def test_kh_uniform_solution(show=False, verbose=False):
"""Test eigenvalue solver using FourierGrid"""
import numpy as np
from psecas import Solver, FourierGrid
from psecas.systems.kh_uniform import KelvinHelmholtzUniform
grid = FourierGrid(N=64, zmin=0, zmax=2)
system = KelvinHelmholtzUniform(grid, beta=1e4, nu=1e-2, kx=3.52615254237)
system.u0 = 1.
solver = Solver(grid, system)
Ns = np.hstack((np.arange(2, 16) * 32, np.arange(2, 12) * 64))
for useOPinv in [True, False]:
omega, v, err = solver.iterate_solver(Ns,
tol=1e-5,
useOPinv=useOPinv,
verbose=verbose)
np.testing.assert_allclose(1.66548246011, omega, atol=1e-5)
if show:
from psecas import plot_solution
plot_solution(system, smooth=True, num=2)
# Check that information converged is set to False when solver does not converge
solver.iterate_solver(np.arange(1, 3) * 16,
tol=1e-16,
useOPinv=useOPinv,
verbose=verbose)
assert solver.system.result['converged'] is False
return err
def test_channel(show=False, verbose=False):
"""Test eigenvalue solver using ChebyshevRationalGrid"""
import numpy as np
from psecas import Solver, ChebyshevRationalGrid, System
grid = ChebyshevRationalGrid(N=199, z='z')
class Channel(System):
def make_background(self):
import numpy as np
zg = self.grid.zg
self.h = np.exp(-zg**2 / 2)
# Create the Channel system
system = Channel(grid, variables='G', eigenvalue='K2')
# Add the first (and only) equation
system.add_equation("-h*K2*G = dz(dz(G)) +z*dz(G)", boundary=True)
solver = Solver(grid, system, do_gen_evp=True)
# Number of modes to test
modes = 3
results = np.zeros(modes, dtype=np.complex128)
checks = np.array([85.08037778, 69.4741069099, 55.4410282999])
def sorting_strategy(E):
"""Sorting strategy for channel modes"""
E[E.real > 100.0] = 0
E[E.real < -10.0] = 0
index = np.argsort(np.real(E))[::-1]
return (E, index)
solver.sorting_strategy = sorting_strategy
if show:
import matplotlib.pyplot as plt
plt.figure(3)
plt.clf()
fig, axes = plt.subplots(num=3, ncols=modes, sharey=True)
for mode in range(modes):
Ns = np.arange(1, 6) * 32
omega, vec, err = solver.iterate_solver(Ns, mode=mode, verbose=True)
results[mode] = omega
if show:
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode)
axes[mode].set_title(r"$\sigma = ${:1.4f}".format(omega.real),
fontsize=10)
axes[mode].plot(grid.zg, system.result['G'].real)
axes[mode].plot(grid.zg, system.result['G'].imag)
axes[mode].set_xlim(-4, 4)
if show:
plt.show()
np.testing.assert_allclose(results, checks, rtol=1e-6)
def f(kx):
grid = FourierGrid(N=64, zmin=0.0, zmax=2.0)
system = KelvinHelmholtzHydroOnly(grid, u0=1.0, delta=1.0, kx=kx)
solver = Solver(grid, system)
Ns = np.hstack((np.arange(1, 5) * 16, np.arange(3, 12) * 32))
omega, v, err = solver.iterate_solver(Ns, verbose=False, tol=1e-8)
return -omega.real
def f(kx):
grid = FourierGrid(N=64, zmin=0.0, zmax=2.0)
system = KelvinHelmholtzUniform(grid, beta=5, nu=0, kx=kx)
solver = Solver(grid, system)
Ns = np.hstack((np.arange(4, 5) * 16, np.arange(3, 16) * 32))
omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-8)
return -omega.real
def f(kx, **kwargs):
