def test_heat_1d_periodic(x_basis_class, Nx, timestepper, dtype):
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c, ))
xb = basis.ComplexFourier(c, size=Nx, bounds=(0, 2 * np.pi))
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb, ), dtype=dtype)
F = field.Field(name='F', dist=d, bases=(xb, ), dtype=dtype)
F['g'] = -np.sin(x)
# Problem
dx = lambda A: operators.Differentiate(A, c)
dt = operators.TimeDerivative
problem = problems.IVP([u])
problem.add_equation((-dt(u) + dx(dx(u)), F))
# Solver
solver = solvers.InitialValueSolver(problem, timestepper)
dt = 1e-5
iter = 10
for i in range(iter):
solver.step(dt)
# Check solution
amp = 1 - np.exp(-solver.sim_time)
u_true = amp * np.sin(x)
assert np.allclose(u['g'], u_true)
def test_waves_1d(x_basis_class, Nx, dtype):
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c, ))
xb = x_basis_class(c, size=Nx, bounds=(0, np.pi))
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb, ), dtype=dtype)
a = field.Field(name='a', dist=d, dtype=dtype)
τ1 = field.Field(name='τ1', dist=d, dtype=dtype)
τ2 = field.Field(name='τ2', dist=d, dtype=dtype)
# Problem
dx = lambda A: operators.Differentiate(A, c)
xb2 = dx(dx(u)).domain.bases[0]
P1 = field.Field(name='P1', dist=d, bases=(xb2, ), dtype=dtype)
P2 = field.Field(name='P2', dist=d, bases=(xb2, ), dtype=dtype)
P1['c'][-1] = 1
P2['c'][-2] = 1
problem = problems.EVP([u, τ1, τ2], a)
problem.add_equation((a * u + dx(dx(u)) + P1 * τ1 + P2 * τ2, 0))
problem.add_equation((u(x=0), 0))
problem.add_equation((u(x=np.pi), 0))
# Solver
solver = solvers.EigenvalueSolver(problem, matrix_coupling=[True])
solver.solve_dense(solver.subproblems[0])
i_sort = np.argsort(solver.eigenvalues)
sorted_eigenvalues = solver.eigenvalues[i_sort]
# Check solution
Nmodes = Nx // 4
k = np.arange(Nmodes) + 1
assert np.allclose(sorted_eigenvalues[:Nmodes], k**2)
def test_waves_1d_first_order(x_basis_class, Nx, dtype):
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c, ))
xb = x_basis_class(c, size=Nx, bounds=(0, np.pi))
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb, ), dtype=dtype)
ux = field.Field(name='ux', dist=d, bases=(xb, ), dtype=dtype)
a = field.Field(name='a', dist=d, dtype=dtype)
τ1 = field.Field(name='τ1', dist=d, dtype=dtype)
τ2 = field.Field(name='τ2', dist=d, dtype=dtype)
# Problem
dx = lambda A: operators.Differentiate(A, c)
xb1 = dx(u).domain.bases[0]
P1 = field.Field(name='P1', dist=d, bases=(xb1, ), dtype=dtype)
P2 = field.Field(name='P2', dist=d, bases=(xb1, ), dtype=dtype)
P1['c'][-1] = 1
P2['c'][-1] = 1
problem = problems.EVP([u, ux, τ1, τ2], a)
problem.add_equation((a * u + dx(ux) + P1 * τ1, 0))
problem.add_equation((ux - dx(u) + P2 * τ2, 0))
problem.add_equation((u(x=0), 0))
problem.add_equation((u(x=np.pi), 0))
# Solver
solver = solvers.EigenvalueSolver(problem, matrix_coupling=[True])
solver.solve_dense(solver.subproblems[0])
i_sort = np.argsort(solver.eigenvalues)
solver.eigenvalues = solver.eigenvalues[i_sort]
