def task18():
""" task 1.8 """
def exactReduced(t, X0):
""" exact solution reduced units"""
sol = np.zeros([np.size(t), 3])
E = 0.1
B = 1
v_ExB = E / B
sol[:, 0] = v_ExB / X0[3] * t + (1 - v_ExB / X0[3]) * np.sin(t)
sol[:, 1] = np.sign(q) * (1 - v_ExB / X0[3]) * (np.cos(t) - 1)
sol[:, 2] = X0[5] / X0[3] * t
return sol
euler = False
midpoint = False
RK4 = True
# initial values
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
B = Bfield(0, 0, 0, 0)
E = Efield(0, 0, 0, 0)
q = elementaryCharge
m = protonMass
h = 0.005
X0 = [0, 0, 0, 1, 0, 1] # initial state [X, V]
omega_c = abs(q) * np.linalg.norm(B) / m
r_L = X0[3] / omega_c
t = np.arange(0, 20, h)
# Exact solution
solution_exact = exactReduced(t, X0)
# Numerical solution
sol = odesolver2(f, X0, t, h)
solution_RK4 = sol.RK4()
feil_RK4 = np.max(
np.linalg.norm(solution_RK4[:, 0:3], axis=1) -
np.linalg.norm(solution_exact, axis=1))
print('Numerical error using RK4: {}'.format(feil_RK4))
fil = h5.File('task18_results_RK4.h5', 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
fil = h5.File('task18_results_exa.h5', 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
def task19():
def kineticEnergy(t, X0, m, q):
""" Kinetic energy constant E and B """
K0 = 0.5 * (X0[3]**2 + X0[5]**2)
E = 0.1
B = 1
v_ExB = E / B
K = K0 + np.sign(q) * E * (1 - v_ExB / X0[3]) * (np.cos(t) - 1)
return K
def kineticEnergyNum(t, X, m):
""" Kinetic energy numerical sol """
v = X[:, 3::]
v = np.linalg.norm(v, axis=1)
return 0.5 * v**2
# initial values
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
B = Bfield(0, 0, 0, 0)
E = Efield(0, 0, 0, 0)
q = elementaryCharge
m = protonMass
h = 0.1
X0 = [0, 0, 0, 1, 0, 1] # initial state [X, V]
t = np.arange(0, 5, h)
omega_c = abs(q) * np.linalg.norm(B) / m
r_L = X0[3] / omega_c
K = kineticEnergy(t, X0, m, q)
# Numerical solution
sol = odesolver2(f, X0, t, h)
solution_RK4 = sol.RK4()
K_num = kineticEnergyNum(t, solution_RK4, m)
solution_euler = sol.euler()
K_num_euler = kineticEnergyNum(t, solution_euler, m)
fig = plt.figure()
plt.plot(t, K, label='Exact')
plt.plot(t, K_num, '.', label='RK4')
plt.plot(t, K_num_euler, 's', alpha=0.2, label='Euler')
plt.xlabel('t')
plt.title('Kinetic Energy')
plt.legend()
plt.show()
def task15(stepsize):
""" Task 1.5 """
euler = True
midpoint = True
RK4 = True
# initial values
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
h = stepsize
X0 = [0, 0, 0, 1, 0, 1] # initial state [X, V]
t = np.arange(0, 20, h)
tic = time.time()
print('h = {} N: {} time: '.format(round(h, 4), np.size(t)), end='\r')
sol = odesolver2(f, X0, t, h)
if euler:
solution_euler = sol.euler()
if midpoint:
solution_mid = sol.midpoint()
if RK4:
solution_RK4 = sol.RK4()
print('h = {} N: {} time: {}'.format(round(h, 4), np.size(t),
round((time.time() - tic), 5)))
if euler:
fil = h5.File('results_euler_h{}.h5'.format(round(h, 4)), 'w')
fil.create_dataset('results', data=solution_euler)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
if midpoint:
fil = h5.File('results_midpoint_h{}.h5'.format(round(h, 4)), 'w')
fil.create_dataset('results', data=solution_mid)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
if RK4:
fil = h5.File('results_RK4_h{}.h5'.format(round(h, 4)), 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
def task112():
""" Task 1.12 """
# initial values
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
h = 0.005
X0 = [0, 0, 0, 1, 0, 0] # initial state [X, V]
t = np.arange(0, 40, h)
sol = odesolver2(f, X0, t, h)
solution_RK4 = sol.RK4()
'''
Man må manulelt gå inn i f() å endre fortegn
'''
fil = h5.File('task112_results_RK4_pos.h5', 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
def earth():
"""earth's radiationbelt"""
def Bfield(x, y, z, t):
"""B-field"""
r = (x * x + y * y + z * z)**(0.5)
Bx = -3 * x * z / (r**5)
By = -3 * y * z / (r**5)
Bz = -(1 / (r**5)) * (2 * z * z - x * x - y * y)
return np.array([Bx, By, Bz])
def f(X, t, dothething=False):
""" X' = f(X, t) """
# X = [x1 x2 x3 x4 x5 x6] = [x y z v_x v_y v_z]
# sign = 1
B = Bfield(X[0], X[1], X[2], t)
