theta=\pi/4$')
plt.plot(rr, q1[Nt / 4, :], label='$\\theta=\pi/2$')
plt.plot(rr, q1[3 * Nt / 8, :], label='$\\theta=3\pi/4$')
plt.plot(rr, q1[Nt / 2, :], label='$\\theta=\pi$')
plt.plot(rr, q1[5 * Nt / 8, :], label='$\\theta=5\pi/4$')
plt.plot(rr, q1[3 * Nt / 4, :], label='$\\theta=3\pi/2$')
plt.plot(rr, q1[7 * Nt / 8, :], label='$\\theta=7\pi/4$')
plt.legend(loc='best')
plt.show()
def test_exact_neumann(osc=0,oscheb=0,etdrk4=0):
L = 10.0
if osc:
N = 128
Ns = 1000 + 1 #20000 + 1
W, u0, x = init_fourier(N, L)
print 'OSC N = ', N, ' Ns = ', Ns-1
#q1, x1 = cheb_mde_osc(W, u0, L, Ns)
osc_solver = OSC(L, N, Ns)
q1, x1 = osc_solver.solve(W, u0)
Q1 = L * simps(q1, dx=1./N)
#data_name = 'benchmark/NBC-NBC/exact/OSS_N'
data_name = 'OSS_N'
data_name = data_name + str(N) + '_Ns' + str(Ns-1)
savemat(data_name,{
'N':N, 'Ns':Ns-1, 'W':W, 'u0':u0, 'Lz':L,
'x':x, 'q':q1, 'Q':Q1})
plt.plot(x1, q1, 'b')
plt.axis([0, 10, 0, 1.15])
plt.xlabel('$z$')
plt.ylabel('$q(z)$')
plt.savefig(data_name, bbox_inches='tight')
#plt.show()
if oscheb:
N = 128
Ns = 200 + 1 #20000 + 1
W, u0, x = init_chebyshev_fredrikson(N, L)
print 'OSCHEB N = ', N, ' Ns = ', Ns-1
#q2 = cheb_mde_neumann_oscheb(W, u0, L, Ns)
oscheb_sovler = OSCHEB(L, N, Ns, bc=BC('Neumann'))
q2, x2 = oscheb_sovler.solve(W, u0)
Q2 = 0.5 * L * cheb_quadrature_clencurt(q2)
#data_name = 'benchmark/NBC-NBC/exact/OSCHEB_N'
data_name = 'OSCHEB_N'
data_name = data_name + str(N) + '_Ns' + str(Ns-1)
savemat(data_name,{
'N':N, 'Ns':Ns-1, 'W':W, 'u0':u0, 'Lz':L,
'x':x2, 'q':q2, 'Q':Q2})
plt.plot(x2, q2, 'g')
plt.axis([0, 10, 0, 1.15])
plt.xlabel('$z$')
plt.ylabel('$q(z)$')
plt.savefig(data_name, bbox_inches='tight')
#plt.show()
if etdrk4:
N = 128
Ns = 200 + 1
algo = 1
scheme = 1
W, u0, x = init_chebyshev_fredrikson(N, L)
print 'ETDRK4 N = ', N, ' Ns = ', Ns-1
#q3, x3 = cheb_mde_neumann_etdrk4(W, u0, L, Ns, None, algo, scheme)
lbc = BC('Neumann')
rbc = BC('Neumann')
etdrk4_solver = ETDRK4(L, N, Ns, h=None, lbc=lbc, rbc=rbc)
q3, x3 = etdrk4_solver.solve(W, u0)
Q3 = 0.5 * L * cheb_quadrature_clencurt(q3)
#if scheme == 0:
# data_name = 'benchmark/NBC-NBC/exact/ETDRK4_Cox_N'
# data_name = data_name + str(N) + '_Ns' + str(Ns-1)
#else:
# data_name = 'benchmark/NBC-NBC/exact/ETDRK4_Krogstad_N'
# data_name = data_name + str(N) + '_Ns' + str(Ns-1)
#savemat(data_name,{
# 'N':N, 'Ns':Ns-1, 'W':W, 'u0':u0, 'Lz':L,
# 'x':x, 'q':q3, 'Q':Q3})
plt.plot(x3, q3, 'r')
plt.axis([0, 10, 0, 1.15])
plt.xlabel('$z$')
plt.ylabel('$q(z)$')
#plt.savefig(data_name, bbox_inches='tight')
plt.show()
def test_etdrk4fxcy():
'''
The test function is
u = e^[f(x,y) - t]
where
f(x,y) = h(x) + g(y)
Assume it is the solution of following PDE
du/dt = (d^2/dx^2 + d^2/dy^2) u - w(x,y)u
in the domain [0,Lx]x[0,Ly] for time t=0 to t=1,
with boundary conditions
u(x+Lx,y,t) = u(x,y,t) # periodic in x direction
d/dy[u(x,y=0,t)] = ka u(y=0)
d/dy[u(x,y=Ly,t)] = kb u(y=Ly)
To generate a suitable solution, we assume
h(x) = sin(x)
h_x = dh/dx = cos(x)
h_xx = d^2h/dx^2 = -sin(x)
since it is periodic in x direction.