solver.eigenvectors = solver.eigenvectors[:, i_sort]
# Check solution
solver.set_state(0, solver.subproblems[0].subsystems[0])
eigenfunction = u['g'] / np.max(u['g'])
sol = np.sin(x) / np.max(np.sin(x))
assert np.allclose(eigenfunction, sol)
def test_sin_nlbvp(Nx, dtype, dealias, basis_class):
ncc_cutoff = 1e-10
tolerance = 1e-10
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c,))
xb = basis_class(c, size=Nx, bounds=(0, 1), dealias=dealias)
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb,), dtype=dtype)
τ = field.Field(name='τ', dist=d, dtype=dtype)
xb1 = xb.clone_with(a=xb.a+1, b=xb.b+1)
P = field.Field(name='P', dist=d, bases=(xb1,), dtype=dtype)
P['c'][-1] = 1
# Problem
dx = lambda A: operators.Differentiate(A, c)
problem = problems.NLBVP([u, τ])
problem.add_equation((dx(u) + τ*P, np.sqrt(1-u*u)))
problem.add_equation((u(x=0), 0))
# Solver
solver = solvers.NonlinearBoundaryValueSolver(problem, ncc_cutoff=ncc_cutoff)
# Initial guess
u['g'] = x
# Iterations
def error(perts):
return np.sum([np.sum(np.abs(pert['c'])) for pert in perts])
err = np.inf
while err > tolerance:
solver.newton_iteration()
err = error(solver.perturbations)
# Check solution
u_true = np.sin(x)
u.change_scales(1)
assert np.allclose(u['g'], u_true)
def test_jacobi_differentiate(N, a, b, k, dtype):
c = coords.Coordinate('x')
d = distributor.Distributor((c,))
b = basis.Jacobi(c, size=N, a0=a, b0=b, a=a+k, b=b+k, bounds=(0, 1))
x = b.local_grid(1)
f = field.Field(dist=d, bases=(b,), dtype=dtype)
f['g'] = x**5
fx = operators.Differentiate(f, c).evaluate()
assert np.allclose(fx['g'], 5*x**4)
def test_jacobi_convert_explicit(N, a, b, k, dtype, layout):
c = coords.Coordinate('x')
d = distributor.Distributor((c,))
b = basis.Jacobi(c, size=N, a0=a, b0=b, a=a+k, b=b+k, bounds=(0, 1))
x = b.local_grid(1)
f = field.Field(dist=d, bases=(b,), dtype=dtype)
f['g'] = x**5
fx = operators.Differentiate(f, c)
f[layout]
g = operators.convert(f, fx.domain.bases).evaluate()
assert np.allclose(g['g'], f['g'])
def test_fourier_differentiate(N, bounds, dtype):
c = coords.Coordinate('x')
d = distributor.Distributor((c, ))
if dtype == np.float64:
b = basis.RealFourier(c, size=N, bounds=bounds)
elif dtype == np.complex128:
b = basis.ComplexFourier(c, size=N, bounds=bounds)
x = b.local_grid(1)
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
k = 4 * np.pi / (bounds[1] - bounds[0])
f['g'] = 1 + np.sin(k * x + 0.1)
fx = operators.Differentiate(f, c).evaluate()
assert np.allclose(fx['g'], k * np.cos(k * x + 0.1))
def test_jacobi_convert_implicit(N, a, b, k, dtype):
c = coords.Coordinate('x')
d = distributor.Distributor((c,))
b = basis.Jacobi(c, size=N, a0=a, b0=b, a=a+k, b=b+k, bounds=(0, 1))
x = b.local_grid(1)
f = field.Field(dist=d, bases=(b,), dtype=dtype)
f['g'] = x**5
fx = operators.Differentiate(f, c).evaluate()
g = field.Field(dist=d, bases=(b,), dtype=dtype)
problem = problems.LBVP([g])
problem.add_equation((g, fx))
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
