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<html>
<head>
<title>
FD1D_HEAT_EXPLICIT - TIme Dependent 1D Heat Equation, Finite Difference, Explicit Time Stepping
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_HEAT_EXPLICIT <br>
Finite Difference Solution of the<br>
Time Dependent 1D Heat Equation<br>
using Explicit Time Stepping
</h1>
<hr>
<p>
<b>FD1D_HEAT_EXPLICIT</b>
is a FORTRAN90 library which
solves the time-dependent 1D heat equation, using the finite difference
method in space, and an explicit version of the method of lines to handle
integration in time.
</p>
<p>
This program solves
<pre>
dUdT - k * d2UdX2 = F(X,T)
</pre>
over the interval [A,B] with boundary conditions
<pre>
U(A,T) = UA(T),
U(B,T) = UB(T),
</pre>
over the time interval [T0,T1] with initial conditions
<pre>
U(X,T0) = U0(X)
</pre>
</p>
<p>
A second order finite difference is used to approximate the
second derivative in space.
</p>
<p>
The solver applies an
explicit forward Euler approximation to the first derivative in time.
</p>
<p>
The resulting finite difference form can be written as
<pre>
U(X,T+dt) - U(X,T) ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) )
------------------ = F(X,T) + k * ------------------------------------
dt dx * dx
</pre>
or, assuming we have solved for all values of U at time T, we have
<pre>
U(X,T+dt) = U(X,T)
+ dt * ( F(X,T)
+ k * ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) ) / dx / dx )
</pre>
</p>
<p>
Other approaches would involve a fully implicit backward Euler approximation or
the Crank-Nicholson approximation. These latter two methods have
improved stability.
</p>
<p>
A second worthwhile change would be to replace the constant
heat conductivity <b>K</b> by a function <b>K(X,T)</b>. The
spatial variation would allow for the modeling of a region
divided into subregions of different materials.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FD1D_HEAT_EXPLICIT</b> is available in
<a href = "../../c_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a C version</a> and
<a href = "../../cpp_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
FD1D_BURGERS_LAX</a>,
a FORTRAN90 program which
applies the finite difference method and the Lax-Wendroff method
to solve the non-viscous time-dependent Burgers equation
in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
FD1D_BURGERS_LEAP</a>,
a FORTRAN90 program which
applies the finite difference method and the leapfrog approach
to solve the non-viscous time-dependent Burgers equation in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a FORTRAN90 program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fd1d_heat_implicit/fd1d_heat_implicit.html">
FD1D_HEAT_IMPLICIT</a>,
a FORTRAN90 program which
uses the finite difference method and implicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a FORTRAN90 program which
uses the finite difference method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fd1d_predator_prey/fd1d_predator_prey.html">
FD1D_PREDATOR_PREY</a>,
a FORTRAN90 program which
uses finite differences to solve a 1D predator prey problem.
</p>
<p>
<a href = "../../f_src/fd1d_wave/fd1d_wave.html">
FD1D_WAVE</a>,
a FORTRAN90 program which
applies the finite difference method to solve the time-dependent
wave equation utt = c * uxx in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fem1d/fem1d.html">
FEM1D</a>,
a FORTRAN90 program which
applies the finite element
method, with piecewise linear basis functions, to a linear
two point boundary value problem;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
George Lindfield, John Penny,<br>
Numerical Methods Using MATLAB,<br>
Second Edition,<br>
Prentice Hall, 1999,<br>
ISBN: 0-13-012641-1,<br>
LC: QA297.P45.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_heat_explicit.f90">fd1d_heat_explicit.f90</a>, the source code.
</li>
<li>
<a href = "fd1d_heat_explicit.sh">fd1d_heat_explicit.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_heat_explicit_prb.f90">fd1d_heat_explicit_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "fd1d_heat_explicit_prb.sh">fd1d_heat_explicit_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fd1d_heat_explicit_prb_output.txt">fd1d_heat_explicit_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>TEST01</b> runs with initial condition 50 everywhere, boundary conditions
of 90 on the left and 70 on the right, and no right hand side source term.
<ul>
<li>
<a href = "plot_test01.png">plot_test01.png</a>,
a PNG image of the solution, using the MATLAB MESH command
to emphasize the method of lines approach underlying the solution.
</li>
<li>
<a href = "h_test01.txt">h_test01.txt</a>,
the computed H data.
</li>
<li>
<a href = "t_test01.txt">t_test01.txt</a>,
the T data.
</li>
<li>
<a href = "x_test01.txt">x_test01.txt</a>,
the X data.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FD1D_HEAT_EXPLICIT:</b> Finite difference solution of 1D heat equation.
</li>
<li>
<b>FD1D_HEAT_EXPLICIT_CFL:</b> compute the Courant-Friedrichs-Loewy coefficient.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_WRITE</b> writes an R8VEC file.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 25 January 2012.
</i>
<!-- John Burkardt -->
</body>
</html>