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numerical_methods.py
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""" Numerical methods other than the 1st order upwind scheme. """
import numpy as np
def apply_numerical_method(r_initial, dr_vec, dp_vec, r0=30 * 695700, alpha=0.15, rh=50 * 695700, add_v_acc=True,
omega_rot=(2 * np.pi) / (25.38 * 86400), numerical_method="upwind_first_maccormack",
flux_function="vanleer"):
"""Apply a numerical method to solve the solar wind problem.
r/phi grid. return and save all radial velocity slices.
:param r_initial: 1d array, initial condition (vr0). units = (km/sec).
:param dr_vec: 1d array, mesh spacing in r. units = (km)
:param dp_vec: 1d array, mesh spacing in p. units = (radians)
:param alpha: float, hyper parameter for acceleration (default = 0.15).
:param rh: float, hyper parameter for acceleration (default r=50*695700). units: (km)
:param r0: float, initial radial location. units = (km).
:param add_v_acc: bool, True will add acceleration boost.
:param omega_rot: differential rotation.
:param numerical_method: specify the numerical method used (str).
:param flux_function: a flux-limiter function for high-low res solutions.
:return: velocity matrix dimensions (nr x np)
"""
v = np.zeros((len(dr_vec) + 1, len(dp_vec) + 1)) # initialize array vr.
v[0, :] = r_initial
if add_v_acc:
v_acc = alpha * (v[0, :] * (1 - np.exp(-r0 / rh)))
v[0, :] = v_acc + v[0, :]
for i in range(len(dr_vec)):
for j in range(len(dp_vec) + 1):
if j == len(dp_vec): # force periodicity
v[i + 1, j] = v[i + 1, 0]
else:
if (omega_rot * dr_vec[i]) / (dp_vec[j] * v[i, j]) > 1:
print(dr_vec[i] - dp_vec[j] * v[i, j] / omega_rot)
print(i, j) # courant condition
elif numerical_method == "second_order_upwind":
if j == len(dp_vec) - 1:
frac1 = (4 * v[i, j + 1] - 3 * v[i, j] - v[i, -1]) / (v[i, j])
else:
frac1 = (4 * v[i, j + 1] - 3 * v[i, j] - v[i, j + 2]) / (v[i, j])
frac2 = (omega_rot * dr_vec[i]) / (2 * dp_vec[j])
v[i + 1, j] = v[i, j] + frac1 * frac2
elif numerical_method == "second_order_upwind_conservative":
if j == len(dp_vec) - 1:
frac1 = (4 * np.log(v[i, j + 1]) - 3 * np.log(v[i, j]) - np.log(v[i, -1]))
else:
frac1 = (4 * np.log(v[i, j + 1]) - 3 * np.log(v[i, j]) - np.log(v[i, j + 2]))
frac2 = (omega_rot * dr_vec[i]) / (2 * dp_vec[j])
v[i + 1, j] = v[i, j] + frac1 * frac2
elif numerical_method == "third_order_upwind":
if j == len(dp_vec) - 1:
frac1 = (-v[i, -2] + 6 * v[i, -1] - 3 * v[i, j] - 2 * v[i, j - 1]) / (v[i, j])
elif j == len(dp_vec) - 2:
frac1 = (-v[i, -1] + 6 * v[i, j + 1] - 3 * v[i, j] - 2 * v[i, j - 1]) / (v[i, j])
else:
frac1 = (-v[i, j + 2] + 6 * v[i, j + 1] - 3 * v[i, j] - 2 * v[i, j - 1]) / (v[i, j])
frac2 = (omega_rot * dr_vec[i]) / (6 * dp_vec[j])
v[i + 1, j] = v[i, j] + frac1 * frac2
elif numerical_method == "third_order_upwind_conservative":
if j == len(dp_vec) - 1:
frac1 = (-np.log(v[i, -2]) + 6 * np.log(v[i, -1]) - 3 * np.log(v[i, j]) - 2 * np.log(
v[i, j - 1]))
elif j == len(dp_vec) - 2:
frac1 = (-np.log(v[i, -1]) + 6 * np.log(v[i, j + 1]) - 3 * np.log(v[i, j]) - 2 * np.log(
v[i, j - 1]))
else:
frac1 = (-np.log(v[i, j + 2]) + 6 * np.log(v[i, j + 1]) - 3 * np.log(v[i, j]) - 2 * np.log(
v[i, j - 1]))
frac2 = (omega_rot * dr_vec[i]) / (6 * dp_vec[j])
v[i + 1, j] = v[i, j] + frac1 * frac2
elif numerical_method == "conservative_upwind":