# Set up a grid
grid = FourierGrid(N=64, zmin=0, zmax=2)
system = KelvinHelmholtzUniform(grid, beta=1e3, nu=0, kx=kx)
if 'nu' in kwargs.keys():
system.nu = kwargs['nu']
# Set up a solver
solver = Solver(grid, system)
# Iteratively solve
Ns = np.hstack((np.arange(1, 5) * 16, np.arange(3, 12) * 32))
omega, v, err = solver.iterate_solver(Ns, verbose=False, tol=1e-4)
return -omega.real
def test_mti_solution(show=False, verbose=False):
"""Test eigenvalue solver using ChebyshevExtremaGrid"""
import numpy as np
from psecas import Solver, ChebyshevExtremaGrid
from psecas.systems.mti import MagnetoThermalInstability
grid = ChebyshevExtremaGrid(N=64, zmin=0, zmax=1)
system = MagnetoThermalInstability(grid, beta=1e5, Kn0=200, kx=4 * np.pi)
system.beta = 1e5
system.Kn0 = 200
assert system.Kn0 == 200
assert system.beta == 1e5
solver = Solver(grid, system)
Ns = np.hstack(np.arange(1, 10) * 16)
mode = 0
omega, vec, err = solver.iterate_solver(Ns,
mode=mode,
tol=1e-8,
verbose=verbose)
system.get_bx_and_by()
if show:
from psecas import plot_solution
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
plot_solution(system, smooth=True, num=1)
np.testing.assert_allclose(1.7814514515967603, omega, atol=1e-8)
return err
a = 3.512831867406509
b = 3.512831875508205
(a, b) = golden_section(f, a, b, tol=1e-8)
# Create initial conditions for Athena simulation
if True:
from psecas import save_system
kxmax = 3.5128319
grid = FourierGrid(N=256, zmin=0.0, zmax=2.0)
system = KelvinHelmholtzHydroOnly(grid, u0=1.0, delta=1.0, kx=kxmax)
solver = Solver(grid, system)
Ns = np.hstack((np.arange(1, 5) * 16, np.arange(3, 12) * 32))
omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-8)
# Normalize eigenmodes
y = np.vstack([
system.result["dvx"].real,
system.result["dvx"].imag,
system.result["dvz"].real,
system.result["dvz"].imag,
])
val = np.max(np.abs(y))
for key in system.variables:
system.result[key] /= val
# Save system pickle object
save_system(system, "./khi_hydro_delta.p")
# Overwrite the default sorting strategy in the Solver class
ch.sorting_strategy = sorting_strategy
plt.figure(1)
plt.clf()
modes = 11
fig, axes = plt.subplots(num=1,
nrows=2,
ncols=modes,
sharex=True,
sharey='row')
for mode in range(modes):
Ns = np.hstack((np.arange(3, 6) * 32, np.arange(4, 12) * 64))
K2, vec, err = ch.iterate_solver(Ns,
mode=mode,
tol=1e-4,
verbose=True,
guess_tol=5e-3)
K = np.sqrt(K2.real)
phi = np.arctan(vec[2].imag / vec[2].real)
A = np.max(np.abs(vec))
ch.keep_result(K2, vec * np.exp(-1j * phi) / A, mode=mode)
axes[0, mode].set_title(r"$\sigma = ${:1.4f}".format(K), fontsize=10)
axes[0, mode].plot(grid.zg, system.result["F"].real)
axes[0, mode].plot(grid.zg, system.result["F"].imag)
axes[0, mode].set_xlim(-4, 4)
axes[1, mode].set_xlabel(r'$z$')
G = grid.der(system.result['F']) / K
axes[1, mode].plot(grid.zg, G.real)
axes[1, mode].plot(grid.zg, G.imag)
from psecas.systems.kh_uniform import KelvinHelmholtzUniform
"""
This example shows how the eigenmodes can be stored in text format which
can be loaded into the MHD code Athena (Stone, J. et al, 2008).
It also shows how to save a system using Python's pickle package.
