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 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 test_fourier_interpolation(show=False):
"""Test the inperpolation routine of FourierGrid"""
import numpy as np
from psecas import FourierGrid
N = 16
zmin = 0
zmax = np.pi * np.sqrt(2)
grid = FourierGrid(N, zmin, zmax)
grid_fine = FourierGrid(N * 4, zmin, zmax)
z = grid_fine.zg
# y = np.sin(5 * 2 * np.pi * grid.zg / grid.L)
# y_fine = np.sin(5 * 2 * np.pi * z / grid.L)
y = (np.sin(5 * 2 * np.pi * grid.zg / grid.L) *
np.cos(2 * np.pi * grid.zg / grid.L)**2)
y_fine = (np.sin(5 * 2 * np.pi * z / grid.L) *
np.cos(2 * np.pi * z / grid.L)**2)
y_interpolated = grid.interpolate(z, y)
if show:
import matplotlib.pyplot as plt
plt.figure(1)
plt.clf()
plt.title("Interpolation with Fourier series")
plt.plot(z, y_fine, "-")
plt.plot(z, y_interpolated, "--")
plt.plot(grid.zg, y, "+")
plt.show()
np.testing.assert_allclose(y_fine, y_interpolated, atol=1e-12)
return (y_fine, y_interpolated)
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_fourier_differentation(show=False):
"""Test the differentation routine of FourierGrid"""
import numpy as np
from psecas import FourierGrid
N = 256
zmin = 0
zmax = 2
grid = FourierGrid(128, -20, 30)
# Use the setter methods
grid.zmin = zmin
grid.zmax = zmax
grid.N = N
assert grid.zmin == zmin
assert grid.zmax == zmax
assert grid.N == N
assert grid.zg[0] < grid.zg[-1]
z = grid.zg
y = np.tanh((z - 1.5) / 0.05) - np.tanh((z - 0.5) / 0.05) + 1.0
yp_exac = (-np.tanh((z - 1.5) / 0.05)**2 / 0.05 + np.tanh(
(z - 0.5) / 0.05)**2 / 0.05)
yp_num = np.matmul(grid.d1, y)
np.testing.assert_allclose(yp_num, grid.der(y), atol=1e-16)
if show:
import matplotlib.pyplot as plt
plt.figure(1)
plt.clf()
plt.title("Differentation with matrix (FourierGrid)")
plt.plot(z, yp_exac, "-")
plt.plot(z, yp_num, "--")
plt.show()
np.testing.assert_allclose(yp_num, yp_exac, atol=1e-7)
return (yp_num, yp_exac)
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
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))
Berlok, T. & Pfrommer, C. (2019). *On the Kelvin-Helmholtz instability
with smooth initial conditions – Linear theory and simulations*, MNRAS,
485, 908
Another reference for the KHI with anisotropic viscosity is
Suzuki, K., Ogawa, T., Matsumoto, Y., & Matsumoto, R. (2013).
Magnetohydrodynamic simulations of the formation of cold fronts in
clusters of galaxies: Effects of anisotropic viscosity. Astrophysical
Journal, 768(2). https://doi.org/10.1088/0004-637X/768/2/175
"""
directory = './data/'
kx_global = np.linspace(3, 4, 5)
kx_local = kx_global[comm.rank :: comm.size]
grid = FourierGrid(N=64, zmin=0, zmax=2)
system = KelvinHelmholtzUniform(grid, beta=1e4, nu=1e-2, kx=0)
io = IO(system, directory, __file__, len(kx_global))
solver = Solver(grid, system)
for i in range(len(kx_local)):
t1 = time.time()
system.kx = kx_local[i]
omega, v = solver.solve()
io.save_system(i)
io.log(i, time.time() - t1, 'kx = {:1.4e}'.format(system.kx))
io.finished()
import numpy as np
from psecas import Solver, FourierGrid
from psecas.systems.kh_hydro import KelvinHelmholtzHydroOnly
import time
import matplotlib.pyplot as plt
# Initialize grid, system and solver for hydrodynamic KHI
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)
# Define a sorting strategy for the eigenvalues
# This function depends on the physical system
def sorting_strategy(E):
"""
The default sorting strategy.
"Large" real and imaginary eigenvalues are removed and the eigenvalues
are sorted from largest to smallest
"""
import numpy as np
# Set large values to zero, as they are presumably numerical artifact
# and unphysical.
E[np.abs(E.real) > 20.0] = 0
E[np.abs(E.imag) > 20.0] = 0
# Sort from largest to smallest eigenvalue
index = np.argsort(np.real(E))[::-1]
return (E, index)
"""
The Mathieu equation is given by
-uₓₓ + 2 q cos(2x) = σ u
where σ is the eigenvalue and q is a parameter.
We solve for u(x) and σ by using the Fourier grid.
This example is taken from the book *Spectral methods in Matlab*
by Lloyd Trefethen and this Python script reproduces the figure on
page 89.
"""
# Create grid
grid = FourierGrid(64, zmin=0, zmax=2*np.pi, z='x')
# Create the system
system = System(grid, variables='u', eigenvalue='sigma')
system.q = 1
# Add the first (and only) equation
system.add_equation("sigma*u = 2*q*np.cos(2*x)*u - dx(dx(u))")
# Create a solver object
solver = Solver(grid, system)
def sorting_strategy(E):
"""Sorting strategy. E is a list of eigenvalues"""
# Sort from smallest to largest eigenvalue
index = np.argsort(E)