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fd1d_advection_diffusion_steady.html
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<html>
<head>
<title>
FD1D_ADVECTION_DIFFUSION_STEADY - Finite Difference Method, Steady 1D Advection Diffusion Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_ADVECTION_DIFFUSION_STEADY <br>
Finite Difference Method<br>
Steady 1D Advection Diffusion Equation
</h1>
<hr>
<p>
<b>FD1D_ADVECTION_DIFFUSION_STEADY</b>
is a C++ program which
applies the finite difference method to solve the
steady advection diffusion equation v*ux-k*uxx=0 in
one spatial dimension, with constant velocity v and diffusivity k,
creating graphics files for processing by GNUPLOT.
</p>
<p>
We solve the steady constant-velocity advection diffusion equation in 1D,
<pre>
v du/dx - k d^2u/dx^2
</pre>
over the interval:
<pre>
0.0 <= x <= 1.0
</pre>
with boundary conditions
<pre>
u(0) = 0, u(1) = 1.
</pre>
</p>
<p>
We do this by discretizing the interval [0,1] into NX nodes.
We write the boundary conditions at the first and last nodes.
At the i-th interior node, we replace derivatives by differences:
<ul>
<li>
du/dx = (u(x+dx)-u(x-dx))/2/dx
</li>
<li>
d^2u/dx^2 = (u(x+dx)-2u(x)+u(x-dx))/dx/dx
</li>
</ul>
This becomes a set of NX linear equations in the NX unknown values of U.
</p>
<p>
The exact solution to this differential equation is known:
<pre>
u = ( 1 - exp ( r * x ) ) / ( 1 - exp ( r ) )
</pre>
where r = v * l / k;
</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_ADVECTION_DIFFUSION_STEADY</b> is available in
<a href = "../../c_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">a C version</a> and
<a href = "../../cpp_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">
FD1D_ADVECTION_FTCS</a>,
a C++ program which
applies the finite difference method to solve the time-dependent
advection equation ut = - c * ux in one spatial dimension, with
a constant velocity, using the forward time, centered space (FTCS)
difference method.
</p>
<p>
<a href = "../../cpp_src/fd1d_advection_lax/fd1d_advection_lax.html">
FD1D_ADVECTION_LAX</a>,
a C++ program which
applies the finite difference method to solve the time-dependent
advection equation ut = - c * ux in one spatial dimension, with
a constant velocity, using the Lax method to treat the time derivative.
</p>
<p>
<a href = "../../cpp_src/fd1d_advection_lax_wendroff/fd1d_advection_lax_wendroff.html">
FD1D_ADVECTION_LAX_WENDROFF</a>,
a C++ program which
applies the finite difference method to solve the time-dependent
advection equation ut = - c * ux in one spatial dimension, with
a constant velocity, using the Lax-Wendroff method to treat the time derivative.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Charles Hall, Thomas Porsching,<br>
Numerical Analysis of Partial Differential Equations,<br>
Prentice-Hall, 1990,<br>
ISBN: 013626557X,<br>
LC: QA374.H29.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_advection_diffusion_steady.cpp">fd1d_advection_diffusion_steady.cpp</a>, the source code.
</li>
<li>
<a href = "fd1d_advection_diffusion_steady.sh">fd1d_advection_diffusion_steady.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_advection_diffusion_steady_output.txt">fd1d_advection_diffusion_steady_output.txt</a>,
the output file.
</li>
<li>
<a href = "fd1d_advection_diffusion_steady_commands.txt">fd1d_advection_diffusion_steady_commands.txt</a>,
Gnuplot commands.
</li>
<li>
<a href = "fd1d_advection_diffusion_steady_data.txt">fd1d_advection_diffusion_steady_data.txt</a>,
data for Gnuplot.
</li>
<li>
<a href = "fd1d_advection_diffusion_steady.png">fd1d_advection_diffusion_steady.png</a>,
a PNG image of the solution.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FD1D_ADVECTION_DIFFUSION_STEADY</b> solves steady advection diffusion equation.
</li>
<li>
<b>R8VEC_LINSPACE_NEW</b> creates a vector of linearly spaced values.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRISOLVE</b> factors and solves a tridiagonal system.
</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 03 May 2014.
</i>
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