"""
# Set up a grid
grid = FourierGrid(N=64, zmin=0, zmax=2)
# Set up the system of equations
kx = 4.627762711864407
# kx = 2*np.pi
system = KelvinHelmholtzUniform(grid, beta=1e3, nu=1e-2, kx=kx)
# Set up a solver
solver = Solver(grid, system)
# Iteratively solve
Ns = np.hstack((np.arange(1, 5) * 32, np.arange(3, 12) * 64))
solver.iterate_solver(Ns, verbose=True, tol=1e-10)
# Write files for loading into Athena
write_athena(system, Nz=256, Lz=2.0)
# Write directly to the Athena directory
write_athena(system, Nz=256, Lz=2.0, path='/Users/berlok/codes/athena/bin/')
save_system(system, '/Users/berlok/codes/athena/bin/kh-visc.p')
The solver class in Psecas contains a different method, iterate_solver, which increases N until one eigenvalue agrees with the value found at the previous value of N (to within relative and absolute tolerances, tol & atol). This method can only find one eigenmode at a time (unlike the solve method which finds all eigenvalues but is unable to check for convergence and can be very expensive to call at high N). Most often, one is only interested in the fastest growing mode (or the first few). This can be controlled with the input 'mode' in the call to iterate_solver. """ # Let us find converged solutions # Define a range of grid resolutions to try Ns = np.hstack((np.arange(1, 5) * 32, np.arange(3, 20) * 64)) # Solve for the fastest growing mode (mode=0) omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-6, mode=0) # Solve for the second fastest growing mode (mode=1) omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-6, mode=1) """ The sorting_strategy function is necessary to weed out the worst of the numerical eigenvalues. For some physical systems, infinite or nan solutions are returned. These are discarded by the sorting_strategy function. The solver class is initialized with a standard sorting_strategy that has worked well on the problems I have considered so far. One should however be careful not to remove physical eigenvalues and for some problems it is probably necessary to define a new sorting_strategy. """
system.add_equation(
"-r*r*sigma*bphi = - 1j*kz*r*r*vphi - 1j*kz*q*r**(2-q)*Aphi - eta*(Drbphi) - lh*va*1j*kz*(DrAphi)"
)
# The boundary conditions
Aphi_bound = 'r**2*dr(dr(Aphi)) + r*dr(Aphi) - Aphi = 0'
system.add_boundary('vr', 'Dirichlet', 'Dirichlet')
system.add_boundary('vphi', 'Dirichlet', 'Dirichlet')
system.add_boundary('vz', 'Neumann', 'Neumann')
system.add_boundary('Aphi', Aphi_bound, Aphi_bound)
system.add_boundary('bphi', 'Dirichlet', 'Dirichlet')
system.add_substitution(
'DrAphi = r*r*dr(dr(Aphi)) + r*dr(Aphi) - Aphi - kz**2*r*r*Aphi')
system.add_substitution(
'Drbphi = r*r*dr(dr(bphi)) + r*dr(bphi) - bphi - kz**2*r*r*bphi')
solver = Solver(grid, system)
mode = 0
Ns = np.arange(1, 32) * 32 - 1
omega, vec, err = solver.iterate_solver(Ns, mode=mode, verbose=True, tol=1e-4)
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
if isinstance(grid, ChebyshevTLnGrid):
plot_solution(system, limits=[0, 1.5])
plt.xlim(0, 1.5)
else:
plot_solution(system)
def test_mri_solution(show=False, verbose=False):
"""Test eigenvalue solver using ChebyshevExtremaGrid"""
import numpy as np
from psecas import Solver, ChebyshevExtremaGrid, System
class HallMRI(System):
def __init__(self, grid, kz, variables, eigenvalue):
# Set parameters
self.q = 1.5
self.eta = 0.003
self.lh = 1
self.h = 0.25
self.va = 0.002
self.kz = kz
super().__init__(grid, variables, eigenvalue)
# Create a grid
grid = ChebyshevExtremaGrid(N=128, zmin=1, zmax=2, z='r')
variables = ['rho', 'vr', 'vphi', 'vz', 'Aphi', 'bphi']
kz = 2 * np.pi
# Create the system
system = HallMRI(grid, kz, variables=variables, eigenvalue='sigma')