if dothething:
normB.append(np.linalg.norm(B))
kinetic_energy.append(X[3]**2 + X[4]**2 + X[5]**2)
newstate = np.array([
X[3], X[4], X[5], alpha * (X[4] * B[2] - X[5] * B[1]),
alpha * (X[5] * B[0] - X[3] * B[2]),
alpha * (X[3] * B[1] - X[4] * B[0])
])
# newstate += np.array([0, 0, 0, E[0], E[1], E[2]])
print('time: {}/{}'.format(round(t, 1), totalTime), end='\r')
if dothething:
vnorm = X[3::] - np.inner(X[3::], B) * B / (normB[-1]**2)
mu.append(-(np.linalg.norm(vnorm)**2) / normB[-1])
return newstate
radiusEarh = 6378000
B0 = 30.7e-6
c0 = 299792458
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
omega = c0 / radiusEarh
omega2 = B0 * elementaryCharge / protonMass
alpha = omega2 / omega
normB = []
mu = []
kinetic_energy = []
#h = 0.01
h = 0.1
totalTime = 10000
X0 = [0, 4, 0, 0.08 * np.sqrt(2) / 2, 0, 0.08 * np.sqrt(2) / 2]
# X0 = [6, 0, 0, 0, np.sqrt(2)/2, np.sqrt(2)/2] # initial state [X, V]
#X0 = [6, 0, 0, 1, 0, 0] # initial state [X, V]
t = np.arange(0, totalTime, h)
tic = time.time()
sol = odesolver2(f, X0, t, h)
solution_RK4 = sol.RK4()
#solution_RK4 = odeint(f, X0, t)
print('time: {}'.format(time.time() - tic))
fil = h5.File('task3_results_RK4_v2.h5', 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
fil = h5.File('task3_magnetic_moment.h5', 'w')
fil.create_dataset('mu', data=mu)
fil.create_dataset('time', data=t)
fil.create_dataset('kinetic_energy', data=kinetic_energy)
fil.close()
def hem2():
"""helmholtz"""
def Bfield(x, y, z, t):
"""Magnetic field, B """
r = (x * x + y * y)**(0.5)
def br(r, z):
y1 = lambda theta: (z - 1) * np.cos(theta) / (
((r - R * np.cos(theta))**2 + R**2 * np.sin(theta)**2 +
(z - 1)**2)**(3 / 2))
y2 = lambda theta: (z + 1) * np.cos(theta) / (
((r - R * np.cos(theta))**2 + R**2 * np.sin(theta)**2 +
(z + 1)**2)**(3 / 2))
I1 = integrate.quad(y1, 0, 2 * np.pi)
I2 = integrate.quad(y2, 0, 2 * np.pi)
return (I1 + I2)
def bz(r, z):
y1 = lambda theta: (1 - (r / R) * np.cos(theta)) / (
((r - R * np.cos(theta))**2 + R**2 * np.sin(theta)**2 +
(z - 1)**2)**(3 / 2))
y2 = lambda theta: (1 - (r / R) * np.cos(theta)) / (
((r - R * np.cos(theta))**2 + R**2 * np.sin(theta)**2 +
(z + 1)**2)**(3 / 2))
I1 = integrate.quad(y1, 0, 2 * np.pi)
I2 = integrate.quad(y2, 0, 2 * np.pi)
return (I1 + I2)
B_r = ((1 + R**2)**(3 / 2)) / (4 * np.pi * R) * br(r, z)[0]
B_z = ((1 + R**2)**(3 / 2)) / (4 * np.pi) * bz(r, z)[0]
return (1 / B0) * np.array([B_r * x / r, B_r * y / r, B_z])
def f(X, t, dothething=False):
""" X' = f(X, t) """
# X = [x1 x2 x3 x4 x5 x6] = [x y z v_x v_y v_z]
sign = 1
B = Bfield(X[0], X[1], X[2], t)
if dothething:
normB.append(np.linalg.norm(B))
kinetic_energy.append(X[3]**2 + X[4]**2 + X[5]**2)
#E = Efield(X[0], X[1], X[2], t)
newstate = np.array([
X[3], X[4], X[5], (X[4] * B[2] - X[5] * B[1]),
(X[5] * B[0] - X[3] * B[2]), (X[3] * B[1] - X[4] * B[0])
])
#newstate += np.array([0, 0, 0, E[0], E[1], E[2]])
print('time: {}/{}'.format(round(t, 1), totalTime), end='\r')
if dothething:
vnorm = X[3::] - np.inner(X[3::], B) * B / (normB[-1]**2)
mu.append(-(np.linalg.norm(vnorm)**2) / normB[-1])
return newstate
protonMass = np.float(1.673e-27)
electronMass = np.float(9.109e-31)
elementaryCharge = np.float(1.602e-19)
h = 0.001
B0 = 0.001
d = 1
R = 0.2
totalTime = 50
omega_c = elementaryCharge * B0 / electronMass
X0 = [1, 0, 0, np.sqrt(2) / 2, 0, np.sqrt(2) / 2]
normB = []
mu = []
kinetic_energy = []
# X0 = [6, 0, 0, 0, np.sqrt(2)/2, np.sqrt(2)/2] # initial state [X, V]
#X0 = [6, 0, 0, 1, 0, 0] # initial state [X, V]
t = np.arange(0, totalTime, h)
sol = odesolver2(f, X0, t, h)
solution_RK4 = sol.RK4()
#solution_RK4 = odeint(f, X0, t)
print('time: {}'.format(time.time() - tic))
fil = h5.File('task21Hem3_results_RK4_v2.h5', 'w')
fil.create_dataset('results', data=solution_RK4)
fil.create_dataset('time', data=t)
fil.create_dataset('stepsize', data=h)
fil.close()
fil = h5.File('task21Hem3_magnetic_moment.h5', 'w')
fil.create_dataset('mu', data=mu)
fil.create_dataset('time', data=t)
fil.create_dataset('kinetic_energy', data=kinetic_energy)
fil.close()