def rule_write ( order, header, x, w, r ): #*****************************************************************************80 # ## RULE_WRITE writes a quadrature rule to a file. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 23 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer ORDER, the order of the rule. # # Input, string HEADER, specifies the output files. # write files 'header_w.txt', 'header_x.txt', 'header_r.txt' defining # weights, abscissas, and region. # # Input, real X(ORDER), the abscissas. # # Input, real W(ORDER), the weights. # # Input, real R(2), the region. # from r8vec_write import r8vec_write filename_x = header + '_x.txt' filename_w = header + '_w.txt' filename_r = header + '_r.txt' print '' print ' Creating quadrature files.' print '' print ' Common header is "%s".' % ( header ) print '' print ' Weight file will be "%s".' % ( filename_w ) print ' Abscissa file will be "%s".' % ( filename_x ) print ' Region file will be "%s".' % ( filename_r ) r8vec_write ( filename_w, order, w ) r8vec_write ( filename_x, order, x ) r8vec_write ( filename_r, 2, r ) return
def rule_write(order, header, x, w, r):
#*****************************************************************************80
#
## RULE_WRITE writes a quadrature rule to a file.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the rule.
#
# Input, string HEADER, specifies the output files.
# write files 'header_w.txt', 'header_x.txt', 'header_r.txt' defining
# weights, abscissas, and region.
#
# Input, real X(ORDER), the abscissas.
#
# Input, real W(ORDER), the weights.
#
# Input, real R(2), the region.
#
from r8vec_write import r8vec_write
filename_x = header + '_x.txt'
filename_w = header + '_w.txt'
filename_r = header + '_r.txt'
print ''
print ' Creating quadrature files.'
print ''
print ' Common header is "%s".' % (header)
print ''
print ' Weight file will be "%s".' % (filename_w)
print ' Abscissa file will be "%s".' % (filename_x)
print ' Region file will be "%s".' % (filename_r)
r8vec_write(filename_w, order, w)
r8vec_write(filename_x, order, x)
r8vec_write(filename_r, 2, r)
return
def fd1d_heat_implicit_test03():
#*****************************************************************************80
#
## FD1D_HEAT_IMPLICIT_TEST03 does a simple test problem.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 November 2014
#
# Author:
#
# John Burkardt
#
from fd1d_heat_implicit import fd1d_heat_implicit
from fd1d_heat_implicit_cfl import fd1d_heat_implicit_cfl
from fd1d_heat_implicit_matrix import fd1d_heat_implicit_matrix
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import axes3d, Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ''
print 'FD1D_HEAT_IMPLICIT_TEST03:'
print ' Compute an approximate solution to the time-dependent'
print ' one dimensional heat equation:'
print ' dH/dt - K * d2H/dx2 = f(x,t)'
print ' Run a simple test case.'
#
# Heat coefficient.
#
k = k_test03()
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = -5.0
x_max = +5.0
dx = (x_max - x_min) / (x_num - 1)
x = np.linspace(x_min, x_max, x_num)
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 81
t_min = 0.0
t_max = 4.0
dt = (t_max - t_min) / (t_num - 1)
t = np.linspace(t_min, t_max, t_num)
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_implicit_cfl(k, t_num, t_min, t_max, x_num, x_min, x_max)
print ''
print ' Number of X nodes = %d' % (x_num)
print ' X interval is [%f,%f]' % (x_min, x_max)
print ' X spacing is %f' % (dx)
print ' Number of T values = %d' % (t_num)
print ' T interval is [%f,%f]' % (t_min, t_max)
print ' T spacing is %f' % (dt)
print ' Constant K = %g' % (k)
print ' CFL coefficient = %g' % (cfl)
#
# Get the system matrix.
#
a = fd1d_heat_implicit_matrix(x_num, cfl)
hmat = np.zeros((x_num, t_num))
for j in range(0, t_num):
if (j == 0):
h = ic_test03(x_num, x, t[j])
h = bc_test03(x_num, x, t[j], h)
else:
h = fd1d_heat_implicit(a, x_num, x, t[j - 1], dt, cfl, rhs_test03,
bc_test03, h)
for i in range(0, x_num):
hmat[i, j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid(t, x)
fig = plt.figure()
ax = Axes3D(fig)
surf = ax.plot_surface(xmat, tmat, hmat)
plt.xlabel('<---X--->')
plt.ylabel('<---T--->')
plt.title('H(X,T)')
plt.savefig('plot_test03.png')
plt.show()
#
# Write the data to files.
#
r8mat_write('h_test03.txt', x_num, t_num, hmat)
r8vec_write('t_test03.txt', t_num, t)
r8vec_write('x_test03.txt', x_num, x)
print ''
print ' H(X,T) written to "h_test03.txt"'
print ' T values written to "t_test03.txt"'
print ' X values written to "x_test3.txt"'
print ''
print 'FD1D_HEAT_IMPLICIT_TEST03:'
print ' Normal end of execution.'
return
def fd1d_heat_implicit_test01 ( ):
#*****************************************************************************80
#
## FD1D_HEAT_IMPLICIT_TEST01 does a simple test problem.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 November 2014
#
# Author:
#
# John Burkardt
#
from fd1d_heat_implicit import fd1d_heat_implicit
from fd1d_heat_implicit_cfl import fd1d_heat_implicit_cfl
from fd1d_heat_implicit_matrix import fd1d_heat_implicit_matrix
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import axes3d, Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ''
print 'FD1D_HEAT_IMPLICIT_TEST01:'
print ' Compute an approximate solution to the time-dependent'
print ' one dimensional heat equation:'
print ' dH/dt - K * d2H/dx2 = f(x,t)'
print ' Run a simple test case.'
#
# Heat coefficient.
#
k = k_test01 ( )
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = ( x_max - x_min ) / ( x_num - 1 )
x = np.linspace ( x_min, x_max, x_num )
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 201
t_min = 0.0
t_max = 80.0
dt = ( t_max - t_min ) / ( t_num - 1 )
t = np.linspace ( t_min, t_max, t_num )
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_implicit_cfl ( k, t_num, t_min, t_max, x_num, x_min, x_max )
print ''
print ' Number of X nodes = %d' % ( x_num )
print ' X interval is [%f,%f]' % ( x_min, x_max )
print ' X spacing is %f' % ( dx )
print ' Number of T values = %d' % ( t_num )
print ' T interval is [%f,%f]' % ( t_min, t_max )
print ' T spacing is %f' % ( dt )
print ' Constant K = %g' % ( k )
print ' CFL coefficient = %g' % ( cfl )
#
# Get the system matrix.
#
a = fd1d_heat_implicit_matrix ( x_num, cfl )
hmat = np.zeros ( ( x_num, t_num ) )
for j in range ( 0, t_num ):
if ( j == 0 ):
h = ic_test01 ( x_num, x, t[j] )
h = bc_test01 ( x_num, x, t[j], h )
else:
h = fd1d_heat_implicit ( a, x_num, x, t[j-1], dt, cfl, rhs_test01, bc_test01, h )
for i in range ( 0, x_num ):
hmat[i,j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid ( t, x )
fig = plt.figure ( )
ax = Axes3D ( fig )
surf = ax.plot_surface ( xmat, tmat, hmat )
plt.xlabel ( '<---X--->' )
plt.ylabel ( '<---T--->' )
plt.title ( 'H(X,T)' )
plt.savefig ( 'plot_test01.png' )
plt.show ( )
#
# Write the data to files.
#
r8mat_write ( 'h_test01.txt', x_num, t_num, hmat )
r8vec_write ( 't_test01.txt', t_num, t )
r8vec_write ( 'x_test01.txt', x_num, x )
print ''
print ' H(X,T) written to "h_test01.txt"'
print ' T values written to "t_test01.txt"'
print ' X values written to "x_test01.txt"'
print ''
print 'FD1D_HEAT_IMPLICIT_TEST01:'
print ' Normal end of execution.'
return
def fd1d_heat_explicit_test02 ( ):
#*****************************************************************************80
#
## FD1D_HEAT_EXPLICIT_TEST02 does a problem with known solution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 06 November 2014
#
# Author:
#
# John Burkardt
#
from fd1d_heat_explicit import fd1d_heat_explicit
from fd1d_heat_explicit_cfl import fd1d_heat_explicit_cfl
from math import sqrt
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ''
print 'FD1D_HEAT_EXPLICIT_TEST02:'
print ' Compute an approximate solution to the time-dependent'
print ' one dimensional heat equation for a problem where we'
print ' know the exact solution.'
print ''
print ' dH/dt - K * d2H/dx2 = f(x,t)'
print ''
print ' Run a simple test case.'
#
# Heat coefficient.
#
k = k_test02 ( )
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = ( x_max - x_min ) / ( x_num - 1 )
x = np.linspace ( x_min, x_max, x_num )
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 26
t_min = 0.0
t_max = 10.0
dt = ( t_max - t_min ) / ( t_num - 1 )
t = np.linspace ( t_min, t_max, t_num )
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_explicit_cfl ( k, t_num, t_min, t_max, x_num, x_min, x_max )
print ''
print ' Number of X nodes = %d' % ( x_num )
print ' X interval is [%f,%f]' % ( x_min, x_max )
print ' X spacing is %f' % ( dx )
print ' Number of T values = %d' % ( t_num )
print ' T interval is [%f,%f]' % ( t_min, t_max )
print ' T spacing is %f' % ( dt )
print ' Constant K = %g' % ( k )
print ' CFL coefficient = %g' % ( cfl )
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
gmat = np.zeros ( ( x_num, t_num ) )
hmat = np.zeros ( ( x_num, t_num ) )
print ''
print ' Step Time RMS Error'
print ''
for j in range ( 0, t_num ):
if ( j == 0 ):
h = ic_test02 ( x_num, x, t[j] )
h = bc_test02 ( x_num, x, t[j], h )
else:
h = fd1d_heat_explicit ( x_num, x, t[j-1], dt, cfl, rhs_test02, bc_test02, h )
g = exact_test02 ( x_num, x, t[j] )
e = 0.0
for i in range ( 0, x_num ):
e = e + ( h[i] - g[i] ) ** 2
e = sqrt ( e ) / sqrt ( x_num )
print ' %4d %14.6g %14.6g' % ( j, t[j], e )
for i in range ( 0, x_num ):
gmat[i,j] = g[i]
hmat[i,j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid ( t, x )
fig = plt.figure ( )
ax = Axes3D ( fig )
surf = ax.plot_surface ( xmat, tmat, hmat )
plt.xlabel ( '<---X--->' )
plt.ylabel ( '<---T--->' )
plt.title ( 'U(X,T)' )
plt.savefig ( 'plot_test02.png' )
plt.show ( )
#
# Write the data to files.
#
r8mat_write ( 'g_test02.txt', x_num, t_num, gmat )
r8mat_write ( 'h_test02.txt', x_num, t_num, hmat )
r8vec_write ( 't_test02.txt', t_num, t )
r8vec_write ( 'x_test02.txt', x_num, x )
print ''
print ' G(X,T) written to "g_test02.txt"'
print ' H(X,T) written to "h_test02.txt"'
print ' T values written to "t_test02.txt"'
print ' X values written to "x_test02.txt"'
return
def fem1d_heat_explicit_test02 ( ):
#*****************************************************************************80
#
## FEM1D_HEAT_EXPLICIT_TEST02 does a problem with known solution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 November 2014
#
# Author:
#
# John Burkardt
#
from assemble_mass import assemble_mass
from fem1d_heat_explicit import fem1d_heat_explicit
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
import numpy as np
from math import sqrt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST02:'
print ' Using the finite element method,'
print ' compute an approximate solution to the time-dependent'
print ' one dimensional heat equation for a problem where we'
print ' know the exact solution.'
print ''
print ' dH/dt - K * d2H/dx2 = f(x,t)'
#
# Set the nodes.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = ( x_max - x_min ) / ( x_num - 1 )
x = np.linspace ( x_min, x_max, x_num )
#
# Set the times.
#
t_num = 51
t_min = 0.0
t_max = 10.0
dt = ( t_max - t_min ) / ( t_num - 1 )
t = np.linspace ( t_min, t_max, t_num )
#
# Set finite element information.
#
element_num = x_num - 1
element_node = np.zeros ( ( 2, element_num ) )
for j in range ( 0, element_num ):
element_node[0,j] = j
element_node[1,j] = j + 1
quad_num = 3
mass = assemble_mass ( x_num, x, element_num, element_node, quad_num )
print ''
print ' Number of X nodes = %d' % ( x_num )
print ' X interval = [ %f, %f ]' % ( x_min, x_max )
print ' X step size = %f' % ( dx )
print ' Number of T steps = %d' % ( t_num )
print ' T interval = [ %f, %f ]' % ( t_min, t_max )
print ' T step size = %f' % ( dt )
print ' Number of elements = %d' % ( element_num )
print ' Number of quadrature points = %d' % ( quad_num )
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
g_mat = np.zeros ( ( x_num, t_num ) )
h_mat = np.zeros ( ( x_num, t_num ) )
print ''
print ' Step Time RMS Error'
print ''
for j in range ( 0, t_num ):
if ( j == 0 ):
h = ic_test02 ( x_num, x, t[j] )
h = bc_test02 ( x_num, x, t[j], h )
else:
h = fem1d_heat_explicit ( x_num, x, t[j-1], dt, k_test02, \
rhs_test02, bc_test02, element_num, element_node, quad_num, mass, h )
g = exact_test02 ( x_num, x, t[j] )
e = 0.0
for i in range ( 0, x_num ):
e = e + ( h[i] - g[i] ) ** 2
e = sqrt ( e / x_num )
print ' %4d %14.6g %14.6g' % ( j, t[j], e )
for i in range ( 0, x_num ):
g_mat[i,j] = g[i]
h_mat[i,j] = h[i]
#
# Make a product grid of T and X for plotting.
#
t_mat, x_mat = np.meshgrid ( t, x )
#
# Plot the data.
#
fig = plt.figure ( )
ax = fig.add_subplot ( 111, projection = '3d' )
surf = ax.plot_surface ( x_mat, t_mat, h_mat )
plt.xlabel ( '<---X--->' )
plt.ylabel ( '<---T--->' )
plt.title ( 'U(X,T)' )
plt.savefig ( 'plot_test02.png' )
plt.show ( )
#
# Write the data to files.
#
r8mat_write ( 'g_test02.txt', x_num, t_num, g_mat )
r8mat_write ( 'h_test02.txt', x_num, t_num, h_mat )
r8vec_write ( 't_test02.txt', t_num, t )
r8vec_write ( 'x_test02.txt', x_num, x )
print ''
print ' G(X,T) written to "g_test02.txt"'
print ' H(X,T) written to "h_test02.txt"'
print ' T values written to "t_test02.txt"'
print ' X values written to "x_test02.txt"'
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST02:'
print ' Normal end of execution.'
return
def fd1d_heat_explicit_test01():
# *****************************************************************************80
#
## FD1D_HEAT_EXPLICIT_TEST01 does a simple test problem
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 06 November 2014
#
# Author:
#
# John Burkardt
#
from fd1d_heat_explicit import fd1d_heat_explicit
from fd1d_heat_explicit_cfl import fd1d_heat_explicit_cfl
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ""
print "FD1D_HEAT_EXPLICIT_TEST01:"
print " Compute an approximate solution to the time-dependent"
print " one dimensional heat equation:"
print ""
print " dH/dt - K * d2H/dx2 = f(x,t)"
print ""
print " Run a simple test case."
#
# Heat coefficient.
#
k = k_test01()
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = (x_max - x_min) / (x_num - 1)
x = np.linspace(x_min, x_max, x_num)
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 201
t_min = 0.0
t_max = 80.0
dt = (t_max - t_min) / (t_num - 1)
t = np.linspace(t_min, t_max, t_num)
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_explicit_cfl(k, t_num, t_min, t_max, x_num, x_min, x_max)
print ""
print " Number of X nodes = %d" % (x_num)
print " X interval is [%f,%f]" % (x_min, x_max)
print " X spacing is %f" % (dx)
print " Number of T values = %d" % (t_num)
print " T interval is [%f,%f]" % (t_min, t_max)
print " T spacing is %f" % (dt)
print " Constant K = %g" % (k)
print " CFL coefficient = %g" % (cfl)
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
hmat = np.zeros((x_num, t_num))
for j in range(0, t_num):
if j == 0:
h = ic_test01(x_num, x, t[j])
h = bc_test01(x_num, x, t[j], h)
else:
h = fd1d_heat_explicit(x_num, x, t[j - 1], dt, cfl, rhs_test01, bc_test01, h)
for i in range(0, x_num):
hmat[i, j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid(t, x)
fig = plt.figure()
# ax = fig.add_subplot ( 111, projection = '3d' )
ax = Axes3D(fig)
surf = ax.plot_surface(xmat, tmat, hmat)
plt.xlabel("<---X--->")
plt.ylabel("<---T--->")
plt.title("U(X,T)")
plt.savefig("plot_test01.png")
plt.show()
#
# Write the data to files.
#
r8mat_write("h_test01.txt", x_num, t_num, hmat)
r8vec_write("t_test01.txt", t_num, t)
r8vec_write("x_test01.txt", x_num, x)
print ""
print ' H(X,T) written to "h_test01.txt"'
print ' T values written to "t_test01.txt"'
print ' X values written to "x_test01.txt"'
#
# Terminate.
#
print ""
print "FD1D_HEAT_EXPLICIT_TEST01:"
print " Normal end of execution."
return
def fem1d_heat_explicit_test02():
#*****************************************************************************80
#
## FEM1D_HEAT_EXPLICIT_TEST02 does a problem with known solution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 November 2014
#
# Author:
#
# John Burkardt
#
from assemble_mass import assemble_mass
from fem1d_heat_explicit import fem1d_heat_explicit
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
import numpy as np
from math import sqrt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST02:'
print ' Using the finite element method,'
print ' compute an approximate solution to the time-dependent'
print ' one dimensional heat equation for a problem where we'
print ' know the exact solution.'
print ''
print ' dH/dt - K * d2H/dx2 = f(x,t)'
#
# Set the nodes.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = (x_max - x_min) / (x_num - 1)
x = np.linspace(x_min, x_max, x_num)
#
# Set the times.
#
t_num = 51
t_min = 0.0
t_max = 10.0
dt = (t_max - t_min) / (t_num - 1)
t = np.linspace(t_min, t_max, t_num)
#
# Set finite element information.
#
element_num = x_num - 1
element_node = np.zeros((2, element_num))
for j in range(0, element_num):
element_node[0, j] = j
element_node[1, j] = j + 1
quad_num = 3
mass = assemble_mass(x_num, x, element_num, element_node, quad_num)
print ''
print ' Number of X nodes = %d' % (x_num)
print ' X interval = [ %f, %f ]' % (x_min, x_max)
print ' X step size = %f' % (dx)
print ' Number of T steps = %d' % (t_num)
print ' T interval = [ %f, %f ]' % (t_min, t_max)
print ' T step size = %f' % (dt)
print ' Number of elements = %d' % (element_num)
print ' Number of quadrature points = %d' % (quad_num)
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
g_mat = np.zeros((x_num, t_num))
h_mat = np.zeros((x_num, t_num))
print ''
print ' Step Time RMS Error'
print ''
for j in range(0, t_num):
if (j == 0):
h = ic_test02(x_num, x, t[j])
h = bc_test02(x_num, x, t[j], h)
else:
h = fem1d_heat_explicit ( x_num, x, t[j-1], dt, k_test02, \
rhs_test02, bc_test02, element_num, element_node, quad_num, mass, h )
g = exact_test02(x_num, x, t[j])
e = 0.0
for i in range(0, x_num):
e = e + (h[i] - g[i])**2
e = sqrt(e / x_num)
print ' %4d %14.6g %14.6g' % (j, t[j], e)
for i in range(0, x_num):
g_mat[i, j] = g[i]
h_mat[i, j] = h[i]
#
# Make a product grid of T and X for plotting.
#
t_mat, x_mat = np.meshgrid(t, x)
#
# Plot the data.
#
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
surf = ax.plot_surface(x_mat, t_mat, h_mat)
plt.xlabel('<---X--->')
plt.ylabel('<---T--->')
plt.title('U(X,T)')
plt.savefig('plot_test02.png')
plt.show()
#
# Write the data to files.
#
r8mat_write('g_test02.txt', x_num, t_num, g_mat)
r8mat_write('h_test02.txt', x_num, t_num, h_mat)
r8vec_write('t_test02.txt', t_num, t)
r8vec_write('x_test02.txt', x_num, x)
print ''
print ' G(X,T) written to "g_test02.txt"'
print ' H(X,T) written to "h_test02.txt"'
print ' T values written to "t_test02.txt"'
print ' X values written to "x_test02.txt"'
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST02:'
print ' Normal end of execution.'
return
def fd1d_heat_explicit_test02 ( ):
# FD1D_HEAT_EXPLICIT_TEST03 does a simple test problem.
from fd1d_heat_explicit import fd1d_heat_explicit
from fd1d_heat_explicit_cfl import fd1d_heat_explicit_cfl
from math import sqrt
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ''
print 'FD1D_HEAT_EXPLICIT_TEST03:'
print ' Compute an approximate solution to the time-dependent'
print ' one dimensional heat equation:'
print ''
print ' dH/dt - K * d2H/dx2 = f(x,t)'
print ''
print ' Run a simple test case.'
#
# Heat coefficient.
#
k = k_test02 ( )
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = -5.0
x_max = +5.0
dx = ( x_max - x_min ) / ( x_num - 1 )
x = np.linspace ( x_min, x_max, x_num )
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 81
t_min = 0.0
t_max = 4.0
dt = ( t_max - t_min ) / ( t_num - 1 )
t = np.linspace ( t_min, t_max, t_num )
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_explicit_cfl ( k, t_num, t_min, t_max, x_num, x_min, x_max )
print ''
print ' Number of X nodes = %d' % ( x_num )
print ' X interval is [%f,%f]' % ( x_min, x_max )
print ' X spacing is %f' % ( dx )
print ' Number of T values = %d' % ( t_num )
print ' T interval is [%f,%f]' % ( t_min, t_max )
print ' T spacing is %f' % ( dt )
print ' Constant K = %g' % ( k )
print ' CFL coefficient = %g' % ( cfl )
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
hmat = np.zeros ( ( x_num, t_num ) )
for j in range ( 0, t_num ):
if ( j == 0 ):
h = ic_test02 ( x_num, x, t[j] )
h = bc_test02 ( x_num, x, t[j], h )
else:
h = fd1d_heat_explicit ( x_num, x, t[j-1], dt, cfl, rhs_test02, bc_test02, h )
for i in range ( 0, x_num ):
hmat[i,j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid ( t, x )
fig = plt.figure ( )
ax = Axes3D ( fig )
surf = ax.plot_surface ( xmat, tmat, hmat )
plt.xlabel ( '<---X--->' )
plt.ylabel ( '<---T--->' )
plt.title ( 'U(X,T)' )
plt.savefig ( 'plot_test02.png' )
plt.show ( )
#
# Write the data to files.
#
r8mat_write ( 'h_test02.txt', x_num, t_num, hmat )
r8vec_write ( 't_test02.txt', t_num, t )
r8vec_write ( 'x_test02.txt', x_num, x )
print ''
print ' H(X,T) written to "h_test02.txt"'
print ' T values written to "t_test02.txt"'
print ' X values written to "x_test3.txt"'
return
def fd1d_heat_explicit_test02():
#*****************************************************************************80
#
## FD1D_HEAT_EXPLICIT_TEST02 does a problem with known solution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 06 November 2014
#
# Author:
#
# John Burkardt
#
from fd1d_heat_explicit import fd1d_heat_explicit
from fd1d_heat_explicit_cfl import fd1d_heat_explicit_cfl
from math import sqrt
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
import numpy as np
print ''
print 'FD1D_HEAT_EXPLICIT_TEST02:'
print ' Compute an approximate solution to the time-dependent'
print ' one dimensional heat equation for a problem where we'
print ' know the exact solution.'
print ''
print ' dH/dt - K * d2H/dx2 = f(x,t)'
print ''
print ' Run a simple test case.'
#
# Heat coefficient.
#
k = k_test02()
#
# X_NUM is the number of equally spaced nodes to use between 0 and 1.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = (x_max - x_min) / (x_num - 1)
x = np.linspace(x_min, x_max, x_num)
#
# T_NUM is the number of equally spaced time points between 0 and 10.0.
#
t_num = 26
t_min = 0.0
t_max = 10.0
dt = (t_max - t_min) / (t_num - 1)
t = np.linspace(t_min, t_max, t_num)
#
# Get the CFL coefficient.
#
cfl = fd1d_heat_explicit_cfl(k, t_num, t_min, t_max, x_num, x_min, x_max)
print ''
print ' Number of X nodes = %d' % (x_num)
print ' X interval is [%f,%f]' % (x_min, x_max)
print ' X spacing is %f' % (dx)
print ' Number of T values = %d' % (t_num)
print ' T interval is [%f,%f]' % (t_min, t_max)
print ' T spacing is %f' % (dt)
print ' Constant K = %g' % (k)
print ' CFL coefficient = %g' % (cfl)
#
# Running the code produces an array H of temperatures H(t,x),
# and vectors x and t.
#
gmat = np.zeros((x_num, t_num))
hmat = np.zeros((x_num, t_num))
print ''
print ' Step Time RMS Error'
print ''
for j in range(0, t_num):
if (j == 0):
h = ic_test02(x_num, x, t[j])
h = bc_test02(x_num, x, t[j], h)
else:
h = fd1d_heat_explicit(x_num, x, t[j - 1], dt, cfl, rhs_test02,
bc_test02, h)
g = exact_test02(x_num, x, t[j])
e = 0.0
for i in range(0, x_num):
e = e + (h[i] - g[i])**2
e = sqrt(e) / sqrt(x_num)
print ' %4d %14.6g %14.6g' % (j, t[j], e)
for i in range(0, x_num):
gmat[i, j] = g[i]
hmat[i, j] = h[i]
#
# Plot the data.
#
tmat, xmat = np.meshgrid(t, x)
fig = plt.figure()
ax = Axes3D(fig)
surf = ax.plot_surface(xmat, tmat, hmat)
plt.xlabel('<---X--->')
plt.ylabel('<---T--->')
plt.title('U(X,T)')
plt.savefig('plot_test02.png')
plt.show()