assert np.allclose(g['g'], fx['g'])
def test_heat_1d_periodic(x_basis_class, Nx, dtype):
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c, ))
xb = x_basis_class(c, size=Nx, bounds=(0, 2 * np.pi))
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb, ), dtype=dtype)
a = field.Field(name='a', dist=d, dtype=dtype)
# Problem
dx = lambda A: operators.Differentiate(A, c)
problem = problems.EVP([u], a)
problem.add_equation((a * u + dx(dx(u)), 0))
# Solver
solver = solvers.EigenvalueSolver(problem, matrix_coupling=[True])
solver.solve_dense(solver.subproblems[0])
# Check solution
k = xb.wavenumbers
if x_basis_class is basis.RealFourier:
k = k[1:] # Drop one k=0 for msin
assert np.allclose(solver.eigenvalues, k**2)
def __init__(self, Nx, Ny, Lx, Ly, dtype, **kw):
self.Nx = Nx
self.Ny = Ny
self.Lx = Lx
self.Ly = Ly
self.dtype = dtype
self.N = Nx * Ny
self.R = 2 * Nx + 2 * Ny
self.M = self.N + self.R
# Bases
self.c = c = coords.CartesianCoordinates('x', 'y')
self.d = d = distributor.Distributor((c, ))
self.xb = xb = basis.ChebyshevT(c.coords[0], Nx, bounds=(0, Lx))
self.yb = yb = basis.ChebyshevT(c.coords[1], Ny, bounds=(0, Ly))
self.x = x = xb.local_grid(1)
self.y = y = yb.local_grid(1)
xb2 = xb._new_a_b(1.5, 1.5)
yb2 = yb._new_a_b(1.5, 1.5)
# Forcing
self.f = f = field.Field(name='f',
dist=d,
bases=(xb2, yb2),
dtype=dtype)
# Boundary conditions
self.uL = uL = field.Field(name='f', dist=d, bases=(yb, ), dtype=dtype)
self.uR = uR = field.Field(name='f', dist=d, bases=(yb, ), dtype=dtype)
self.uT = uT = field.Field(name='f', dist=d, bases=(xb, ), dtype=dtype)
self.uB = uB = field.Field(name='f', dist=d, bases=(xb, ), dtype=dtype)
# Fields
self.u = u = field.Field(name='u', dist=d, bases=(xb, yb), dtype=dtype)
self.tx1 = tx1 = field.Field(name='tx1',
dist=d,
bases=(xb2, ),
dtype=dtype)
self.tx2 = tx2 = field.Field(name='tx2',
dist=d,
bases=(xb2, ),
dtype=dtype)
self.ty1 = ty1 = field.Field(name='ty1',
dist=d,
bases=(yb2, ),
dtype=dtype)
self.ty2 = ty2 = field.Field(name='ty2',
dist=d,
bases=(yb2, ),
dtype=dtype)
# Problem
Lap = lambda A: operators.Laplacian(A, c)
self.problem = problem = problems.LBVP([u, tx1, tx2, ty1, ty2])
problem.add_equation((Lap(u), f))
problem.add_equation((u(y=Ly), uT))
problem.add_equation((u(x=Lx), uR))
problem.add_equation((u(y=0), uB))
problem.add_equation((u(x=0), uL))
# Solver
self.solver = solver = solvers.LinearBoundaryValueSolver(
problem,
bc_top=False,
tau_left=False,
store_expanded_matrices=False,
**kw)
# Tau entries
L = solver.subproblems[0].L_min.tolil()
# Taus
for nx in range(Nx):
L[Ny - 1 + nx * Ny, Nx * Ny + 0 * Nx + nx] = 1 # tx1 * Py1
L[Ny - 2 + nx * Ny, Nx * Ny + 1 * Nx + nx] = 1 # tx2 * Py2
for ny in range(Ny - 2):
L[(Nx - 1) * Ny + ny,
Nx * Ny + 2 * Nx + 0 * Ny + ny] = 1 # ty1 * Px1
L[(Nx - 2) * Ny + ny,
Nx * Ny + 2 * Nx + 1 * Ny + ny] = 1 # ty2 * Px2
# BC taus not resolution safe
if Nx != Ny:
raise ValueError("Current implementation requires Nx == Ny.")
