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<html>
<head>
<title>
FD1D_BVP - Finite Difference Method - 1D Boundary Value Problem
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_BVP <br>
Finite Difference Method<br>
1D Boundary Value Problem
</h1>
<hr>
<p>
<b>FD1D_BVP</b>
is a C++ program which
applies the finite difference method to solve
a two point boundary value problem in one spatial dimension.
</p>
<p>
The boundary value problem (BVP) that is to be solved has the form:
<pre>
- d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)
</pre>
in the interval X[0] < x < X[N-1]. The functions a(x), c(x), and f(x) are
given functions, and a formula for a'(x) is also available.
</p>
<p>
Boundary conditions are applied at the endpoints, and in this case,
these are assumed to have the form:
<pre>
u(X[0]) = 0.0;
u(X[N-1]) = 0.0.
</pre>
</p>
<p>
To compute a finite difference approximation, a set of n
nodes is defined over the interval, and, at each interior node,
a discretized version of the BVP is written, with u''(x) and u'(x)
approximated by central differences.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>fd1d_bvp</b> ( <i>n</i>, <i>a</i>,
<i>aprime</i>, <i>c</i>, <i>f</i>, <i>x</i>, <i>u</i> )
</blockquote>
where
<ul>
<li>
<i>n</i> the number of nodes.
</li>
<li>
<i>a</i> the function which evaluates a(x);
</li>
<li>
<i>aprime</i> the function which evaluates a'(x);
</li>
<li>
<i>c</i> the function which evaluates c(x);
</li>
<li>
<i>f</i> the function which evaluates f(x).
</li>
<li>
<i>x</i> the vector of n nodes, which may be nonuniformly spaced.
</li>
<li>
<i>u</i> the output vector of solution values at the nodes.
</li>
</ul>
</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_BVP</b> is available in
<a href = "../../c_src/fd1d_bvp/fd1d_bvp.html">a C version</a> and
<a href = "../../cpp_src/fd1d_bvp/fd1d_bvp.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_bvp/fd1d_bvp.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_bvp/fd1d_bvp.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_bvp/fd1d_bvp.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
FD1D_BURGERS_LAX</a>,
a C++ 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 = "../../cpp_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
FD1D_BURGERS_LEAP</a>,
a C++ 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 = "../../m_src/fd1d_display/fd1d_display.html">
FD1D_DISPLAY</a>,
a MATLAB program which
reads a pair of files defining a 1D finite difference model, and plots the data.
</p>
<p>
<a href = "../../cpp_src/fd1d_heat_explicit/fd1d_heat_explicit.html">
FD1D_HEAT_EXPLICIT</a>,
a C++ program which
uses the finite difference method and explicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../cpp_src/fd1d_heat_implicit/fd1d_heat_implicit.html">
FD1D_HEAT_IMPLICIT</a>,
a C++ program which
uses the finite difference method and implicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../cpp_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a C++ program which
uses the finite difference method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../cpp_src/fd1d_wave/fd1d_wave.html">
FD1D_WAVE</a>,
a C++ 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 = "../../cpp_src/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a linear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
FEM1D_BVP_LINEAR</a>,
a C++ program which
applies the finite element method, with piecewise linear elements,
to a two point boundary value problem in one spatial dimension.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dianne O'Leary,<br>
Finite Differences and Finite Elements: Getting to Know You,<br>
Computing in Science and Engineering,<br>
Volume 7, Number 3, May/June 2005.
</li>
<li>
Dianne O'Leary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313..
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_bvp.cpp">fd1d_bvp.cpp</a>, the source code.
</li>
<li>
<a href = "fd1d_bvp.hpp">fd1d_bvp.hpp</a>, the include file.
</li>
<li>
<a href = "fd1d_bvp.sh">fd1d_bvp.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_bvp_prb.cpp">fd1d_bvp_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "fd1d_bvp_prb.sh">fd1d_bvp_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "fd1d_bvp_prb_output.txt">fd1d_bvp_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "fd1d_bvp_test01_nodes.txt">fd1d_bvp_test01_nodes.txt</a>,
the nodes for test 1.
</li>
<li>
<a href = "fd1d_bvp_test01_values.txt">fd1d_bvp_test01_values.txt</a>,
the computed and exact values for test 1.
</li>
<li>
<a href = "fd1d_bvp_test01.png">fd1d_bvp_test01.png</a>,
a PNG image of a plot of the computed and exact values.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "fd1d_bvp_test02_nodes.txt">fd1d_bvp_test02_nodes.txt</a>,
the nodes for test 2.
</li>
<li>
<a href = "fd1d_bvp_test02_values.txt">fd1d_bvp_test02_values.txt</a>,
the computed and exact values for test 2.
</li>
<li>
<a href = "fd1d_bvp_test02.png">fd1d_bvp_test02.png</a>,
a PNG image of a plot of the computed and exact values.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "fd1d_bvp_test03_nodes.txt">fd1d_bvp_test03_nodes.txt</a>,
the nodes for test 3.
</li>
<li>
<a href = "fd1d_bvp_test03_values.txt">fd1d_bvp_test03_values.txt</a>,
the computed and exact values for test 3.
</li>
<li>
<a href = "fd1d_bvp_test03.png">fd1d_bvp_test03.png</a>,
a PNG image of a plot of the computed and exact values.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "fd1d_bvp_test04_nodes.txt">fd1d_bvp_test04_nodes.txt</a>,
the nodes for test 4
</li>
<li>
<a href = "fd1d_bvp_test04_values.txt">fd1d_bvp_test04_values.txt</a>,
the computed and exact values for test 4.
</li>
<li>
<a href = "fd1d_bvp_test04.png">fd1d_bvp_test04.png</a>,
a PNG image of a plot of the computed and exact values.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "fd1d_bvp_test05_nodes.txt">fd1d_bvp_test05_nodes.txt</a>,
the nodes for test 5.
</li>
<li>
<a href = "fd1d_bvp_test05_values.txt">fd1d_bvp_test05_values.txt</a>,
the computed and exact values for test 5.
</li>
<li>
<a href = "fd1d_bvp_test05.png">fd1d_bvp_test05.png</a>,
a PNG image of a plot of the computed and exact values.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FD1D_BVP</b> solves a two point boundary value problem.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R83NP_FS</b> factors and solves an R83NP system.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_EVEN</b> returns N real values, evenly spaced between ALO and AHI.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 17 May 2009.
</i>
<!-- John Burkardt -->
</body>
</html>