The corresponding w(x,y) is
w(x,y) = h_xx + g_yy + (h_x)^2 + (g_y)^2 + 1
1. For homogeneous NBC (ka=kb=0), a suitable g(y) is
g(y) = Ay^2(2y-3)/6
g_y = A(y^2-y) # g_y(y=0)=0, g_y(y=1)=0
g_yy = A(2*y-1)
where A is a positive constant.
Lx = 2*pi, Ly = 1.0, Nx = 64, Ny =32, Ns = 21
is a good parameter set. Note the time step ds = 1/(Ns-1) = 0.05 is very
large.
2. For homogeneous DBC, an approximate g(y) is
g(y) = -A(y-1)^2
g_y = -2A(y-1)
g_yy = -2A
where A is a positive and large constant.
Lx = 2*pi, Ly = 2.0, Nx = 64, Ny =32, Ns = 101
is a good parameter set.
3. For RBC, g(y) is given by
g(y) = -Ay
g_y = -A # ka=kb=-A
g_yy = 0
A is a positive constant.
Numerical result is different than the analytical one.
'''
Lx = 2 * np.pi # x [0, Lx]
Nx = 64
Ly = 1.0 # y [0, Ly]
Ny = 127
Ns = 101
ds = 1. / (Ns - 1)
# Periodic in x direction, Fourier
xx = np.arange(Nx) * Lx / Nx
# Non-periodic in y direction, Chebyshev
ii = np.arange(Ny + 1)
yy = np.cos(np.pi * ii / Ny) # yy [-1, 1]
yy = 0.5 * (yy + 1) * Ly # mapping to [0, Ly]
w = np.zeros([Nx, Ny + 1])
A = 1.0
q = np.zeros([Ns, Nx, Ny + 1])
q_exact = np.zeros([Nx, Ny + 1])
#q[0] = 1.
