python - RK4 giving wrong result -


i'm trying numerically solve simple second order de x'' = -x. used new variable x'=v, have 2 equations. while seems simple, somehow produces result that's far of correct result.

def f(x):     return -x  def rk4(f=f,h=2*pi/100,x0=1,v0=0,t0=0,n=100):     '''rk4'''     v=[v0]     x=[x0]     in range(n-1):         v1= h*f(x[-1])         x1= h *(v[-1])         v2= h*f(x[-1]+1/2*x1)         x2= h *(v[-1]+1/2*v1)         v3= h*f(x[-1]+1/2*x2)         x3= h *(v[-1]+1/2*v2)         v4= h*f(x[-1]+x3)         x4= h *f(v[-1]+v3)         v.append(v[-1]+1/6 *(v1+2*v2+2*v3+v4))         x.append(x[-1]+1/6 *(x1+2*x2+2*x3+x4))     return x,v

the weird thing if use rk45 coefficients, works fine. idea on wrong?

def rk4(f=f,h=2*pi/100,x0=1,v0=0,t0=0,n=100):     '''rk4'''     v=[v0]     x=[x0]      in range(n-1):         v1=h*f(x[-1])         x1=h *v[-1]         v2=h*f(x[-1]+1/4*x1)         x2=h *(v[-1]+1/4*v1)         v3=h*f(x[-1]+9/32*x2+3/32*x1)         x3=h *(v[-1]+9/32*v2+3/32*v1)         v4=h*f(x[-1]+7296/2197*x3 - 7200/2197 * x2 + 1932 / 2197 * x1)         x4=h *(v[-1]+7296/2197*v3 - 7200/2197 * v2 + 1932 / 2197 * v1)         v5=h*f(x[-1]-845/4104 * x4 +3680/513*x3 - 8 * x2 + 439 / 216 * x1)         x5=h *(v[-1]-845/4104 * v4 +3680/513*v3 - 8 * v2 + 439 / 216 * v1)         v.append(v[-1]+25/216*v1+1408/2565*v3+2197/4104*v4-1/5*v5)         x.append(x[-1]+25/216*x1+1408/2565*x3+2197/4104*x4-1/5*x5)     return x,v

it simple copy-paste error. in line x4=... there should no f, in x1=..., x2=..., x3=....

then looks nice should.

this 1 reason stay close possible cook-book implementation, use vector-valued functions , vector arithmetic.

by way, if want last point have time 2*pi, should iterate (in both methods) on range(n) have n integration steps , t=n*(2*pi/n)=2*pi.


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