else:
# Remember L is left preconditoined but not right preconditioned
L[-7, Nx * Ny + 2 * Nx + 0 * Ny + Ny - 2] = 1 # Right -2
L[-5, Nx * Ny + 2 * Nx + 0 * Ny + Ny - 1] = 1 # Right -1
L[-3, Nx * Ny + 2 * Nx + 1 * Ny + Ny - 2] = 1 # Left -2
L[-1, Nx * Ny + 2 * Nx + 1 * Ny + Ny - 1] = 1 # Left -1
solver.subproblems[0].L_min = L.tocsr()
# Neumann operators
ux = operators.Differentiate(u, c.coords[0])
uy = operators.Differentiate(u, c.coords[1])
self.duL = -ux(x='left')
self.duR = ux(x='right')
self.duT = uy(y='right')
self.duB = -uy(y='left')
# Neumann matrix
duT_mat = self.duT.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duR_mat = self.duR.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duB_mat = self.duB.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duL_mat = self.duL.expression_matrices(solver.subproblems[0],
vars=[u])[u]
self.interior_to_neumann_matrix = sp.vstack(
[duT_mat, duR_mat, duB_mat, duL_mat], format='csr')
c = coords.CartesianCoordinates('x', 'z')
d = distributor.Distributor((c,))
if dtype == np.complex128:
xb = basis.ComplexFourier(c.coords[0], size=Nx, bounds=(0, 2*np.pi), dealias=3/2)
elif dtype == np.float64:
xb = basis.RealFourier(c.coords[0], size=Nx, bounds=(0, 2*np.pi), dealias=3/2)
zb = basis.ChebyshevT(c.coords[1], size=Nz, bounds=(0, Lz),dealias=3/2)
x = xb.local_grid(1)
z = zb.local_grid(1)
zb1 = basis.ChebyshevU(c.coords[1], size=Nz, bounds=(0, Lz), alpha0=0)
t1 = field.Field(name='t1', dist=d, bases=(xb,), dtype=dtype)
t2 = field.Field(name='t2', dist=d, bases=(xb,), dtype=dtype)
P1 = field.Field(name='P1', dist=d, bases=(zb1,), dtype=dtype)
if rank == 0:
P1['c'][0,-1] = 1
dz = lambda A: operators.Differentiate(A, c.coords[1])
P2 = dz(P1).evaluate()
# Fields
u = field.Field(name='u', dist=d, bases=(xb,zb), dtype=dtype)
F = field.Field(name='F', dist=d, bases=(xb,zb), dtype=dtype)
u_true = 0
for n in np.arange(1,Nz//8):
u_current_n = 0
for m in np.arange(Nx//4):
u_current_m = np.sin(n*z)*np.cos(m*x)
u_current_n += u_current_m
F['g'] += -m**2*u_current_m
u_true += u_current_n
F['g'] += -n**2*u_current_n
# Problem
# Fields
u = field.Field(dist=d, bases=(zb, b), tensorsig=(c,),dtype=dtype)
w = field.Field(dist=d, bases=(zb, b), dtype=dtype)
p = field.Field(dist=d, bases=(zb, b), dtype=dtype)
tau_u = field.Field(dist=d, bases=(cb,), tensorsig=(c,), dtype=dtype)
tau_w = field.Field(dist=d, bases=(cb,), dtype=dtype)
# Parameters and operators
lap = lambda A: operators.Laplacian(A, c)
div = lambda A: operators.Divergence(A)
grad = lambda A: operators.Gradient(A, c)
curl = lambda A: operators.Curl(A)
dot = lambda A,B: arithmetic.DotProduct(A, B)
cross = lambda A,B: arithmetic.CrossProduct(A, B)
dt = lambda A: operators.TimeDerivative(A)
dz = lambda A: operators.Differentiate(A, zc)
# Problem
def eq_eval(eq_str):
return [eval(expr) for expr in split_equation(eq_str)]
problem = problems.IVP([u, w, p, tau_u, tau_w])
problem.add_equation(eq_eval("dt(u) - nu*(lap(u) + dz(dz(u))) - grad(p) = 0."), condition='nphi != 0')
problem.add_equation(eq_eval("dt(w) - nu*(lap(w) + dz(dz(w))) - dz(p) = 0."), condition='nphi != 0')