for i in xrange(Nx):
for j in xrange(Ny + 1):
x = xx[i]
y = yy[j]
# RBC
#q_exact[i,j] = np.exp(-A*y + np.sin(x) - 1)
#q[0,i,j] = np.exp(-A*y + np.sin(x))
#w[i,j] = np.cos(x)**2 - np.sin(x) + A**2 + 1
# homogeneous NBC
q_exact[i, j] = np.exp(A * y**2 * (2 * y - 3) / 6 + np.sin(x) - 1)
q[0, i, j] = np.exp(A * y**2 * (2 * y - 3) / 6 + np.sin(x))
w[i, j] = (
A * y *
(y - 1))**2 + np.cos(x)**2 - np.sin(x) + A * (2 * y - 1) + 1
# homogeneous DBC
#q[0,i,j] = np.exp(-A*(y-1)**2 + np.sin(x))
#q_exact[i,j] = np.exp(-A*(y-1)**2 + np.sin(x) + 1)
#w[i, j] = np.cos(x)**2 - np.sin(x) + 4*A**2 + (2*A*(y-1))**2 + 1
# Fredrickson
#sech = 1. / np.cosh(0.25*(6*y[j]-3*Ly))
#w[i,j] = (1 - 2*sech**2)*(np.sin(2*np.pi*x[i]/Lx)+1)
#w[i,j] = (1 - 2*sech**2)
x = xx
y = yy
plt.imshow(w)
plt.xlabel('w')
plt.show()
plt.plot(x, w[:, Ny / 2])
plt.xlabel('w(x)')
plt.show()
plt.plot(y, w[Nx / 4, :])
plt.xlabel('w(y)')
plt.show()
# DBC
#lbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
#rbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
# RBC
#lbc = BC(ROBIN, [1.0, A, 0.0])
#rbc = BC(ROBIN, [1.0, A, 0.0])
# NBC
lbc = BC(ROBIN, [1.0, 0, 0.0])
rbc = BC(ROBIN, [1.0, 0, 0.0])
#q_solver = ETDRK4FxCy(Lx, Ly, Nx, Ny, Ns, h=ds, lbc=lbc, rbc=rbc)
q_solver = ETDRK4FxCy2(Lx, Ly, Nx, Ny, Ns, h=ds, lbc=lbc, rbc=rbc)
M = 100 # Took 1117.6 x 4 seconds for cpu one core
with Timer() as t:
for m in xrange(M):
q1 = q_solver.solve(w, q[0], q)
print "100 runs took ", t.secs, " seconds."
print 'Error =', np.max(np.abs(q1 - q_exact))
plt.imshow(q[0])
plt.xlabel('q_0')
plt.show()
plt.imshow(q1)
plt.xlabel('q_solution')
plt.show()
plt.imshow(q_exact)
plt.xlabel('q_exact')
plt.show()
plt.plot(x, q[0, :, Ny / 2], label='q0')
plt.plot(x, q1[:, Ny / 2], label='q_solution')
plt.plot(x, q_exact[:, Ny / 2], label='q_exact')
plt.legend(loc='best')
plt.xlabel('q[:,Ny/2]')
plt.show()
plt.plot(y, q[0, Nx / 4, :], label='q0')
plt.plot(y, q1[Nx / 4, :], label='q_solution')
plt.plot(y, q_exact[Nx / 4, :], label='q_exact')
plt.legend(loc='best')
plt.xlabel('q[Nx/4,:]')
plt.show()
plt.plot(y, q[0, Nx * 3 / 4, :], label='q0')
plt.plot(y, q1[Nx * 3 / 4, :], label='q_solution')
plt.plot(y, q_exact[Nx * 3 / 4, :], label='q_exact')
plt.legend(loc='best')
plt.xlabel('q[Nx*3/4,:]')
plt.show()
exit()
# Check with ETDRK4
sech = 1. / np.cosh(0.25 * (6 * y - 3 * Ly))
w1 = 1 - 2 * sech**2
plt.plot(y, w1)
plt.show()
q = np.zeros([Ns, Ny + 1])
q[0] = 1.
q_solver = ETDRK4(Ly, Ny, Ns, h=ds, lbc=lbc, rbc=rbc)
q1, y = q_solver.solve(w1, q[0], q)
plt.plot(y, q1)
plt.show()
def test_etdrk4fxycz():
'''
Assume the following PDE in cylindrical coordinates,
du/dt = (d^2/dx^2 + d^2/dy^2 + d^2/dz^2) u - w u
with u = u(x,y,z), w=w(x,y,z) in the domain
[0, Lx] x [0, Ly] x [0, Lz]
for time t=0 to t=1,
with boundary conditions
u(x,y,z,t) = u(x+Lx,y,z,t) # periodic in x direction
u(x,y,z,t) = u(x,y+Ly,z,t) # periodic in y direction
d/dr[u(z=0,y,z,t)] = ka u(z=0)
d/dr[u(z=Lz,y,z,t)] = kb u(z=Lz)
Test: PASSED 2013.8.16.