frac1 = (np.log(v[i, j + 1]) - np.log(v[i, j]))
frac2 = (omega_rot * dr_vec[i]) / (dp_vec[j])
v[i + 1, j] = v[i, j] + frac2 * frac1
elif numerical_method == "maccormack":
nu = (omega_rot * dr_vec[i]) / (dp_vec[j])
v_star_curr = v[i, j] + nu * (np.log(v[i, j + 1]) - np.log(v[i, j]))
v_star_prev = v[i, j - 1] + nu * (np.log(v[i, j]) - np.log(v[i, j - 1]))
v[i + 1, j] = 0.5 * (v[i, j] + v_star_curr) + (nu / 2) * (np.log(v_star_curr) - np.log(v_star_prev))
elif numerical_method == "lax_wendroff":
# coefficient
nu = (omega_rot * dr_vec[i]) / (dp_vec[j])
# v(j + 1/2)
v_star_curr = 0.5 * (v[i, j + 1] + v[i, j]) + (nu / 2) * (np.log(v[i, j + 1]) - np.log(v[i, j]))
# v(j - 1/2)
v_star_prev = 0.5 * (v[i, j] + v[i, j - 1]) + (nu / 2) * (np.log(v[i, j]) - np.log(v[i, j - 1]))
v[i + 1, j] = v[i, j] + nu * (np.log(v_star_curr) - np.log(v_star_prev))
elif numerical_method == "lax_friedrichs":
nu = (omega_rot * dr_vec[i]) / (dp_vec[j])
v[i + 1, j] = 0.5 * (v[i, j - 1] + v[i, j + 1]) + \
(nu / 2) * (np.log(v[i, j + 1]) - np.log(v[i, j - 1]))
elif numerical_method == "upwind_first_maccormack":
# first order upwind method (conservative)
f_lower_curr = -omega_rot * np.log(v[i, j + 1])
f_lower_prev = -omega_rot * np.log(v[i, j])
# McCormack's method
nu = (omega_rot * dr_vec[i]) / (dp_vec[j])
v_star_curr = v[i, j] + nu * (np.log(v[i, j + 1]) - np.log(v[i, j]))
v_star_prev = v[i, j - 1] + nu * (np.log(v[i, j]) - np.log(v[i, j - 1]))
f_upper_curr = 0.5 * (f_lower_curr - omega_rot * np.log(v_star_curr))
f_upper_prev = 0.5 * (f_lower_prev - omega_rot * np.log(v_star_prev))
# evaluate the smoothness of the current wave.
theta = (v[i, j] - v[i, j - 1]) / (v[i, j + 1] - v[i, j])
# limiter function
phi = limiter_function(theta=theta, limiter=flux_function)
final_flux_curr = f_lower_curr + phi * (f_upper_curr - f_lower_curr)
final_flux_prev = f_lower_prev + phi * (f_upper_prev - f_lower_prev)
v[i + 1, j] = v[i, j] - (dr_vec[i] / dp_vec[j]) * (final_flux_curr - final_flux_prev)
elif numerical_method == "upwind_first_lax_wendroff":
# first order upwind method (conservative)
f_lower_curr = -omega_rot * np.log(v[i, j + 1])
f_lower_prev = -omega_rot * np.log(v[i, j])
# Lax-Wendroff method
nu = (omega_rot * dr_vec[i]) / (dp_vec[j])
# v(j + 1/2)
v_star_curr = 0.5 * (v[i, j + 1] + v[i, j]) + (nu / 2) * (np.log(v[i, j + 1]) - np.log(v[i, j]))
# v(j - 1/2)
v_star_prev = 0.5 * (v[i, j] + v[i, j - 1]) + (nu / 2) * (np.log(v[i, j]) - np.log(v[i, j - 1]))
f_upper_curr = -omega_rot * np.log(v_star_curr)
f_upper_prev = -omega_rot * np.log(v_star_prev)
# evaluate the smoothness of the current wave.
theta = (v[i, j] - v[i, j - 1]) / (v[i, j + 1] - v[i, j])
# limiter function "superbee"
phi = limiter_function(theta=theta, limiter=flux_function)
final_flux_curr = f_lower_curr + phi * (f_upper_curr - f_lower_curr)
final_flux_prev = f_lower_prev + phi * (f_upper_prev - f_lower_prev)
v[i + 1, j] = v[i, j] - (dr_vec[i] / dp_vec[j]) * (final_flux_curr - final_flux_prev)
return v
def limiter_function(theta, limiter="minmod"):
""" return a flux-limiter-function result."""
if limiter == "vanleer":
return (np.abs(theta) + theta) / (1 + np.abs(theta))
elif limiter == "minmod":
return max(0., min(1., theta))
elif limiter == "superbee":
return max(0., min(1., 2*theta), min(theta, 2))
elif limiter == "mc":
return max(0., min((1 + theta)/2, 2, 2*theta))