# Fourth order Runge-Kutta numerical integration :: 02 Jan 2009

Here’s a Python implementation of RK4, hardcoded for double-integrating the second derivative (acceleration up to position). For a more generalized solution, see my other implementation. I tried to keep this as simple as I could, so people can easily see the relation between the ‘math form’ and ‘code form’ of the algorithm.

def rk4(x, v, a, dt):
"""Returns final (position, velocity) tuple after
time dt has passed.

x: initial position (number-like object)
v: initial velocity (number-like object)
a: acceleration function a(x,v,dt) (must be callable)
dt: timestep (number)"""
x1 = x
v1 = v
a1 = a(x1, v1, 0)

x2 = x + 0.5*v1*dt
v2 = v + 0.5*a1*dt
a2 = a(x2, v2, dt/2.0)

x3 = x + 0.5*v2*dt
v3 = v + 0.5*a2*dt
a3 = a(x3, v3, dt/2.0)

x4 = x + v3*dt
v4 = v + a3*dt
a4 = a(x4, v4, dt)

xf = x + (dt/6.0)*(v1 + 2*v2 + 2*v3 + v4)
vf = v + (dt/6.0)*(a1 + 2*a2 + 2*a3 + a4)

return xf, vf


Here is an example usage of the function and a comparison to Euler integration:

def accel(x, v, dt):
"""Determines acceleration from current position,
velocity, and timestep. This particular acceleration
function models a spring."""
stiffness = 1
damping = -0.005
return -stiffness*x - damping*v

t = 0
dt = 1.0/40 # Timestep of 1/40 second
state = 50, 5 # Position, velocity
euler = 50, 5 # For comparison with Euler integration

print "Initial    -position: %6.2f, velocity: %6.2f"%state

# Run for 100 seconds
while t < 100:
t += dt
state = rk4(state, state, accel, dt)

# Integrate using Euler's method
euler = (
euler + euler*dt,
euler + accel(euler,euler,dt)*dt
)

print "Final RK4  -position: %6.2f, velocity: %6.2f"%state
print "Final Euler-position: %6.2f, velocity: %6.2f"%euler


The output of this really shows how much more accurate RK4 integration can be:

Initial    -position:  50.00, velocity:   5.00
Final RK4  -position:  52.18, velocity:  38.05
Final Euler-position: 178.38, velocity: 137.62


As the timestep is decreased (meaning more computation), Euler approaches RK4 (shown at timestep of 1/400 seconds):

Initial    -position:  50.00, velocity:   5.00
Final RK4  -position:  52.28, velocity:  37.92
Final Euler-position:  59.20, velocity:  43.02