'''
Lx = 2.0 * np.pi # x [0, Lx]
Nx = 32
Ly = 2.0 * np.pi # y [0, Ly]
Ny = 32
Lz = 3.0 # z [0, Lz]
Nz = 63
Np = 64
Ns = 101
ds = 1. / (Ns - 1)
show3d = False
# Periodic in x direction, Fourier
xx = np.arange(Nx) * Lx / Nx
yy = np.arange(Ny) * Ly / Ny
# Non-periodic in z direction, Chebyshev
ii = np.arange(Nz + 1)
zz = np.cos(np.pi * ii / Nz) # zz [-1, 1]
zz = 0.5 * (zz + 1) * Lz # mapping to [0, Lz]
zzp = np.linspace(0, Lz, Np)
x, y, z = np.meshgrid(xx, yy, zz)
#xp, yp, zp = np.meshgrid(xx, yy, zzp)
xp, yp, zp = np.mgrid[0:Lx * (1 - 1. / Nx):Nx * 1j,
0:Ly * (1 - 1. / Ny):Ny * 1j, 0:Lz:Np * 1j]
A = 1.0
w = np.sin(x + 2 * y) + np.cos(z)**2
q = np.zeros([Ns, Nx, Ny, Nz + 1])
q[0] = np.exp(np.sin(x * y) - A * z)
q0p = np.exp(np.sin(xp * yp) - A * zp) # the exact
#q0 = q[0]
#plt.imshow(q0[Nx/2,:,:])
#plt.show()
#plt.imshow(q0[:,Ny/2,:])
#plt.show()
#plt.imshow(q0[:,:,Nz/2])
#plt.show()
if show3d:
q0p2 = cheb_interp3d_z(q[0], zzp)
print np.mean(np.abs(q0p2 - q0p)) # the error of interpolation
#plt.plot(zz, q[0, Nx/2, Ny/2], '.')
#plt.plot(zzp, q0p[Nx/2, Ny/2])
#plt.plot(zzp, q0p2[Nx/2, Ny/2], '^')
#plt.show()
mlab.clf()
#s = mlab.pipeline.scalar_field(xp,yp,zp,q0p2)
#mlab.pipeline.iso_surface(s,
#contours=[q0p.min()+0.1*q0p.ptp(),],
# opacity=0.3)
mlab.contour3d(xp,
yp,
zp,
q0p2,
contours=16,
opacity=0.5,
transparent=True,
colormap='Spectral')
mlab.show()
#lbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
#rbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
lbc = BC(ROBIN, [1.0, 0.1, 0.0])
rbc = BC(ROBIN, [1.0, -0.1, 0.0])
q_solver = ETDRK4FxyCz(Lx, Ly, Lz, Nx, Ny, Nz, Ns, h=ds, lbc=lbc, rbc=rbc)
print 'Initiate done!'
M = 10
# 10 runs took 131.901 seconds for cpu one core (32x32x32)
with Timer() as t:
for m in xrange(M):
q1 = q_solver.solve(w, q[0], q)
print "10 runs took", t.secs, " seconds."
print 'Solve done!'