problem.add_equation(eq_eval("u = 0"), condition='nphi == 0')
problem.add_equation(eq_eval("w = 0"), condition='nphi == 0')
problem.add_equation(eq_eval("div(u) + dz(w) = 0"), condition='nphi != 0')
problem.add_equation(eq_eval("p = 0"), condition='nphi == 0')
problem.add_equation(eq_eval("u(r=1) = 0"), condition='nphi != 0')
problem.add_equation(eq_eval("tau_u = 0"), condition='nphi == 0')
problem.add_equation(eq_eval("w(r=1) = 0"), condition='nphi != 0')
def main(N, alpha, method, tau):
# Bases
c = coords.Coordinate('x')
d = distributor.Distributor((c,))
xb = basis.ChebyshevT(c, size=N, bounds=(-1, 1))
x = xb.local_grid(1)
# Fields
u = field.Field(name='u', dist=d, bases=(xb,), dtype=np.float64)
ux = field.Field(name='ux', dist=d, bases=(xb,), dtype=np.float64)
f = field.Field(name='f', dist=d, bases=(xb,), dtype=np.float64)
t1 = field.Field(name='t1', dist=d, dtype=np.float64)
t2 = field.Field(name='t2', dist=d, dtype=np.float64)
pi, sin, cos = np.pi, np.sin, np.cos
f['g'] = 64*pi**3*(4*pi*sin(4*pi*x)**2*sin(4*pi*cos(4*pi*x)) + cos(4*pi*x)*cos(4*pi*cos(4*pi*x))) + alpha*sin(4*pi*cos(4*pi*x))
# Tau polynomials
xb1 = xb._new_a_b(xb.a+1, xb.b+1)
xb2 = xb._new_a_b(xb.a+2, xb.b+2)
if tau == 0:
# First-order classical Chebyshev tau: T[-1], dx(T[-1])
p1 = field.Field(name='p1', dist=d, bases=(xb,), dtype=np.float64)
p2 = field.Field(name='p2', dist=d, bases=(xb,), dtype=np.float64)
p1['c'][-1] = 1
p2['c'][-1] = 1
elif tau == 1:
# First-order ultraspherical tau: U[-1], dx(U[-1])
p1 = field.Field(name='p1', dist=d, bases=(xb1,), dtype=np.float64)
p2 = field.Field(name='p2', dist=d, bases=(xb1,), dtype=np.float64)
p1['c'][-1] = 1
p2['c'][-1] = 1
# Problem
dx = lambda A: operators.Differentiate(A, c)
problem = problems.LBVP([u, ux, t1, t2])
problem.add_equation((alpha*u - dx(ux) + t1*p1, f))
problem.add_equation((ux - dx(u) + t2*p2, 0))
problem.add_equation((u(x=-1), 0))
problem.add_equation((u(x=+1), 0))
solver = solvers.LinearBoundaryValueSolver(problem)
# Methods
if method == 0:
# Condition number
L_exp = solver.subproblems[0].L_exp
result = np.linalg.cond(L_exp.A)
elif method == 1:
# Roundtrip roundoff with uniform u
A = solver.subproblems[0].L_exp
v = np.random.rand(A.shape[1])
f = A * v
u = solver.subproblem_matsolvers[solver.subproblems[0]].solve(f)
result = np.max(np.abs(u-v))
elif method == 2:
# Manufactured solution
from mpi4py_fft import fftw as mpi4py_fftw
solver.solve()
ue = np.sin(4*np.pi*np.cos(4*np.pi*x))
d = mpi4py_fftw.aligned(N, fill=0)
k = 2*(1 + np.arange((N-1)//2))
d[::2] = (2./N)/np.hstack((1., 1.-k*k))
w = mpi4py_fftw.aligned_like(d)
dct = mpi4py_fftw.dctn(w, axes=(0,), type=3)
weights = dct(d, w)
result = np.sqrt(np.sum(weights*(u['g']-ue)**2))
elif method == 3:
# Roundtrip roundoff with uniform f
A = solver.subproblems[0].L_exp
g = np.random.rand(A.shape[0])
v = solver.subproblem_matsolvers[solver.subproblems[0]].solve(g)
f = A * v
u = solver.subproblem_matsolvers[solver.subproblems[0]].solve(f)
result = np.max(np.abs(u-v))
return result