File:HeatEquationExplicitApproximate.svg
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Summary
DescriptionHeatEquationExplicitApproximate.svg |
English: Explicit method to approximate the heat equation , where . |
Date | |
Source | Own work |
Author | Shiyu Ji |
Python/Matplotlib Code
# Explicit method to solve the head equation.
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as pl
import numpy as np
import matplotlib.patches as mpatches
t0 = 0.0
t_final = 1.0
n_grid = 20
dt = t_final / n_grid
dx = 1.0 / n_grid
T = np.arange(t0, t_final+dt, dt)
X = np.arange(0, 1+dx, dx)
ax = pl.figure().add_subplot(111, projection='3d')
ax.set_xlabel('t')
ax.set_ylabel('x')
ax.set_zlabel('U')
ax.set_xlim([t0,t_final])
ax.set_ylim([0, 1])
Umax= .15
ax.set_zlim([0, Umax])
# Exact solution
def exact(t, x):
return np.exp(-t)*np.sin(np.pi*x)/(np.pi*np.pi)
for t in T:
for x in X:
if t>t0:
ax.plot([t-dt, t],[x, x],[exact(t-dt, x), exact(t, x)], 'b-', linewidth=0.5)
if x>0.0:
ax.plot([t, t],[x-dx, x],[exact(t, x-dx), exact(t,x)], 'b-', linewidth=0.5)
# Explicit method
r = dx/(dt*dt)/(np.pi*np.pi)
U = [[0.0 for _ in X] for _ in T]
for i in range(len(T)):
for j in range(len(X)):
if i==0:
U[i][j] = exact(T[i], X[j])
elif j==0 or j==len(X)-1:
U[i][j] = 0
else:
U[i][j] = (1-2*r)*U[i-1][j] + r*U[i-1][j-1] + r*U[i-1][j+1]
for i in range(len(T)):
for j in range(len(X)):
if 0 < U[i][j]: # select the approximate values within the bound to display
if i>0:
ax.plot([T[i-1], T[i]],[X[j], X[j]],[U[i-1][j], U[i][j]], 'r-', linewidth=0.5)
if j>0:
ax.plot([T[i], T[i]],[X[j-1], X[j]],[U[i][j-1], U[i][j]], 'r-', linewidth=0.5)
bluePatch = mpatches.Patch(color='blue', label='Exact')
redPatch = mpatches.Patch(color='red', label='Explicit')
pl.legend(handles = [bluePatch, redPatch], loc=2)
pl.show()
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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depicts
16 November 2016
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 11:02, 16 November 2016 | 720 × 540 (346 KB) | Shiyu Ji | User created page with UploadWizard |
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Width | 576pt |
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Height | 432pt |