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 go():
xrk2, vrk2 = 5, -1
t, h = 0.0, 0.05
tt = []
xrk2_array, vrk2_array = [], []
while t < 10:
xrk2_array.append(xrk2)
vrk2_array.append(vrk2)
xrk2, vrk2 = ode.rk2(electron_ode, [xrk2, vrk2], t, h)
tt.append(t)
t += h
plt.figure(1)
plt.plot(tt, xrk2_array, '.', label='RK2')
plt.xlabel('Time (s)')
plt.ylabel('x (m)')
plt.legend(loc='best')
plt.show()
plt.figure(2)
plt.plot(tt, vrk2_array, '.', label='RK2')
plt.xlabel('Time (s)')
plt.ylabel('v (m/s)')
plt.legend(loc='best')
plt.show()
plt.figure(3)
plt.plot(xrk2_array, vrk2_array, '.', label='RK2')
plt.xlabel('x (m)')
plt.ylabel('v (m/s)')
plt.legend(loc='best')
plt.show()
dydt[0] = y[2]
dydt[1] = y[3]
dydt[2] = 0
dydt[3] = -g
return dydt
ball = vp.sphere(pos=vp.vector(-20, 0, 0), radius=0.5, color=vp.color.yellow)
floor = vp.box(pos=vp.vector(0, -5, 0), length=50, height=0.2, width=4)
t = 0
h = 0.01
vx = 10 * np.cos(np.deg2rad(30))
vy = 10 * np.sin(np.deg2rad(30))
g = 9.8
while True:
vp.rate(200)
#y = ode.Euler(freefall, [ball.pos.y, v], t, h)
y = ode.rk2(projectile, [ball.pos.x, ball.pos.y, vx, vy], t, h)
ball.pos.x = y[0]
ball.pos.y = y[1]
vx = y[2]
vy = y[3]
if ball.pos.y > floor.pos.y + ball.radius:
vy = vy - 9.8 * h
else:
vy = -vy
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))
abserr_rk4.append(abs(yy_rk4[0] - exs))
return dydt
t = 0
h = 0.01
x = 0
y = 0
vx = 10 * np.cos(np.deg2rad(30))
vy = 10 * np.sin(np.deg2rad(30))
g = 9.8
tt, posx, posy, velx, vely = [], [], [], [], []
while t <= 0.5:
#y = ode.Euler(freefall, [ball.pos.y, v], t, h)
yy = ode.rk2(projectile, [x, y, vx, vy], t, h)
tt.append(t)
posx.append(x)
posy.append(y)
velx.append(vx)
vely.append(vy)
x = yy[0]
y = yy[1]
vx = yy[2]
vy = yy[3]
t += h
if y > -5.5:
vy = vy - 9.8 * h
floor = vp.box(pos = vp.vector(0, -5, 0), length = 8, height = 0.2, width = 4)
m = 1
t = 0
h = 0.01
v = 0.0
g = 9.8
pe = []
ke = []
tt = []
while (t <= 1):
vp.rate(400)
#y = ode.Euler(freefall, [ball.pos.y, v], t, h)
y = ode.rk2(freefall, [ball.pos.y, v], t, h)
t = t + h/2
ball.pos.y = y[0]
v = y[1]
tt.append(t)
pe.append(m*g*ball.pos.y)
ke.append(0.5*m*v**2)
if ball.pos.y > floor.pos.y + ball.radius:
v = v - 9.8*h
else:
v = -v
plt.figure()
plt.plot(tt, pe, label = 'Potential Energy')