使用Runge-Kutta-4方法在Python（物理）中模拟轨道

Question

I'm trying to implement an RK4 method to solve for the orbit of a rocket around the Earth. Eventually this code will be used for more complex solar system simulations, but I'm just trying to get it working in this simple system first. My code is below - I'm hoping someone can tell me what is wrong with it. My trouble-shooting efforts have been long and unfruitful, but I'll summarise what I've found:

• I believe the acceleration function is fine and correct, as it gives believable values and agrees with my calculator/brain

• It appears as though the problem lies somewhere in the calculation of the next "r" value - when you run this code, an xy graph will appear, showing that the rocket initially falls in towards the Earth, then bounces away again, then back. I printed all the relevant values at this point, and found that "v" and "a" were negative in both components, despite the rocket clearly moving in the positive y direction. This makes me think that the calculation of the new "r" is in disagreement with the physics.

• The rocket is falling to Earth much faster than it should, which is also suspicious (technically it shouldn't fall into Earth at all, since the initial velocity is set to the required orbital velocity)

Either way I would greatly appreciate if anyone could find the error, as I have been unable up to this point.

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt


mE = 5.9742e24      #earth mass
mM = 7.35e22        #moon mass
dM = 379728240.5    #distance from moon to barycentre
dE = 4671759.5      #distance from earth to barycentre

s = 6.4686973e7     #hypothesised distance from moon to Lagrange-2 point
sr = 6.5420e7       #alternate L2 distance

def Simulate(iterations):

    x = dM                                                  #initialise     rocket positions
    y = 0
    a = 10                                                  #set the time step
    xdot = 0.                                                #initialise rocket velocity
    ydot = -((6.6726e-11)*mE/x)**0.5
    rocket_history_x, rocket_history_y = [[] for _ in range(2)] 
    history_mx, history_my = [[] for _ in range(2)]
    history_ex, history_ey = [[] for _ in range(2)]
    sep_history, step_history = [[] for _ in range(2)]      #create lists to store data in
    history_vx, history_vy = [[] for _ in range(2)]
    history_ax, history_ay = [[] for _ in range(2)]
    n = 1500
    m = 10000                                               #n,m,p are for storing every nth, mth and pth value to the lists
    p = 60000
    r = np.array((x,y))                                        #create rocket position vector
    v = np.array((xdot, ydot))                                 #create rocket velocity vector

    for i in range(iterations):

        xe, ye = 0, 0                                            #position of earth
        re = np.array((xe,ye))                                     #create earth position vector

        phi = np.arctan2((r[1]-ye),(r[0]-xe))                       #calculate phi, the angle between the rocket and the earth, as measured from the earth
        r_hat_e = np.array((np.cos(phi), np.sin(phi)))             #create vector along which earth's acceleration acts

        def acc(r):                                                             #define the acceleration vector function
            return ((-6.6726e-11)*(mE)/abs(np.dot((r-re),(r-re))))*r_hat_e

        k1v = acc(r)                                             #use RK4 method
        k1r = v
        k2v = acc(r + k1r*(a/2))            #acc(r + (a/2)*v)
        k2r = v * (a/2) * k1v               # v*acc(r) 
        k3v = acc(r + k2r*(a/2))            #acc(r + (a/2)*v*acc(r))
        k3r = v * k2v * (a/2)               #v*(a/2)*acc(r + (a/2)*v)
        k4v = acc(r + k3r*a)                #acc(r + (a**2/2)*v*acc(r + (a/2)*v))
        k4r = v * k3v * a                   #v*a*acc(r + (a/2)*v*acc(r))  

        v = v + (a/6) * (k1v + 2*k2v + 2*k3v + k4v)              #update v
        r = r + (a/6) * (k1r + 2*k2r + 2*k3r + k4r)              #update r

        sep = np.sqrt(np.dot((r-re),(r-re)))                    #separation of rocket and earth, useful for visualisation/trouble-shooting


        if i% n == 0: # Check for the step
            rocket_history_x.append(r[0])
            rocket_history_y.append(r[1])
            history_ex.append(xe)
            history_ey.append(ye)
            sep_history.append(sep)                             #putting data into lists for plotting and troubleshooting
            step_history.append(i)
            history_ax.append(acc(r)[0])
            history_ay.append(acc(r)[1])
            history_vx.append(v[0])
            history_vy.append(v[1])

        #if i% m == 0: # Check for the step
            #print r
            #print acc(r)
        #if i% p == 0: # Check for the step
            #print ((a/6)*(k1v + 2*k2v + 2*k3v + k4v))
            #print ((a/6)*(k1r + 2*k2r + 2*k3r + k4r))
            #print k1v, k2v, k3v, k4v
            #print k1r, k2r, k3r, k4r


    return rocket_history_x, rocket_history_y, history_ex, history_ey, history_mx, history_my, sep_history, step_history, history_ax, history_ay, history_vx, history_vy



x , y, xe, ye, mx, my, sep, step, ax, ay, vx, vy = Simulate(130000)


#print x,y,vx,vy,ax,ay,step

print ("Plotting graph...")



plt.figure()
plt.subplot(311)

plt.plot(x, y, linestyle='--', color = 'green')
#plt.plot(mx, my, linestyle='-', color = 'blue')
plt.plot(xe, ye, linestyle='-', color = 'red')
#plt.plot(xm, ym)
plt.xlabel("Orbit X")
plt.ylabel("Orbit Y")
'''
plt.plot(step, vy)
plt.ylabel("vy")
'''
plt.subplot(312)
plt.plot(step, sep)
plt.xlabel("steps")
plt.ylabel("separation")

plt.subplot(313)
plt.plot(step, ay)
plt.ylabel("ay")



plt.show()



print("Simulation Complete")

Answer 1

Your most grave error is that in the computation of the v slopes, you used multiplication instead of addition.

    k1v = acc(r)                         #use RK4 method
    k1r =     v
    k2v = acc(r + (a/2) * k1r)            
    k2r =     v + (a/2) * k1v            
    k3v = acc(r + (a/2) * k2r)            
    k3r =     v + (a/2) * k2v          
    k4v = acc(r +  a    * k3r)                
    k4r =     v +  a    * k3v                

A second error is that you use a value from a different state in the acceleration computation of the changed states. This might reduce the order of the method down to 1. Which might not change this plot visibly but will have larger errors over longer integration periods. Use

     def acc(r):                                                             #define the acceleration vector function                                                
         return ((-6.6726e-11)*(mE)/abs(np.dot((r-re),(r-re)))**1.5)*(r-re)