X = ST.mesh()
# Get f on quad points and exact solution
fj = Array(ST, buffer=fe)
uj = Array(ST, buffer=ue)
# Compute right hand side
f_hat = Function(ST)
f_hat = inner(v, fj, output_array=f_hat)
# Solve Poisson equation
A = inner(grad(v), grad(u))
u_hat = Function(ST)
u_hat = A.solve(-f_hat, u_hat)
uq = ST.backward(u_hat)
u_hat = ST.forward(uq, u_hat, fast_transform=False)
uq = ST.backward(u_hat, uq, fast_transform=False)
assert np.allclose(uj, uq)
point = np.array([0.1, 0.2])
p = ST.eval(point, u_hat)
assert np.allclose(p, lambdify(x, ue)(point))
if 'pytest' not in os.environ:
import matplotlib.pyplot as plt
plt.figure()
plt.plot(X, uj.real)
plt.title("U")
plt.figure()
f_hat = inner(v, fj, output_array=f_hat)
if family == 'legendre':
f_hat *= -1.
# Get left hand side of Poisson equation
if family == 'chebyshev':
A = inner(v, div(grad(u))) + alpha*inner(v, grad(u))
else:
A = inner(grad(v), grad(u)) - alpha*inner(v, grad(u))
if family == 'chebyshev':
f_hat[0] -= 0.5*np.pi*alpha*a
f_hat[0] += 0.5*np.pi*alpha*b
elif family == 'legendre':
f_hat[0] += alpha*a
f_hat[0] -= alpha*b
f_hat[:-2] = A.solve(f_hat[:-2])
f_hat[-2] = a
f_hat[-1] = b
uj = SD.backward(f_hat)
# Compare with analytical solution
ua = ul(X)
print("Error=%2.16e" %(np.linalg.norm(uj-ua)))
assert np.allclose(uj, ua)
point = np.array([0.1, 0.2])
p = SD.eval(point, f_hat)
assert np.allclose(p, ul(point))
f_hat = Function(SD, buffer=inner(v, fj))
if family == 'legendre':
f_hat *= -1.
# Get left hand side of Poisson equation
if family == 'chebyshev':
A = inner(v, div(grad(u)))
else:
A = inner(grad(v), grad(u))
f_hat = A.solve(f_hat)
# Solve and transform to real space
u = np.zeros(N) # Solution real space
u = SD.backward(f_hat, u)
# Compare with analytical solution
uj = ul(X)
print(abs(uj-u).max())
assert np.allclose(uj, u)
if not plt is None and not 'pytest' in os.environ:
plt.figure()
plt.plot(X, u)
plt.figure()
plt.plot(X, uj)
plt.figure()
plt.plot(X, u-uj)
def main(N, dt=0.005, end_time=2, dealias_initial=True, plot_result=False):
SD = Basis(N, 'F', dtype='D')
X = SD.points_and_weights()[0]
v = TestFunction(SD)
U = Array(SD)
dU = Function(SD)
U_hat = Function(SD)
U_hat0 = Function(SD)
U_hat1 = Function(SD)
w0 = Function(SD)
a = [1./6., 1./3., 1./3., 1./6.] # Runge-Kutta parameter
b = [0.5, 0.5, 1.] # Runge-Kutta parameter
nu = 0.2
k = SD.wavenumbers().astype(float)
# initialize
U[:] = 3./(5.-4.*np.cos(X))
if not dealias_initial:
U_hat = SD.forward(U, U_hat)
else:
U_hat[:] = 2**(-abs(k))
def compute_rhs(rhs, u_hat, w0):
rhs.fill(0)
w0.fill(0)
rhs = inner(v, nu*div(grad(u_hat)), output_array=rhs)
rhs -= inner(v, grad(u_hat), output_array=w0)
return rhs
# Integrate using a 4th order Rung-Kutta method
t = 0.0
tstep = 0
if plot_result is True:
im = plt.figure()
ca = im.gca()
ca.plot(X, U.real, 'b')
plt.draw()
plt.pause(1e-6)
while t < end_time-1e-8:
t += dt
tstep += 1
U_hat1[:] = U_hat0[:] = U_hat
for rk in range(4):
dU = compute_rhs(dU, U_hat, w0)
if rk < 3:
U_hat[:] = U_hat0 + b[rk]*dt*dU
U_hat1 += a[rk]*dt*dU
U_hat[:] = U_hat1
if tstep % (200) == 0 and plot_result is True:
ca.plot(X, U.real)
plt.pause(1e-6)
#plt.savefig('Ginzburg_Landau_pad_{}_real_{}.png'.format(N[0], int(np.round(t))))
U = SD.backward(U_hat, U)
Ue = np.zeros_like(U)
for k in range(-100, 101):
Ue += 2**(-abs(k))*np.exp(1j*k*(X-t) - nu*k**2*t)
err = np.sqrt(2*np.pi/N*np.linalg.norm(U-Ue.real)**2)
return 1./N, err
# Compute right hand side of Poisson equation
f_hat = Array(SD)
f_hat = inner(v, fj, output_array=f_hat)
# Get left hand side of Poisson equation
if family == 'chebyshev':
A = inner(v, -div(grad(u)))
B = inner(v, alfa * u)
else:
A = inner(grad(v), grad(u))
B = inner(v, alfa * u)
H = Solver(A, B)
u_hat = Function(SD) # Solution spectral space
u_hat = H(u_hat, f_hat)
uj = SD.backward(u_hat)
# Compare with analytical solution
ua = ul(X)
if family == 'chebyshev':
# Compute L2 error norm using Clenshaw-Curtis integration
from shenfun import clenshaw_curtis1D
error = clenshaw_curtis1D((uj - ua)**2, quad=SD.quad)
print("Error=%2.16e" % (error))
else:
x, w = SD.points_and_weights()
print("Error=%2.16e" % (np.sqrt(np.sum((uj - ua)**2 * w))))
assert np.allclose(uj, ua)
# Some work required for inhomogeneous boundary conditions only
if SD.has_nonhomogeneous_bcs:
bc_mats = extract_bc_matrices([matrices])
# Add boundary terms to the known right hand side
SD.bc.set_boundary_dofs(u_hat, final=True) # Fixes boundary dofs in u_hat
w0 = np.zeros_like(u_hat)
for m in bc_mats:
f_hat -= m.matvec(u_hat, w0)
# Create linear algebra solver
H = Solver(*matrices)
u_hat = H(u_hat, f_hat)
uj = Array(SD)
uj = SD.backward(u_hat, uj)
uh = uj.forward()
# Compare with analytical solution
uq = Array(SD, buffer=ue)
print("Error=%2.16e" % (np.linalg.norm(uj - uq)))
assert np.linalg.norm(uj - uq) < 1e-8
if 'pytest' not in os.environ:
import matplotlib.pyplot as plt
plt.figure()
plt.plot(X, uq)
plt.figure()
plt.plot(X, uj)
# Compute right hand side of biharmonic equation
f_hat = inner(v, fj)
# Get left hand side of biharmonic equation (no integration by parts)
S = inner(v, a*Dx(u, 0, 4))
A = inner(v, b*Dx(u, 0, 2))
B = inner(v, c*u)
# Create linear algebra solver
H = Solver(S, A, B)
# Solve and transform to real space
u_hat = Function(SD) # Solution spectral space
u_hat = H(u_hat, f_hat) # Solve
u = Array(SD)
u = SD.backward(u_hat, u)
uh = u.forward()
# Compare with analytical solution
uj = ul(X)
print("Error=%2.16e" %(np.linalg.norm(uj-u)))
assert np.allclose(uj, u)
point = np.array([0.1, 0.2])
p = u_hat.eval(point)
assert np.allclose(p, ul(point))
if 'pytest' not in os.environ:
import matplotlib.pyplot as plt
plt.figure()
plt.plot(X, u)
