class allencahn_imex(ptype):
"""
Example implementing Allen-Cahn equation in 2-3D using mpi4py-fft for solving linear parts, IMEX time-stepping
mpi4py-fft: https://mpi4py-fft.readthedocs.io/en/latest/
Attributes:
fft: fft object
X: grid coordinates in real space
K2: Laplace operator in spectral space
dx: mesh width in x direction
dy: mesh width in y direction
"""
def __init__(self,
problem_params,
dtype_u=parallel_mesh,
dtype_f=parallel_imex_mesh):
"""
Initialization routine
Args:
problem_params (dict): custom parameters for the example
dtype_u: fft data type (will be passed to parent class)
dtype_f: fft data type wuth implicit and explicit parts (will be passed to parent class)
"""
if 'L' not in problem_params:
problem_params['L'] = 1.0
if 'init_type' not in problem_params:
problem_params['init_type'] = 'circle'
if 'comm' not in problem_params:
problem_params['comm'] = None
if 'dw' not in problem_params:
problem_params['dw'] = 0.0
# these parameters will be used later, so assert their existence
essential_keys = ['nvars', 'eps', 'L', 'radius', 'dw', 'spectral']
for key in essential_keys:
if key not in problem_params:
msg = 'need %s to instantiate problem, only got %s' % (
key, str(problem_params.keys()))
raise ParameterError(msg)
if not (isinstance(problem_params['nvars'], tuple)
and len(problem_params['nvars']) > 1):
raise ProblemError('Need at least two dimensions')
# Creating FFT structure
ndim = len(problem_params['nvars'])
axes = tuple(range(ndim))
self.fft = PFFT(problem_params['comm'],
list(problem_params['nvars']),
axes=axes,
dtype=np.float,
collapse=True)
# get test data to figure out type and dimensions
tmp_u = newDistArray(self.fft, problem_params['spectral'])
# invoke super init, passing the communicator and the local dimensions as init
super(allencahn_imex,
self).__init__(init=(tmp_u.shape, problem_params['comm'],
tmp_u.dtype),
dtype_u=dtype_u,
dtype_f=dtype_f,
params=problem_params)
L = np.array([self.params.L] * ndim, dtype=float)
# get local mesh
X = np.ogrid[self.fft.local_slice(False)]
N = self.fft.global_shape()
for i in range(len(N)):
X[i] = (X[i] * L[i] / N[i])
self.X = [np.broadcast_to(x, self.fft.shape(False)) for x in X]
# get local wavenumbers and Laplace operator
s = self.fft.local_slice()
N = self.fft.global_shape()
k = [np.fft.fftfreq(n, 1. / n).astype(int) for n in N[:-1]]
k.append(np.fft.rfftfreq(N[-1], 1. / N[-1]).astype(int))
K = [ki[si] for ki, si in zip(k, s)]
Ks = np.meshgrid(*K, indexing='ij', sparse=True)
Lp = 2 * np.pi / L
for i in range(ndim):
Ks[i] = (Ks[i] * Lp[i]).astype(float)
K = [np.broadcast_to(k, self.fft.shape(True)) for k in Ks]
K = np.array(K).astype(float)
self.K2 = np.sum(K * K, 0, dtype=float)
# Need this for diagnostics
self.dx = self.params.L / problem_params['nvars'][0]
self.dy = self.params.L / problem_params['nvars'][1]
def eval_f(self, u, t):
"""
Routine to evaluate the RHS
Args:
u (dtype_u): current values
t (float): current time
Returns:
dtype_f: the RHS
"""
f = self.dtype_f(self.init)
if self.params.spectral:
f.impl = -self.K2 * u
if self.params.eps > 0:
tmp = self.fft.backward(u)
tmpf = - 2.0 / self.params.eps ** 2 * tmp * (1.0 - tmp) * (1.0 - 2.0 * tmp) - \
6.0 * self.params.dw * tmp * (1.0 - tmp)
f.expl[:] = self.fft.forward(tmpf)
else:
u_hat = self.fft.forward(u)
lap_u_hat = -self.K2 * u_hat
f.impl[:] = self.fft.backward(lap_u_hat, f.impl)
if self.params.eps > 0:
f.expl = - 2.0 / self.params.eps ** 2 * u * (1.0 - u) * (1.0 - 2.0 * u) - \
6.0 * self.params.dw * u * (1.0 - u)
