I want to use the spectral method to solve partial differential equations. The equations like that, formula,the initial condition is u(t=0,x)=(a^2)*sech(x),u'_t (t=0)=0.
To solve it, I use the python with the spectral method. Following is the code,
import numpy as np
from scipy.integrate import solve_ivp
from scipy.fftpack import diff as psdiff
#RHS of equations
def f(t,u):
uxx= psdiff(u[N:],period=L,order=2)
du1dt=u[:N]
du2dt =a**2*uxx
dudt=np.append(du1dt,du2dt)
return dudt
a=1
amin=-40;bmax=40
L=bmax-amin;N=256
deltax=L/N
x=np.arange(amin,bmax,deltax)
u01 = 2*np.cosh(x)**(-1)
u02=np.zeros(N)
# y0
inital=np.append(u01,u02)
sola1 = solve_ivp(f, t_span=[0,40],y0=inital,args=(a,))
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.plot(x,sola1.y[:N,5])
plt.show()
Following is my expected result,
My python code can run,but I can't get the expected result,and can't find the problem.Following is the result from my python code, my result
-----------------------------Update---------------------------------------------- I also try a new code,but still can't solve
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff
from itertools import chain
def lambdifide_odes(y,t,a):
# uxx =- (1j)**2*k**2*u[:N]
u1=y[::2]
u2=y[1::2]
dudt=np.empty_like(y)
du1dt=dudt[::2]
du2dt=dudt[1::2]
du1dt=u2
uxx=psdiff(u1,order=2,period=L)
du2dt=a**2*uxx
return dudt
a=1
amin=-40;bmax=40
L=bmax-amin;N=256
deltax=L/N
x=np.arange(amin,bmax,deltax)
u01 = 2*np.cosh(x)**(-1)
u02=np.zeros(N)
initial=np.array(list(chain.from_iterable(zip(u01,u02))))
t0=np.linspace(0,40,100)
sola1 = odeint(lambdifide_odes,y0=initial,t=t0,args=(a,))
fig, ax = plt.subplots()
ax.plot(x,sola1[20,::2])
plt.show()