#plt.imshow(q1[Nx/2,:,:])
#plt.show()
#plt.imshow(q1[:,Ny/2,:])
#plt.show()
#plt.imshow(q1[:,:,Nz/2])
#plt.show()
if show3d:
q1p = cheb_interp3d_z(q1, zzp)
mlab.clf()
mlab.contour3d(xp, yp, zp, q1p, contours=16, transparent=True)
#mlab.contour3d(q1)
mlab.show()
def test_etdrk4cylind():
'''
Assume the following PDE in cylindrical coordinates,
du/dt = (d^2/dr^2 + (1/r)d/dr + (1/r^2)d^2/d^theta + d^2/dz^2) u -
w*u
with u = u(r,theta,z), w=w(r,theta,z) in the domain
[0, R] x [0, 2pi] x [0, Lz]
for time t=0 to t=1,
with boundary conditions
d/dr[u(r=R,theta,z,t)] = ka u(r=R)
u(r,theta,z,t) = u(r,theta+2pi,z,t) # periodic in theta direction
u(r,theta,z,t) = u(r,theta,z+lz,t) # periodic in z direction
'''
Lt = 2*np.pi
Nt = 32 # theta
Lz = 4.0
Nz = 64
R = 2.0 # r [0, R]
Nr = 23 # Nr must be odd
N2 = (Nr + 1) / 2
Nxp = 20
Nyp = 20
Nzp = 15
Ns = 101
ds = 1. / (Ns - 1)
# Periodic in x and z direction, Fourier
tt = np.arange(Nt) * Lt / Nt
zz = np.arange(Nz) * Lz / Nz
# Non-periodic in r direction, Chebyshev
ii = np.arange(Nr+1)
rr = np.cos(np.pi * ii / Nr) # rr [-1, 1]
rr = rr[:N2] # rr in (0, 1] with r[0] = 1
rrp = np.linspace(0, R, Nxp)
A = 1.0
q = np.zeros([Ns, Nt, Nz, N2])
z, t, r = np.meshgrid(zz, tt, rr)
zp, tp, rp = np.meshgrid(zz, tt, rrp)
print tt.shape, zz.shape, rr.shape
print t.shape, z.shape, r.shape
print q[0].shape
q[0] = r**2 * np.cos(t) + A*z
xp = rp * np.cos(tp)
yp = rp * np.sin(tp)
print xp.max(), xp.min()
print yp.max(), yp.min()
print zp.max(), zp.min()
q0 = cheb_interp3d_r(q[0], rrp)
xpp, ypp, zpp = np.mgrid[-R:R:Nxp*1j,
-R:R:Nyp*1j,
0:Lz*(1-1./Nz):Nzp*1j]
q0p = griddata((xp.ravel(),yp.ravel(),zp.ravel()), q0.ravel(),
(xpp, ypp, zpp), method='linear')
print q0p.shape
mlab.contour3d(xpp, ypp, zpp, q0p,
contours=64, transparent=True, colormap='Spectral')
mlab.show()
exit()
q0p[:-1,:] = q[0]
q0p[-1,:] = q[0,0,:]
print rp.shape, tp.shape, q0p.shape
scft_contourf(rp*np.cos(tp), rp*np.sin(tp), q0p)
plt.xlabel('q[0]')
plt.show()
#plt.contourf(r*np.cos(t), r*np.sin(t), w)
w = np.zeros([Nt, Nz, N2])
w = np.cos(t)
wp = np.zeros([Nt+1,N2])
wp[:-1,:] = w
wp[-1,:] = w[0,:]
scft_contourf(rp*np.cos(tp), rp*np.sin(tp), wp)
plt.xlabel('w')
plt.show()
#lbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
#rbc = BC(DIRICHLET, [0.0, 1.0, 0.0])
lbc = BC(ROBIN, [1.0, -0.1, 0.0])
rbc = BC(ROBIN, [1.0, 0.1, 0.0])
q_solver = ETDRK4Cylind(R, Nr, Nt, Ns, h=ds, lbc=lbc, rbc=rbc)
q1 = q_solver.solve(w, q[0], q)
#plt.contourf(r*np.cos(t), r*np.sin(t), q1)
q1p = np.zeros([Nt+1,N2])
q1p[:-1,:] = q1
q1p[-1,:] = q1[0,:]
scft_contourf(rp*np.cos(tp), rp*np.sin(tp), q1p)
plt.xlabel('q_solution')
plt.show()
plt.plot(rr, q1[0,:], label='$\\theta=0$')
plt.plot(rr, q1[Nt/8,:], label='$\\theta=\pi/4$')
plt.plot(rr, q1[Nt/4,:], label='$\\theta=\pi/2$')
plt.plot(rr, q1[3*Nt/8,:], label='$\\theta=3\pi/4$')
plt.plot(rr, q1[Nt/2,:], label='$\\theta=\pi$')
plt.plot(rr, q1[5*Nt/8,:], label='$\\theta=5\pi/4$')
plt.plot(rr, q1[3*Nt/4,:], label='$\\theta=3\pi/2$')
plt.plot(rr, q1[7*Nt/8,:], label='$\\theta=7\pi/4$')
plt.legend(loc='best')
plt.show()