return f
def solve_system(self, rhs, factor, u0, t):
"""
Simple FFT solver for the diffusion part
Args:
rhs (dtype_f): right-hand side for the linear system
factor (float) : abbrev. for the node-to-node stepsize (or any other factor required)
u0 (dtype_u): initial guess for the iterative solver (not used here so far)
t (float): current time (e.g. for time-dependent BCs)
Returns:
dtype_u: solution as mesh
"""
if self.params.spectral:
me = rhs / (1.0 + factor * self.K2)
else:
me = self.dtype_u(self.init)
rhs_hat = self.fft.forward(rhs)
rhs_hat /= (1.0 + factor * self.K2)
me[:] = self.fft.backward(rhs_hat)
return me
def u_exact(self, t):
"""
Routine to compute the exact solution at time t
Args:
t (float): current time
Returns:
dtype_u: exact solution
"""
assert t == 0, 'ERROR: u_exact only valid for t=0'
me = self.dtype_u(self.init, val=0.0)
if self.params.init_type == 'circle':
r2 = (self.X[0] - 0.5)**2 + (self.X[1] - 0.5)**2
if self.params.spectral:
tmp = 0.5 * (1.0 + np.tanh((self.params.radius - np.sqrt(r2)) /
(np.sqrt(2) * self.params.eps)))
me[:] = self.fft.forward(tmp)
else:
me[:] = 0.5 * (1.0 + np.tanh(
(self.params.radius - np.sqrt(r2)) /
(np.sqrt(2) * self.params.eps)))
elif self.params.init_type == 'circle_rand':
ndim = len(me.shape)
L = int(self.params.L)
# get random radii for circles/spheres
np.random.seed(1)
lbound = 3.0 * self.params.eps
ubound = 0.5 - self.params.eps
rand_radii = (ubound - lbound) * np.random.random_sample(
size=tuple([L] * ndim)) + lbound
# distribute circles/spheres
tmp = newDistArray(self.fft, False)
if ndim == 2:
for i in range(0, L):
for j in range(0, L):
# build radius
r2 = (self.X[0] + i - L + 0.5)**2 + (self.X[1] + j -
L + 0.5)**2
# add this blob, shifted by 1 to avoid issues with adding up negative contributions
tmp += np.tanh((rand_radii[i, j] - np.sqrt(r2)) /
(np.sqrt(2) * self.params.eps)) + 1
# normalize to [0,1]
tmp *= 0.5
assert np.all(tmp <= 1.0)
if self.params.spectral:
me[:] = self.fft.forward(tmp)
else:
me[:] = tmp[:]
else:
raise NotImplementedError(
'type of initial value not implemented, got %s' %
self.params.init_type)
return me
class nonlinearschroedinger_imex(ptype):
"""
Example implementing the nonlinear Schrödinger equation in 2-3D using mpi4py-fft for solving linear parts,
IMEX time-stepping
mpi4py-fft: https://mpi4py-fft.readthedocs.io/en/latest/
Attributes:
fft: fft object
X: grid coordinates in real space
K2: Laplace operator in spectral space
"""
def __init__(self,
problem_params,
dtype_u=parallel_mesh,
dtype_f=parallel_imex_mesh):
"""
Initialization routine
Args:
problem_params (dict): custom parameters for the example
dtype_u: fft data type (will be passed to parent class)
dtype_f: fft data type wuth implicit and explicit parts (will be passed to parent class)
"""
if 'L' not in problem_params:
problem_params['L'] = 2.0 * np.pi
# if 'init_type' not in problem_params:
# problem_params['init_type'] = 'circle'
if 'comm' not in problem_params:
problem_params['comm'] = MPI.COMM_WORLD
if 'c' not in problem_params:
problem_params['c'] = 1.0
if not problem_params['L'] == 2.0 * np.pi:
raise ProblemError(
f'Setup not implemented, L has to be 2pi, got {problem_params["L"]}'
)
if not (problem_params['c'] == 0.0 or problem_params['c'] == 1.0):
raise ProblemError(
f'Setup not implemented, c has to be 0 or 1, got {problem_params["c"]}'
)
# these parameters will be used later, so assert their existence
essential_keys = ['nvars', 'c', 'L', 'spectral']
for key in essential_keys:
if key not in problem_params:
msg = 'need %s to instantiate problem, only got %s' % (
key, str(problem_params.keys()))
raise ParameterError(msg)
if not (isinstance(problem_params['nvars'], tuple)
and len(problem_params['nvars']) > 1):
raise ProblemError('Need at least two dimensions')
# Creating FFT structure
self.ndim = len(problem_params['nvars'])
axes = tuple(range(self.ndim))
self.fft = PFFT(problem_params['comm'],
list(problem_params['nvars']),
axes=axes,
dtype=np.complex128,
collapse=True)
# get test data to figure out type and dimensions
tmp_u = newDistArray(self.fft, problem_params['spectral'])
# invoke super init, passing the communicator and the local dimensions as init
super(nonlinearschroedinger_imex,
self).__init__(init=(tmp_u.shape, problem_params['comm'],
tmp_u.dtype),
dtype_u=dtype_u,
dtype_f=dtype_f,
params=problem_params)
self.L = np.array([self.params.L] * self.ndim, dtype=float)
# get local mesh
X = np.ogrid[self.fft.local_slice(False)]
N = self.fft.global_shape()
for i in range(len(N)):
X[i] = (X[i] * self.L[i] / N[i])
self.X = [np.broadcast_to(x, self.fft.shape(False)) for x in X]
# get local wavenumbers and Laplace operator
s = self.fft.local_slice()
N = self.fft.global_shape()
k = [np.fft.fftfreq(n, 1. / n).astype(int) for n in N]
K = [ki[si] for ki, si in zip(k, s)]
Ks = np.meshgrid(*K, indexing='ij', sparse=True)
Lp = 2 * np.pi / self.L
for i in range(self.ndim):
Ks[i] = (Ks[i] * Lp[i]).astype(float)
K = [np.broadcast_to(k, self.fft.shape(True)) for k in Ks]
K = np.array(K).astype(float)
self.K2 = np.sum(K * K, 0, dtype=float)
# Need this for diagnostics
self.dx = self.params.L / problem_params['nvars'][0]
self.dy = self.params.L / problem_params['nvars'][1]
def eval_f(self, u, t):
"""
Routine to evaluate the RHS
Args:
u (dtype_u): current values
t (float): current time
Returns:
dtype_f: the RHS
"""
f = self.dtype_f(self.init)
if self.params.spectral:
f.impl = -self.K2 * 1j * u
tmp = self.fft.backward(u)
tmpf = self.ndim * self.params.c * 2j * np.absolute(tmp)**2 * tmp
f.expl[:] = self.fft.forward(tmpf)
else:
u_hat = self.fft.forward(u)
lap_u_hat = -self.K2 * 1j * u_hat
f.impl[:] = self.fft.backward(lap_u_hat, f.impl)
f.expl = self.ndim * self.params.c * 2j * np.absolute(u)**2 * u
return f
def solve_system(self, rhs, factor, u0, t):
"""
Simple FFT solver for the diffusion part
Args:
rhs (dtype_f): right-hand side for the linear system
factor (float) : abbrev. for the node-to-node stepsize (or any other factor required)
u0 (dtype_u): initial guess for the iterative solver (not used here so far)
t (float): current time (e.g. for time-dependent BCs)
Returns:
dtype_u: solution as mesh
"""
if self.params.spectral:
me = rhs / (1.0 + factor * self.K2 * 1j)
else:
me = self.dtype_u(self.init)
rhs_hat = self.fft.forward(rhs)
rhs_hat /= (1.0 + factor * self.K2 * 1j)
me[:] = self.fft.backward(rhs_hat)
return me
def u_exact(self, t):
"""
Routine to compute the exact solution at time t, see (1.3) https://arxiv.org/pdf/nlin/0702010.pdf for details
Args:
t (float): current time
Returns:
dtype_u: exact solution
"""
def nls_exact_1D(t, x, c):
ae = 1.0 / np.sqrt(2.0) * np.exp(1j * t)
if c != 0:
u = ae * ((np.cosh(t) + 1j * np.sinh(t)) /
(np.cosh(t) - 1.0 / np.sqrt(2.0) * np.cos(x)) - 1.0)
else:
u = np.sin(x) * np.exp(-t * 1j)
return u
me = self.dtype_u(self.init, val=0.0)
if self.params.spectral:
tmp = nls_exact_1D(self.ndim * t, sum(self.X), self.params.c)
me[:] = self.fft.forward(tmp)
else:
me[:] = nls_exact_1D(self.ndim * t, sum(self.X), self.params.c)
return me
uex[:] = -2.0 * (2.0 * np.pi)**2 * np.sin(2 * np.pi * X[0]) * np.sin(
2 * np.pi * X[1])
u_hat = fft.forward(u)
lap_u_hat = -K2 * u_hat
lap_u = np.zeros_like(u)
lap_u = fft.backward(lap_u_hat, lap_u)
local_error = np.amax(abs(lap_u - uex))
err = MPI.COMM_WORLD.allreduce(local_error, MPI.MAX)
print('Laplace error:', err)
ratio = 2
Nc = np.array([nvars // ratio] * ndim, dtype=int)
fftc = PFFT(MPI.COMM_WORLD, Nc, axes=axes, dtype=np.float, slab=True)
print(Nc, fftc.global_shape())
Xc = get_local_mesh(fftc, L)
uex = newDistArray(fft, False)
uexc = newDistArray(fftc, False)
r2 = (X[0] - 0.5)**2 + (X[1] - 0.5)**2
uex[:] = 0.5 * (1.0 + np.tanh((0.25 - np.sqrt(r2)) / (np.sqrt(2) * 0.04)))
r2c = (Xc[0] - 0.5)**2 + (Xc[1] - 0.5)**2
uexc[:] = 0.5 * (1.0 + np.tanh((0.25 - np.sqrt(r2c)) / (np.sqrt(2) * 0.04)))
uc = uex[::ratio, ::ratio]
local_error = np.amax(abs(uc - uexc))
err = MPI.COMM_WORLD.allreduce(local_error, MPI.MAX)
print('Restriction error real:', err)
uexcs = fftc.forward(uexc)
class grayscott_imex_diffusion(ptype):
"""
Example implementing the Gray-Scott equation in 2-3D using mpi4py-fft for solving linear parts,
IMEX time-stepping (implicit diffusion, explicit reaction)
mpi4py-fft: https://mpi4py-fft.readthedocs.io/en/latest/
Attributes:
fft: fft object
X: grid coordinates in real space
ndim: number of spatial dimensions
Ku: Laplace operator in spectral space (u component)
Kv: Laplace operator in spectral space (v component)
"""
def __init__(self,
problem_params,
dtype_u=parallel_mesh,
dtype_f=parallel_imex_mesh):
"""
Initialization routine
Args:
problem_params (dict): custom parameters for the example
dtype_u: fft data type (will be passed to parent class)
dtype_f: fft data type wuth implicit and explicit parts (will be passed to parent class)
"""
if 'L' not in problem_params:
problem_params['L'] = 2.0
# if 'init_type' not in problem_params:
# problem_params['init_type'] = 'circle'
if 'comm' not in problem_params:
problem_params['comm'] = MPI.COMM_WORLD
# these parameters will be used later, so assert their existence
essential_keys = ['nvars', 'Du', 'Dv', 'A', 'B', 'spectral']
for key in essential_keys:
if key not in problem_params:
msg = 'need %s to instantiate problem, only got %s' % (
key, str(problem_params.keys()))
raise ParameterError(msg)
if not (isinstance(problem_params['nvars'], tuple)
and len(problem_params['nvars']) > 1):
raise ProblemError('Need at least two dimensions')
# Creating FFT structure
self.ndim = len(problem_params['nvars'])
axes = tuple(range(self.ndim))
self.fft = PFFT(problem_params['comm'],
list(problem_params['nvars']),
axes=axes,
dtype=np.float64,
collapse=True,
backend='fftw')
# get test data to figure out type and dimensions
tmp_u = newDistArray(self.fft, problem_params['spectral'])
# add two components to contain field and temperature
self.ncomp = 2
sizes = tmp_u.shape + (self.ncomp, )
# invoke super init, passing the communicator and the local dimensions as init
super(grayscott_imex_diffusion,
self).__init__(init=(sizes, problem_params['comm'], tmp_u.dtype),
dtype_u=dtype_u,
dtype_f=dtype_f,
params=problem_params)
L = np.array([self.params.L] * self.ndim, dtype=float)
# get local mesh
X = np.ogrid[self.fft.local_slice(False)]
N = self.fft.global_shape()
for i in range(len(N)):
X[i] = -L[i] / 2 + (X[i] * L[i] / N[i])
self.X = [np.broadcast_to(x, self.fft.shape(False)) for x in X]
# get local wavenumbers and Laplace operator
s = self.fft.local_slice()
