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<html>
<head>
<title>
FD1D_HEAT_IMPLICIT - TIme Dependent 1D Heat Equation, Finite Difference, Implicit Time Stepping
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_HEAT_IMPLICIT <br>
Finite Difference Solution of the<br>
Time Dependent 1D Heat Equation<br>
using Implicit Time Stepping
</h1>
<hr>
<p>
<b>FD1D_HEAT_IMPLICIT</b>
is a FORTRAN90 program which
solves the time-dependent 1D heat equation, using the finite difference
method in space, and an implicit 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
implicit backward 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,+dtT) - 2 U(X,+dtT) + U(X+dx,+dtT) )
------------------ = F(X,T+dt) + k * ---------------------------------------------
dt dx * dx
</pre>
or, assuming we have solved for all values of U at time T, we have
<pre>
- k * dt / dx / dx * U(X-dt,T+dt)
+ ( 1 + 2 * k * dt / dx / dx ) * U(X, T+dt)
- k * dt / dx / dx * U(X+dt,T+dt)
= dt * F(X, T+dt)
+ U(X, T)
</pre>
which can be written as A*x=b, where A is a tridiagonal matrix whose
entries are the same for every time step.
</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_IMPLICIT</b> is available in
<a href = "../../c_src/fd1d_heat_implicit/fd1d_heat_implicit.html">a C version</a> and
<a href = "../../cpp_src/fd1d_heat_implicit/fd1d_heat_implicit.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_heat_implicit/fd1d_heat_implicit.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_heat_implicit/fd1d_heat_implicit.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_heat_implicit/fd1d_heat_implicit.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_explicit/fd1d_heat_explicit.html">
FD1D_HEAT_EXPLICIT</a>,
a FORTRAN90 program which
uses the finite difference method to solve the time dependent
heat equation in 1D, using an explicit time step method.
</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 = "../../m_src/fem_50_heat/fem_50_heat.html">
FEM_50_HEAT</a>,
a MATLAB program which
applies the finite element method to solve the 2D heat equation.
</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>
<p>
<a href = "../../f_src/fem2d_heat/fem2d_heat.html">
FEM2D_HEAT</a>,
a FORTRAN90 program which
applies the finite element method to solve the 2D heat equation.
</p>
<p>
<a href = "../../m_src/hot_pipe/hot_pipe.html">
HOT_PIPE</a>,
a MATLAB program which
uses <b>FEM_50_HEAT</b> to solve a heat problem in a pipe.
</p>
<p>
<a href = "../../m_src/hot_point/hot_point.html">
HOT_POINT</a>,
a MATLAB program which
uses <b>FEM_50_HEAT</b> to solve a heat problem with a point source.
</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_implicit.f90">fd1d_heat_implicit.f90</a>, the source code.
</li>
<li>
<a href = "fd1d_heat_implicit.sh">fd1d_heat_implicit.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "x,txt">x.txt</a>
a file containing the spatial coordinates.
</li>
<li>
<a href = "t,txt">t.txt</a>
a file containing the times.
</li>
<li>
<a href = "u,txt">u.txt</a>
a file containing the values of the computed solution at each point X and time T.
</li>
<li>
<a href = "solution.png">solution.png</a>
a PNG image of the MATLAB graphics created by the commands:
<pre>
x = load ( 'x.txt' );
t = load ( 't.txt' );
u = load ( 'u.txt' );
[ xg, tg ] = meshgrid ( x, t );
mesh ( xg, tg, u );
</pre>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for FD1D_HEAT_IMPLICIT.
</li>
<li>
<b>F</b> returns the right hand side of the heat equation.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R83_NP_FA</b> factors an R83 matrix without pivoting.
</li>
<li>
<b>R83_NP_SL</b> solves an R83 system factored by R83_NP_FA.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>U0</b> returns the initial condition at the starting time.
</li>
<li>
<b>UA</b> returns the Dirichlet boundary condition at the left endpoint.
</li>
<li>
<b>UB</b> returns the Dirichlet boundary condition at the right endpoint.
</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 30 May 2009.
</i>
<!-- John Burkardt -->
</body>
</html>