Cannot get RK4 to solve for position of orbiting body in Python
Physics
The Newton law gives you a second order ODE u''=F(u)
with u=[x,y]
. Using v=[x',y']
you get the first order system
u' = v
v' = F(u)
which is 4-dimensional and has to be solved using a 4 dimensional state. The only reduction available is to use the Kepler laws which allow to reduce the system to a scalar order one ODE for the angle. But that is not the task here.
But to get the scales correct, for a circular orbit of radius R
with angular velocity w
one gets the identity w^2*R^3=G*M
which implies that the speed along the orbit is w*R=sqrt(G*M/R)
and period T=2*pi*sqrt(R^3/(G*M))
. With the data given, R ~ 10
, w ~ 1
, thus G*M ~ 1000
for a close-to-circular orbit, so with M=20
this would require G
between 50
and 200
, with an orbital period of about 2*pi ~ 6
. The time span of 10 could represent one half to about 2 or 3 orbits.
Euler method
You correctly implemented the Euler method to calculate values in the last loop of your code. That it may look un-physical can be because the Euler method continuously increases the orbit, as it moves to the outside of convex trajectories following the tangent. In your implementation this outward spiral can be seen for G=100
.
This can be reduced in effect by choosing a smaller step size, such as dt=0.001
.
You should select the integration time to be a good part of a full orbit to get a presentable result, with above parameters you get about 2 loops, which is good.
RK4 implementation
You made several errors. Somehow you lost the velocities, the position updates should be based on the velocities.
Then you should have halted at fx(x + .5*kx1, y + .5*kx1, t + .5*dt)
to reconsider your approach as that is inconsistent with any naming convention. The consistent, correct variant is
fx(x + .5*kx1, y + .5*ky1, t + .5*dt)
which shows that you can not decouple the integration of a coupled system, as you need the y
updates alongside the x
updates. Further, the function values are the accelerations, thus update the velocities. The position updates use the velocities of the current state. Thus the step should start as
kx1 = dt * fx(x,y,t) # vx update
mx1 = dt * vx # x update
ky1 = dt * fy(x,y,t) # vy update
my1 = dt * vy # y update
kx2 = dt * fx(x + 0.5*mx1, y + 0.5*my1, t + 0.5*dt)
mx2 = dt * (vx + 0.5*kx1)
ky2 = dt * fy(x + 0.5*mx1, y + 0.5*my1, t + 0.5*dt)
my2 = dt * (vy + 0.5*ky1)
etc.
However, as you see, this already starts to become unwieldy. Assemble the state into a vector and use a vector valued function for the system equations
M, G = 20, 100
def orbitsys(u):
x,y,vx,vy = u
r = np.hypot(x,y)
f = G*M/r**3
return np.array([vx, vy, -f*x, -f*y]);
Then you can use a cook-book implementation of the Euler or Runge-Kutta step
def Eulerstep(f,u,dt): return u+dt*f(u)
def RK4step(f,u,dt):
k1 = dt*f(u)
k2 = dt*f(u+0.5*k1)
k3 = dt*f(u+0.5*k2)
k4 = dt*f(u+k3)
return u + (k1+2*k2+2*k3+k4)/6
and combine them into an integration loop
def Eulerintegrate(f, y0, tspan):
y = np.zeros([len(tspan),len(y0)])
y[0,:]=y0
for k in range(1, len(tspan)):
y[k,:] = Eulerstep(f, y[k-1], tspan[k]-tspan[k-1])
return y
def RK4integrate(f, y0, tspan):
y = np.zeros([len(tspan),len(y0)])
y[0,:]=y0
for k in range(1, len(tspan)):
y[k,:] = RK4step(f, y[k-1], tspan[k]-tspan[k-1])
return y
and invoke them with your given problem
dt = .1
t = np.arange(0,10,dt)
y0 = np.array([10, 0.0, 10, 10])
sol_euler = Eulerintegrate(orbitsys, y0, t)
x,y,vx,vy = sol_euler.T
plt.plot(x,y)
sol_RK4 = RK4integrate(orbitsys, y0, t)
x,y,vx,vy = sol_RK4.T
plt.plot(x,y)
You are not using rkx
, rky
functions anywhere!
There are two return
at the end of function definition you should use
return [(kx1 + 2*kx2 + 2*kx3 + kx4)/6, (mx1 + 2*mx2 + 2*mx3 + mx4)/6]
(as pointed out by @eapetcho). Also, your implementation of Runge-Kutta is not clear to me.
You have dv/dt
so you solve for v
and then update r
accordingly.
for n in range(1,len(t)): #solve using RK4 functions
vx[n] = vx[n-1] + rkx(vx[n-1],vy[n-1],t[n-1])*dt
vy[n] = vy[n-1] + rky(vx[n-1],vy[n-1],t[n-1])*dt
x[n] = x[n-1] + vx[n-1]*dt
y[n] = y[n-1] + vy[n-1]*dt
Here is my version of the code
import numpy as np
#constants
G=1
M=1
h=0.1
#initiating variables
rt = np.arange(0,10,h)
vx = np.zeros(len(rt))
vy = np.zeros(len(rt))
rx = np.zeros(len(rt))
ry = np.zeros(len(rt))
#initial conditions
vx[0] = 10 #initial x velocity
vy[0] = 10 #initial y velocity
rx[0] = 10 #initial x position
ry[0] = 0 #initial y position
def fx(x,y): #x acceleration
return -G*M*x/((x**2+y**2)**(3/2))
def fy(x,y): #y acceleration
return -G*M*y/((x**2+y**2)**(3/2))
def rk4(xj, yj):
k0 = h*fx(xj, yj)
l0 = h*fx(xj, yj)
k1 = h*fx(xj + 0.5*k0 , yj + 0.5*l0)
l1 = h*fy(xj + 0.5*k0 , yj + 0.5*l0)
k2 = h*fx(xj + 0.5*k1 , yj + 0.5*l1)
l2 = h*fy(xj + 0.5*k1 , yj + 0.5*l1)
k3 = h*fx(xj + k2, yj + l2)
l3 = h*fy(xj + k2, yj + l2)
xj1 = xj + (1/6)*(k0 + 2*k1 + 2*k2 + k3)
yj1 = yj + (1/6)*(l0 + 2*l1 + 2*l2 + l3)
return (xj1, yj1)
for t in range(1,len(rt)):
nv = rk4(vx[t-1],vy[t-1])
[vx[t],vy[t]] = nv
rx[t] = rx[t-1] + vx[t-1]*h
ry[t] = ry[t-1] + vy[t-1]*h
I suspect there are issues with fx(x,y,t) and fy(x,y,t)
This is the case, I just checked my code for fx=3
and fy=y
and I got a nice trajectory.
Here is the ry
vs rx
plot: