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PHY03, Heat flow via time stepping

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Contents

Heat Flow via Time-Stepping (Leapfrog)

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The Parabolic Heat Equation (Theory)

Figure_1

Main code
# EqHeat.py : solves heat equation via finite differences , 3D plot
from numpy import *
import matplotlib.pylab as p
from mpl_toolkits.mplot3d import Axes3D

Nx = 101; Nt = 3000; Dx = 0.03; Dt = 0.9
KAPPA = 210.; SPH = 900.; RHO = 2700.         # Conductivity , specf heat , density
T = zeros((Nx, 2), float); Tpl = zeros((Nx, 31), float)

print("Working , wait for figure after count to 10")

for ix in range(1, Nx - 1):
    T[ix, 0] = 100.0;                         # Initial T
T[0, 0] = 0.0; T[0, 1] = 0.                   # 1st & last T = 0
T [Nx-1, 0] = 0.; T[Nx-1, 1] = 0.0
cons = KAPPA / (SPH*RHO)*Dt / (Dx*Dx);        # constant
m = 1                                         # counter

for t in range(1, Nt):
    for ix in range(1 , Nx-1):
        T[ix, 1] = T[ix, 0] + cons * (T[ix+1, 0] + T[ix-1, 0] - 2.*T[ix, 0])
    if t %300 == 0 or t == 1:                 # Every 300 steps
        for ix in range(1, Nx-1, 2):
            Tpl[ix, m] = T[ix, 1]
        print(m)
        m = m + 1
    for ix in range(1 , Nx-1) :
        T[ix , 0] = T[ix , 1]
x = list(range(1, Nx-1, 2))                   # Plot alternate pts
y = list(range(1, 30))
X, Y = p.meshgrid(x, y)

def functz (Tpl):
    z = Tpl[X, Y]
    return z

Z = functz (Tpl)
fig = p.figure()                              # Create figure
ax = Axes3D(fig)
ax . plot_wireframe(X, Y, Z, color='r')
ax . set_xlabel('Position')
ax . set_ylabel('time')
ax . set_zlabel('Temperature')
p . show( )
print ('finished')

Assessment and Visualization

Improved Heat Flow: Crank–Nicolson Method

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Reference

  • Rubin H. Landau, ManuelJ. Páez, Cristian C. Bordeianu - Computational Physics, Problem Solving with Python, 3rd completely revised edition
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