#
# Write the data to files.
#
r8mat_write('g_test02.txt', x_num, t_num, gmat)
r8mat_write('h_test02.txt', x_num, t_num, hmat)
r8vec_write('t_test02.txt', t_num, t)
r8vec_write('x_test02.txt', x_num, x)
print ''
print ' G(X,T) written to "g_test02.txt"'
print ' H(X,T) written to "h_test02.txt"'
print ' T values written to "t_test02.txt"'
print ' X values written to "x_test02.txt"'
return
def fem1d_heat_explicit_test01 ( ):
#*****************************************************************************80
#
## FEM1D_HEAT_EXPLICIT_TEST01 runs a simple test.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 06 November 2014
#
# Author:
#
# John Burkardt
#
from assemble_mass import assemble_mass
from fem1d_heat_explicit import fem1d_heat_explicit
from r8mat_write import r8mat_write
from r8vec_write import r8vec_write
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import matplotlib.pyplot as plt
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST01:'
print ' The time dependent 1D heat equation is'
print ''
print ' Ut - k * Uxx = f(x,t)'
print ''
print ' for space interval A <= X <= B with boundary conditions'
print ''
print ' U(A,t) = UA(t)'
print ' U(B,t) = UB(t)'
print ''
print ' and time interval T0 <= T <= T1 with initial condition'
print ''
print ' U(X,T0) = U0(X).'
print ''
print ' To compute an approximate solution:'
print ' the interval [A,B] is replace by a discretized mesh Xi'
print ' a set of finite element functions PSI(X) are determined,'
print ' the solution U is written as a weighted sum of the basis functions,'
print ' the weak form of the differential equation is written,'
print ' a time grid Tj is imposed, and time derivatives replaced by'
print ' an explicit forward Euler first difference,'
print ''
print ' The continuous PDE has now been transformed into a set of algebraic'
print ' equations for the coefficients C(Xi,Tj).'
#
# Set the nodes.
#
x_num = 21
x_min = 0.0
x_max = 1.0
dx = ( x_max - x_min ) / ( x_num - 1 )
x = np.linspace ( x_min, x_max, x_num )
#
# Set the times.
#
t_num = 401
t_min = 0.0
t_max = 80.0
dt = ( t_max - t_min ) / ( t_num - 1 )
t = np.linspace ( t_min, t_max, t_num )
#
# Set finite element information.
#
element_num = x_num - 1
element_node = np.zeros ( ( 2, element_num ) )
for j in range ( 0, element_num ):
element_node[0,j] = j
element_node[1,j] = j + 1
quad_num = 3
mass = assemble_mass ( x_num, x, element_num, element_node, quad_num )
print ''
print ' Number of X nodes = %d' % ( x_num )
print ' X interval = [ %f, %f ]' % ( x_min, x_max )
print ' X step size = %f' % ( dx )
print ' Number of T steps = %d' % ( t_num )
print ' T interval = [ %f, %f ]' % ( t_min, t_max )
print ' T step size = %f' % ( dt )
print ' Number of elements = %d' % ( element_num )
print ' Number of quadrature points = %d' % ( quad_num )
u_mat = np.zeros ( ( x_num, t_num ) )
for j in range ( 0, t_num ):
if ( j == 0 ):
u = ic_test01 ( x_num, x, t[j] )
u = bc_test01 ( x_num, x, t[j], u )
else:
u = fem1d_heat_explicit ( x_num, x, t[j-1], dt, k_test01, \
rhs_test01, bc_test01, element_num, element_node, quad_num, mass, u )
for i in range ( 0, x_num ):
u_mat[i,j] = u[i]
#
# Make a product grid of T and X for plotting.
#
t_mat, x_mat = np.meshgrid ( t, x )
#
# Make a mesh plot of the solution.
#
fig = plt.figure ( )
ax = fig.add_subplot ( 111, projection = '3d' )
surf = ax.plot_surface ( x_mat, t_mat, u_mat )
plt.xlabel ( '<---X--->' )
plt.ylabel ( '<---T--->' )
plt.title ( 'U(X,T)' )
plt.savefig ( 'plot_test01.png' )
plt.show ( )
#
# Write the data to files.
#
r8mat_write ( 'h_test01.txt', x_num, t_num, u_mat )
r8vec_write ( 't_test01.txt', t_num, t )
r8vec_write ( 'x_test01.txt', x_num, x )
print ''
print ' H(X,T) written to "h_test01.txt"'
print ' T values written to "t_test01.txt"'
print ' X values written to "x_test01.txt"'
print ''
print 'FEM1D_HEAT_EXPLICIT_TEST01:'
print ' Normal end of execution.'
return