def eulerstep(phi, qn, vn, Dt, M):
"""
This function takes one timestep across a set of equations dq/dt = v,
dv/dt = -dphi(q)/dq using the Euler method, where q is a generalised
spatial coordinate, v the velocity, t is time and phi is the potential.
Parameters
phi: A function defining the potential of the system.
qn: The initial spatial coordinate at the start of the timestep.
vn: The initial velocity at the start of the timestep.
Dt: The length of the time step.
M: The mass of the particle acting under a force.
Returns
The function returns the value of the spatial coordinate and velocity
after taking a timestep of length Dt using the Euler method.
"""
qn1 = qn + Dt*vn
# ensuring h is an exact machine represenable number (NRiC)
# note using the symmetrized for of the numerical derivative (see NRiC)
# may want to consider Richardson extrapolation for serious work, again see
# NRiC pp. 231
# vn1 = vn - Dt*((phi(qn + h) - phi(qn - h))/(2*h))
useRidders = True
if useRidders:
dphidq, err = deriv.derivative(phi, qn, np.abs(qn1 - qn), useRidders)
else:
dphidq = deriv.derivative(phi, qn, np.abs(qn1 - qn), useRidders)
vn1 = vn - (1/M)*Dt*dphidq
return qn1, vn1
def rk4step(phi, qn, vn, Dt, M):
"""
This function takes one timestep across a set of equations dq/dt = v,
M*dv/dt = -dphi(q)/dq using a fourth order Runge-Kutta method, where q is a
generalised spatial coordinate, v the velocity, t is time and phi is the
potential.
Parameters
phi: A function defining the potential of the system.
qn: The initial spatial coordinate at the start of the timestep.
vn: The initial velocity at the start of the timestep.
Dt: The length of the time step.
M: The mass of the particle acting under a force.
Returns
The function returns the value of the spatial coordinate and velocity
after taking a timestep of length Dt using a fourth order Runge-Kutta
method.
"""
# Z1
Q1 = qn
V1 = vn
dphidQ1, err = deriv.derivative(phi, Q1, np.abs(Dt*vn))
# Z2
Q2 = qn + 0.5*Dt*V1
V2 = vn - 0.5*Dt*1.0/M*dphidQ1
dphidQ2, err = deriv.derivative(phi, Q2, np.abs(Dt*vn))
# Z3
Q3 = qn + 0.5*Dt*V2
V3 = vn - 0.5*Dt*1.0/M*dphidQ2
dphidQ3, err = deriv.derivative(phi, Q3, np.abs(Dt*vn))
# Z4
Q4 = qn + Dt*V3
V4 = vn - Dt*1.0/M*dphidQ3
dphidQ4, err = deriv.derivative(phi, Q4, np.abs(Dt*vn))
# final step
qn1 = qn + Dt/6.0*(V1 + 2.0*V2 + 2.0*V3 + V4)
vn1 = vn - Dt/(6.0*M)*(dphidQ1 + 2.0*dphidQ2 + 2.0*dphidQ3 + dphidQ4)
return qn1, vn1
def eulerBstep(phi, qn, vn, Dt, M):
"""
This function takes one timestep across a set of equations dq/dt = v,
M*dv/dt = -dphi(q)/dq using the Euler B method, where q is a generalised
spatial coordinate, v the velocity, t is time and phi is the potential.
Parameters
phi: A function defining the potential of the system.
qn: The initial spatial coordinate at the start of the timestep.
vn: The initial velocity at the start of the timestep.
Dt: The length of the time step.
M: The mass of the particle acting under a force.
Returns
The function returns the value of the spatial coordinate and velocity
after taking a timestep of length Dt using the Euler B method.
"""
dphidq, err = deriv.derivative(phi, qn, np.abs(Dt*vn))
vn1 = vn + (1/M)*Dt*dphidq
qn1 = qn - Dt*vn1
return qn1, vn1
e2norm = np.zeros(N + 1, dtype = np.float64)
energyError = np.zeros(N + 1, dtype = np.float64)
phi = np.zeros(N + 1, dtype = np.float64)
dphidq_analytic = np.zeros(N + 1, dtype = np.float64)
dphidq_numeric = np.zeros(N + 1, dtype = np.float64)
# initial conditions
q[0] = 1.9
v[0] = -0.0001
energyConst = 0.5*v[0] + LJ(q[0])
# calcualte phase space
for ii in range(0, N):
t[ii + 1] = t[ii] + Dt
q[ii + 1], v[ii + 1] = step.eulerstep(LJ, q[ii], v[ii], Dt, M)
dphidq = -deriv.derivative(LJ, q[ii], np.abs(q[ii + 1] - q[ii]), False)[0]
d2phidq2 = -deriv.derivative(LJdiff, q[ii], np.abs(q[ii + 1] - q[ii]), False)[0]
a[ii] = np.max([1, d2phidq2])
E = 0.5*v[ii]**2 + LJ(q[ii])
b[ii] = 0.5*np.sqrt(dphidq**2 + 2*d2phidq2**2*(E - LJ(q[ii])))
energyError[ii] = E - energyConst
# plot
plt.figure(1)
plt.clf()
ax1 = plt.subplot2grid((3, 2), (0, 0), rowspan = 3)
ax1.plot(q, v)
plt.title('Phase Plot')
plt.xlabel('Generalised Coordainte, $q$')
plt.ylabel('Generalised Velocity, $v(q)$')
plt.axis([0.5, 2.5, -0.5, 0.5])
