def lane_emden(Nr, m=1.5, n_rho=3, radius=1,
ncc_cutoff = 1e-10, tolerance = 1e-10, dtype=np.complex128, comm=None):
# TO-DO: clean this up and make work for ncc ingestion in main script in np.float64 rather than np.complex128
c = de.SphericalCoordinates('phi', 'theta', 'r')
d = de.Distributor((c,), comm=comm, dtype=dtype)
b = de.BallBasis(c, (1, 1, Nr), radius=radius, dtype=dtype)
br = b.radial_basis
phi, theta, r = b.local_grids()
# Fields
f = d.Field(name='f', bases=b)
R = d.Field(name='R')
τ = d.Field(name='τ', bases=b.S2_basis(radius=radius))
# Parameters and operators
lap = lambda A: de.Laplacian(A, c)
lift_basis = b.clone_with(k=2) # match laplacian
lift = lambda A: de.LiftTau(A, lift_basis, -1)
problem = de.NLBVP([f, R, τ])
problem.add_equation((lap(f) + lift(τ), - R**2 * f**m))
problem.add_equation((f(r=0), 1))
problem.add_equation((f(r=radius), np.exp(-n_rho/m, dtype=dtype))) # explicit typing to match domain
# Solver
solver = problem.build_solver(ncc_cutoff=ncc_cutoff)
# Initial guess
f['g'] = np.cos(np.pi/2 * r)**2
R['g'] = 5
# Iterations
logger.debug('beginning Lane-Emden NLBVP iterations')
pert_norm = np.inf
while pert_norm > tolerance:
solver.newton_iteration()
pert_norm = sum(pert.allreduce_data_norm('c', 2) for pert in solver.perturbations)
logger.debug(f'Perturbation norm: {pert_norm:.3e}')
T = d.Field(name='T', bases=br)
ρ = d.Field(name='ρ', bases=br)
lnρ = d.Field(name='lnρ', bases=br)
T['g'] = f['g']
ρ['g'] = f['g']**m
lnρ['g'] = np.log(ρ['g'])
structure = {'T':T,'lnρ':lnρ}
for key in structure:
structure[key].require_scales(1)
structure['r'] = r
structure['problem'] = {'c':c, 'b':b, 'problem':problem}
return structure
zb = de.ChebyshevT(c.coords[1], size=nz, bounds=(0, Lz), dealias=dealias) b = (xb, zb) x = xb.local_grid(1) z = zb.local_grid(1) # Fields T = d.Field(name='T', bases=b) Υ = d.Field(name='Υ', bases=b) s = d.Field(name='s', bases=b) u = d.VectorField(c, name='u', bases=b) # Taus zb1 = zb.clone_with(a=zb.a + 1, b=zb.b + 1) zb2 = zb.clone_with(a=zb.a + 2, b=zb.b + 2) lift_basis = zb.clone_with(a=1 / 2, b=1 / 2) # First derivative basis lift = lambda A, n: de.LiftTau(A, lift_basis, n) τs1 = d.Field(name='τs1', bases=xb) τs2 = d.Field(name='τs2', bases=xb) τu1 = d.VectorField(c, name='τu1', bases=(xb, )) τu2 = d.VectorField(c, name='τu2', bases=(xb, )) # Parameters and operators div = lambda A: de.Divergence(A, index=0) lap = lambda A: de.Laplacian(A, c) grad = lambda A: de.Gradient(A, c) #curl = lambda A: de.operators.Curl(A) dot = lambda A, B: de.DotProduct(A, B) cross = lambda A, B: de.CrossProduct(A, B) trace = lambda A: de.Trace(A) trans = lambda A: de.TransposeComponents(A) dt = lambda A: de.TimeDerivative(A)
xb = de.RealFourier(c.coords[0], size=nx, bounds=(0, Lx), dealias=dealias) zb = de.ChebyshevT(c.coords[1], size=nz, bounds=(0, Lz), dealias=dealias) b = (xb, zb) x = xb.local_grid(1) z = zb.local_grid(1) # Fields θ = d.Field(name='θ', bases=b) Υ = d.Field(name='Υ', bases=b) s = d.Field(name='s', bases=b) u = d.VectorField(c, name='u', bases=b) # Taus zb1 = zb.clone_with(a=zb.a + 1, b=zb.b + 1) zb2 = zb.clone_with(a=zb.a + 2, b=zb.b + 2) lift1 = lambda A, n: de.LiftTau(A, zb1, n) lift = lambda A, n: de.LiftTau(A, zb2, n) τs1 = d.Field(name='τs1', bases=xb) τs2 = d.Field(name='τs2', bases=xb) τu1 = d.VectorField(c, name='τu1', bases=(xb, )) τu2 = d.VectorField(c, name='τu2', bases=(xb, )) # Parameters and operators div = lambda A: de.Divergence(A, index=0) lap = lambda A: de.Laplacian(A, c) grad = lambda A: de.Gradient(A, c) #curl = lambda A: de.operators.Curl(A) dot = lambda A, B: de.DotProduct(A, B) cross = lambda A, B: de.CrossProduct(A, B) trace = lambda A: de.Trace(A) trans = lambda A: de.TransposeComponents(A)