# The linearized equations
system.add_equation("-r*sigma*rho = r*dr(vr) + vr + 1j*kz*r*vz")
system.add_equation(
"-r*r*sigma*vr = - 2*r**(2-q)*vphi + h**2*r*r*dr(rho) + va**2*(DrAphi)"
)
system.add_equation("-sigma*vphi = + (2-q)*r**(-q)*vr - va**2*1j*kz*bphi")
system.add_equation("-sigma*vz = h**2*1j*kz*rho")
system.add_equation(
"-r*r*sigma*Aphi = + r*r*vr - eta*(DrAphi) + lh*va*1j*kz*r*r*bphi")
system.add_equation(
"-r*r*sigma*bphi = - 1j*kz*r*r*vphi - 1j*kz*q*r**(2-q)*Aphi - eta*(Drbphi) - lh*va*1j*kz*(DrAphi)"
)
# The boundary conditions
Aphi_bound = 'r**2*dr(dr(Aphi)) + r*dr(Aphi) - Aphi = 0'
system.add_boundary('vr', 'Dirichlet', 'Dirichlet')
system.add_boundary('vphi', 'Dirichlet', 'Dirichlet')
system.add_boundary('vz', 'Neumann', 'Neumann')
system.add_boundary('Aphi', Aphi_bound, Aphi_bound)
system.add_boundary('bphi', 'Dirichlet', 'Dirichlet')
# Short hands for long expressions for derivatives
system.add_substitution(
'DrAphi = r*r*dr(dr(Aphi)) + r*dr(Aphi) - Aphi - kz**2*r*r*Aphi')
system.add_substitution(
'Drbphi = r*r*dr(dr(bphi)) + r*dr(bphi) - bphi - kz**2*r*r*bphi')
solver = Solver(grid, system)
mode = 0
Ns = np.hstack(np.arange(1, 10) * 32)
omega, vec, err = solver.iterate_solver(Ns,
mode=mode,
tol=1e-8,
verbose=verbose)
if show:
from psecas import plot_solution
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
plot_solution(system, smooth=True, num=1)
np.testing.assert_allclose(0.09892641, omega, atol=1e-8)
return err
self.dvdz = lambdify(z, dvdz_sym)(zg)
self.d2vdz = lambdify(z, d2vdz_sym)(zg)
if __name__ == '__main__':
import numpy as np
from psecas import Solver, LegendreExtremaGrid
import matplotlib.pyplot as plt
from psecas import plot_solution
grid = LegendreExtremaGrid(N=200, zmin=-0.5, zmax=0.5)
system = KelvinHelmholtzMiura(grid, kx=2*np.pi, B=0.2, z1=0, a=1/25.0)
solver = Solver(grid, system)
Ns = np.arange(1, 8) * 64
omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-8)
print('\nB={:1.2f} has gammma*2a = {:1.6f}'.format(system.B,
omega.real*system.a*2))
plot_solution(system, num=1, filename='Ryu_and_Frank1996.pdf')
# Make figure 4 in Miura & Pritchett (1982)
# This calculation takes some time.
if True:
# The iterative solver is more precise but slow at high kx
# Set fast=True to instead run with a fixed grid size and get a figure
# much faster. The result looks okay but is imprecise at high k!
fast = False
return -omega.real
(a, b) = golden_section(f, 3.512295, 3.513135, tol=1e-5)
# Create initial conditions for Athena simulation
if False:
from psecas import write_athena, save_system
kxmax = 3.5128286141291243
grid = FourierGrid(N=64, zmin=0.0, zmax=2.0)
system = KelvinHelmholtzHydroOnly(grid, u0=1.0, delta=1.0, kx=kxmax)
solver = Solver(grid, system)
Ns = np.hstack((np.arange(1, 5) * 16, np.arange(3, 12) * 32))
omega, v, err = solver.iterate_solver(Ns, verbose=False, tol=1e-6)
# Write files for loading into Athena
# write_athena(system, Nz=256, Lz=2.0)
# Write directly to the Athena directory
write_athena(
system, Nz=256, Lz=2.0, path='/Users/berlok/codes/athena/bin/'
)
save_system(system, '/Users/berlok/codes/athena/bin/kh-with-delta.p')
plt.figure(1)
plt.plot(kxmax, omega.real, '+')
Lx = 2 * np.pi / system.kx
"""
grid = ChebyshevRationalGrid(N=32, C=0.4)
u0 = 1.0
delta = 1.0
kx = 5.1540899
kx = 3.5128310
system = KelvinHelmholtzHydroOnlySlab(grid, u0, delta, kx)
system.boundaries = [True, True, True, True]
solver = Solver(grid, system)
Ns = np.arange(1, 32) * 32 - 1
omega, vec, err = solver.iterate_solver(Ns, mode=0, verbose=True)
xmin = 0
xmax = 2 * np.pi / kx
zmin = -4
zmax = 4
Nx = 256
Nz = 1024
plt.rc("image", origin="lower", cmap="RdBu", interpolation="None")
extent = [xmin, xmax, zmin, zmax]
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=0)