N = self.fft.global_shape()
k = [np.fft.fftfreq(n, 1. / n).astype(int) for n in N[:-1]]
k.append(np.fft.rfftfreq(N[-1], 1. / N[-1]).astype(int))
K = [ki[si] for ki, si in zip(k, s)]
Ks = np.meshgrid(*K, indexing='ij', sparse=True)
Lp = 2 * np.pi / L
for i in range(self.ndim):
Ks[i] = (Ks[i] * Lp[i]).astype(float)
K = [np.broadcast_to(k, self.fft.shape(True)) for k in Ks]
K = np.array(K).astype(float)
self.K2 = np.sum(K * K, 0, dtype=float)
self.Ku = -self.K2 * self.params.Du
self.Kv = -self.K2 * self.params.Dv
# Need this for diagnostics
self.dx = self.params.L / problem_params['nvars'][0]
self.dy = self.params.L / problem_params['nvars'][1]
def eval_f(self, u, t):
"""
Routine to evaluate the RHS
Args:
u (dtype_u): current values
t (float): current time
Returns:
dtype_f: the RHS
"""
f = self.dtype_f(self.init)
if self.params.spectral:
f.impl[..., 0] = self.Ku * u[..., 0]
f.impl[..., 1] = self.Kv * u[..., 1]
tmpu = newDistArray(self.fft, False)
tmpv = newDistArray(self.fft, False)
tmpu[:] = self.fft.backward(u[..., 0], tmpu)
tmpv[:] = self.fft.backward(u[..., 1], tmpv)
tmpfu = -tmpu * tmpv**2 + self.params.A * (1 - tmpu)
tmpfv = tmpu * tmpv**2 - self.params.B * tmpv
f.expl[..., 0] = self.fft.forward(tmpfu)
f.expl[..., 1] = self.fft.forward(tmpfv)
else:
u_hat = self.fft.forward(u[..., 0])
lap_u_hat = self.Ku * u_hat
f.impl[..., 0] = self.fft.backward(lap_u_hat, f.impl[..., 0])
u_hat = self.fft.forward(u[..., 1])
lap_u_hat = self.Kv * u_hat
f.impl[..., 1] = self.fft.backward(lap_u_hat, f.impl[..., 1])
f.expl[...,
0] = -u[..., 0] * u[...,
1]**2 + self.params.A * (1 - u[..., 0])
f.expl[...,
1] = u[..., 0] * u[..., 1]**2 - self.params.B * u[..., 1]
return f
def solve_system(self, rhs, factor, u0, t):
"""
Simple FFT solver for the diffusion part
Args:
rhs (dtype_f): right-hand side for the linear system
factor (float) : abbrev. for the node-to-node stepsize (or any other factor required)
u0 (dtype_u): initial guess for the iterative solver (not used here so far)
t (float): current time (e.g. for time-dependent BCs)
Returns:
dtype_u: solution as mesh
"""
me = self.dtype_u(self.init)
if self.params.spectral:
me[..., 0] = rhs[..., 0] / (1.0 - factor * self.Ku)
me[..., 1] = rhs[..., 1] / (1.0 - factor * self.Kv)
else:
rhs_hat = self.fft.forward(rhs[..., 0])
rhs_hat /= (1.0 - factor * self.Ku)
me[..., 0] = self.fft.backward(rhs_hat, me[..., 0])
rhs_hat = self.fft.forward(rhs[..., 1])
rhs_hat /= (1.0 - factor * self.Kv)
me[..., 1] = self.fft.backward(rhs_hat, me[..., 1])
return me
def u_exact(self, t):
"""
Routine to compute the exact solution at time t=0, see https://www.chebfun.org/examples/pde/GrayScott.html
Args:
t (float): current time
Returns:
dtype_u: exact solution
"""
assert t == 0.0, 'Exact solution only valid as initial condition'
assert self.ndim == 2, 'The initial conditions are 2D for now..'
me = self.dtype_u(self.init, val=0.0)
# This assumes that the box is [-L/2, L/2]^2
if self.params.spectral:
tmp = 1.0 - np.exp(-80.0 * ((self.X[0] + 0.05)**2 +
(self.X[1] + 0.02)**2))
me[..., 0] = self.fft.forward(tmp)
tmp = np.exp(-80.0 * ((self.X[0] - 0.05)**2 +
(self.X[1] - 0.02)**2))
me[..., 1] = self.fft.forward(tmp)
else:
me[..., 0] = 1.0 - np.exp(-80.0 * ((self.X[0] + 0.05)**2 +
(self.X[1] + 0.02)**2))
me[..., 1] = np.exp(-80.0 * ((self.X[0] - 0.05)**2 +
(self.X[1] - 0.02)**2))
# tmpu = np.load('data/u_0001.npy')
# tmpv = np.load('data/v_0001.npy')
#
# me[..., 0] = self.fft.forward(tmpu)
# me[..., 1] = self.fft.forward(tmpv)
return me