def go():
xlf, vlf = 1.5, 0.0
xe, ve = 1.5, 0.0
xrk2, vrk2 = 1.5, 0.0
t, h = 0.0, 0.1
tt = []
xlf_array, vlf_array = [], []
xe_array, ve_array = [], []
xrk2_array, vrk2_array = [], []
energy_lf, energy_e, energy_rk2 = [], [], []
while t < 4 * np.pi:
xlf_array.append(xlf)
vlf_array.append(vlf)
energy_lf.append(0.5 * m * vlf**2 + 0.5 * m * w**2 * xlf**2)
xlf, vlf = ode.leapfrog(oscillator, xlf, vlf, t, h)
xe_array.append(xe)
ve_array.append(ve)
energy_e.append(0.5 * m * ve**2 + 0.5 * m * w**2 * xe**2)
xe, ve = ode.Euler(oscillator_rk, [xe, ve], t, h)
xrk2_array.append(xrk2)
vrk2_array.append(vrk2)
energy_rk2.append(0.5 * m * vrk2**2 + 0.5 * m * w**2 * xrk2**2)
xrk2, vrk2 = ode.rk2(oscillator_rk, [xrk2, vrk2], t, h)
tt.append(t)
t += h
plt.figure(1)
plt.plot(xlf_array, vlf_array, '--', label='Leapfrog')
plt.plot(xe_array, ve_array, label='Euler')
plt.plot(xrk2_array, vrk2_array, '.', label='RK2')
plt.xlabel('x (m)')
plt.ylabel('v (m/s)')
plt.legend(loc='best')
plt.show()
plt.figure(2)
plt.semilogy(tt, energy_lf, '--', label='Leapfrog')
plt.semilogy(tt, energy_e, label='Euler')
plt.semilogy(tt, energy_rk2, '.', label='RK2')
plt.xlabel('Time (s)')
plt.ylabel('Total Energy (J)')
plt.legend(loc='best')
plt.grid()
plt.show()
def test_euler():
t_euler_raw, x_euler_raw = ode.euler(oscillator_1st_deriv,
xzero=[0, 1],
timerange=[0, 5],
timestep=0.1)
p_euler_raw, v_euler_raw = x_euler_raw
t_euler = [round(x, 6) for x in t_euler_raw]
p_euler = [round(x, 6) for x in p_euler_raw]
v_euler = [round(x, 6) for x in v_euler_raw]
t_ieuler_raw, x_ieuler_raw = zip(*list(
ode.Euler(
oscillator_1st_deriv, xzero=[0, 1], timerange=[0, 5
], timestep=0.1)))
p_ieuler_raw, v_ieuler_raw = zip(*x_ieuler_raw)
t_ieuler = [round(x, 6) for x in t_ieuler_raw]
p_ieuler = [round(x, 6) for x in p_ieuler_raw]
v_ieuler = [round(x, 6) for x in v_ieuler_raw]
assert (t_euler, p_euler, v_euler) == (t_ieuler, p_ieuler, v_ieuler)
import sys, os
sys.path.append(os.path.join(os.path.dirname(os.getcwd()), 'copads'))
import ode
initial_nuclei = 10000.0
decay_constant = 0.2
def decay(t, y):
return -decay_constant * y[0]
print('Solving using Euler method......')
for i in [x for x in ode.Euler([decay], 0.0, [initial_nuclei], 0.1, 50.0)]:
print(','.join([str(x) for x in i]))
print('')
print('Solving using RK4 method......')
for i in [x for x in ode.RK4([decay], 0.0, [initial_nuclei], 0.1, 50.0)]:
print(','.join([str(x) for x in i]))
print('')
print('Solving using RK3 method......')
for i in [x for x in ode.RK3([decay], 0.0, [initial_nuclei], 0.1, 50.0)]:
print(','.join([str(x) for x in i]))
print('')
print("Solving using Heun's method......")
for i in [x for x in ode.Heun([decay], 0.0, [initial_nuclei], 0.1, 50.0)]:
print(','.join([str(x) for x in i]))
print('')
def exact_sol(t):
return 1000 * np.exp(-lamb * t)
t = 0
h = 0.5
y_euler, y_rk2, y_rk4, y_rk45n = 1000, 1000, 1000, 1000
tt, exsol = [], []
nuclear_euler, nuclear_rk2, nuclear_rk4, nuclear_rk45n = [], [], [], []
abserr_euler, abserr_rk2, abserr_rk4, abserr_rk45n = [], [], [], []
clf_euler, clf_rk2, clf_rk4, clf_rk45n = [], [], [], []
while t <= 8:
yy_euler = ode.Euler(decay, [y_euler], t, h)
yy_rk2 = ode.rk2(decay, [y_rk2], t, h)
yy_rk4 = ode.rk4(decay, [y_rk4], t, h)
yy_rk45n = ode.RK45n(decay, [y_rk45n], t, h)
exs = exact_sol(t)
tt.append(t)
exsol.append(exs)
nuclear_euler.append(yy_euler[0])
nuclear_rk2.append(yy_rk2[0])
nuclear_rk4.append(yy_rk4[0])
nuclear_rk45n.append(yy_rk45n[0])
abserr_euler.append(abs(yy_euler[0] - exs))
abserr_rk2.append(abs(yy_rk2[0] - exs))