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How to implement boundary value in Python using spectral methods

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1

1

$begingroup$

I want to solve the heat equation $$u_t=au_xx,quad 0leq t, 0leq xleqpi$$ in Python using spectral methods. I set $u(0,x)=sin(x)$ as initial value for the time and choose $a=2$. My Code:

import numpy as np
import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff

def GenerateID(x):
 u = np.sin(x)
 return u

def hEq(u, t, L, a):
 uxx = psdiff(u, period=L, order=2) 
 dudt = a*uxx
 return dudt

def hEq_solution(u0, t, L):
 sol = odeint(hEq, u0, t, args=(L,a,), mxstep=5000)
 return sol

if __name__ == "__main__":
 L = np.pi
 N = 1000
 x = np.linspace(0, L, N)
 a = 2
 u0 = GenerateID(x)
 Tfinal = 5.0
 t = np.linspace(0.0, Tfinal, 1000)
 sol = hEq_solution(u0, t, L)
 fig = plt.figure()
 ax = fig.add_subplot(111, projection='3d')
 sx, st = np.meshgrid(x, t)
 ax.plot_surface(sx, st, sol, cmap='jet')
 plt.xlabel('x')
 plt.ylabel('t')
 plt.show()

Which produces
This seems kind of right. Now my question: If I want to introduce the boundary condition $$u(t,0)=u(t,pi)=0,$$ how would I do that? I know that considering all above conditions, the exact solution should be $$u(x,t)=exp[-2t]sin(x).$$

ordinary-differential-equations pde python fast-fourier-transform

share|cite|improve this question

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  In a spectral method the boundary conditions are baked into the choice of the basis functions ($sin(nx)$ in your case). Thus really the question is whether your fft is set up to do a sine transform or not which is now really a programming question.
  $endgroup$
  – Ian
  Mar 13 at 7:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please use the Math forum only for Math questions
  $endgroup$
  – asdf
  Mar 13 at 13:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, it is a math question now. @Ian I don't understand how the basis would do that. Also, can I influence that by calculating the fft and change the wave numbers?
  $endgroup$
  – schoeni
  Mar 14 at 5:36
  
  

  
  
  
  

add a comment | 

1

1

$begingroup$

I want to solve the heat equation $$u_t=au_xx,quad 0leq t, 0leq xleqpi$$ in Python using spectral methods. I set $u(0,x)=sin(x)$ as initial value for the time and choose $a=2$. My Code:

import numpy as np
import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff

def GenerateID(x):
 u = np.sin(x)
 return u

def hEq(u, t, L, a):
 uxx = psdiff(u, period=L, order=2) 
 dudt = a*uxx
 return dudt

def hEq_solution(u0, t, L):
 sol = odeint(hEq, u0, t, args=(L,a,), mxstep=5000)
 return sol

if __name__ == "__main__":
 L = np.pi
 N = 1000
 x = np.linspace(0, L, N)
 a = 2
 u0 = GenerateID(x)
 Tfinal = 5.0
 t = np.linspace(0.0, Tfinal, 1000)
 sol = hEq_solution(u0, t, L)
 fig = plt.figure()
 ax = fig.add_subplot(111, projection='3d')
 sx, st = np.meshgrid(x, t)
 ax.plot_surface(sx, st, sol, cmap='jet')
 plt.xlabel('x')
 plt.ylabel('t')
 plt.show()

Which produces
This seems kind of right. Now my question: If I want to introduce the boundary condition $$u(t,0)=u(t,pi)=0,$$ how would I do that? I know that considering all above conditions, the exact solution should be $$u(x,t)=exp[-2t]sin(x).$$

ordinary-differential-equations pde python fast-fourier-transform

share|cite|improve this question

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  In a spectral method the boundary conditions are baked into the choice of the basis functions ($sin(nx)$ in your case). Thus really the question is whether your fft is set up to do a sine transform or not which is now really a programming question.
  $endgroup$
  – Ian
  Mar 13 at 7:48
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please use the Math forum only for Math questions
  $endgroup$
  – asdf
  Mar 13 at 13:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, it is a math question now. @Ian I don't understand how the basis would do that. Also, can I influence that by calculating the fft and change the wave numbers?
  $endgroup$
  – schoeni
  Mar 14 at 5:36
  
  

  
  
  
  

add a comment | 

$begingroup$

I want to solve the heat equation $$u_t=au_xx,quad 0leq t, 0leq xleqpi$$ in Python using spectral methods. I set $u(0,x)=sin(x)$ as initial value for the time and choose $a=2$. My Code:

import numpy as np
import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff

def GenerateID(x):
 u = np.sin(x)
 return u

def hEq(u, t, L, a):
 uxx = psdiff(u, period=L, order=2) 
 dudt = a*uxx
 return dudt

def hEq_solution(u0, t, L):
 sol = odeint(hEq, u0, t, args=(L,a,), mxstep=5000)
 return sol

if __name__ == "__main__":
 L = np.pi
 N = 1000
 x = np.linspace(0, L, N)
 a = 2
 u0 = GenerateID(x)
 Tfinal = 5.0
 t = np.linspace(0.0, Tfinal, 1000)
 sol = hEq_solution(u0, t, L)
 fig = plt.figure()
 ax = fig.add_subplot(111, projection='3d')
 sx, st = np.meshgrid(x, t)
 ax.plot_surface(sx, st, sol, cmap='jet')
 plt.xlabel('x')
 plt.ylabel('t')
 plt.show()

Which produces
This seems kind of right. Now my question: If I want to introduce the boundary condition $$u(t,0)=u(t,pi)=0,$$ how would I do that? I know that considering all above conditions, the exact solution should be $$u(x,t)=exp[-2t]sin(x).$$

ordinary-differential-equations pde python fast-fourier-transform

share|cite|improve this question

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

$endgroup$

import numpy as np
import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
from scipy.fftpack import diff as psdiff

def GenerateID(x):
 u = np.sin(x)
 return u

def hEq(u, t, L, a):
 uxx = psdiff(u, period=L, order=2) 
 dudt = a*uxx
 return dudt

def hEq_solution(u0, t, L):
 sol = odeint(hEq, u0, t, args=(L,a,), mxstep=5000)
 return sol

if __name__ == "__main__":
 L = np.pi
 N = 1000
 x = np.linspace(0, L, N)
 a = 2
 u0 = GenerateID(x)
 Tfinal = 5.0
 t = np.linspace(0.0, Tfinal, 1000)
 sol = hEq_solution(u0, t, L)
 fig = plt.figure()
 ax = fig.add_subplot(111, projection='3d')
 sx, st = np.meshgrid(x, t)
 ax.plot_surface(sx, st, sol, cmap='jet')
 plt.xlabel('x')
 plt.ylabel('t')
 plt.show()

share|cite|improve this question

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

share|cite|improve this question

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

edited Mar 13 at 12:33

Brian

677115

edited Mar 13 at 12:33

Brian

677115

asked Mar 13 at 7:41

schoenischoeni

185

asked Mar 13 at 7:41

schoenischoeni

185