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<html>
<head>
<title>
FD1D_ADVECTION_FTCS - Finite Difference Method, 1D Advection Equation, Forward Time, Centered Space
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_ADVECTION_FTCS <br>
Finite Difference Method<br>
1D Advection Equation<br>
Forward Time Difference, Centered Space Difference
</h1>
<hr>
<p>
<b>FD1D_ADVECTION_FTCS</b>
is 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 FTCS method, forward time difference,
centered space difference, writing graphics files for processing by gnuplot.
</p>
<p>
We solve the constant-velocity advection equation in 1D,
<pre>
du/dt = - c du/dx
</pre>
over the interval:
<pre>
0.0 <= x <= 1.0
</pre>
with periodic boundary conditions, and
with a given initial condition
<pre>
u(0,x) = (10x-4)^2 (6-10x)^2 for 0.4 <= x <= 0.6
= 0 elsewhere.
</pre>
</p>
<p>
We use a method known as FTCS:
<ul>
<li>
FT: Forward Time : du/dt = (u(t+dt,x)-u(t,x))/dt
</li>
<li>
CS: Centered Space: du/dx = (u(t,x+dx)-u(t,x-dx))/2/dx
</li>
</ul>
</p>
<p>
The FTCS method is <i>unstable</i> for the advection problem.
One purpose of this example is to demonstrate that fact.
</p>
<p>
For our simple case, the advection velocity is constant
in time and space. Therefore, (given our periodic boundary conditions),
the solution should simply move smoothly from left to right, returning
on the left again. Instead, because of the instabilities, we see
that the solution quickly becomes dominated by erroneous oscillations.
</p>
<p>
There are more sophisticated methods for the advection problem,
which do not exhibit this behavior.
</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_FTCS</b> is available in
<a href = "../../c_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">a C version</a> and
<a href = "../../cpp_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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_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 = "../../cpp_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a C++ program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</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_predator_prey/fd1d_predator_prey.html">
FD1D_PREDATOR_PREY</a>,
a C++ program which
implements a finite difference algorithm for predator-prey system
with spatial variation 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/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which
illustrate how a program can write data and command files
so that gnuplot can create plots of the program results.
</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_advection_ftcs.cpp">fd1d_advection_ftcs.cpp</a>, the source code.
</li>
<li>
<a href = "fd1d_advection_ftcs.sh">fd1d_advection_ftcs.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_ftcs_output.txt">fd1d_advection_ftcs_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
Graphical output for this program was created using GNUPLOT. Data at selected
time steps was written to a "data" file, and the appropriate GNUPLOT commands were
written to a "command" file. The plot can be created by the command
<pre>
gnuplot < advection_commands.txt
</pre>
<ul>
<li>
<a href = "advection_data.txt">advection_data.txt</a>,
the solution data.
</li>
<li>
<a href = "advection_commands.txt">advection_commands.txt</a>,
gnuplot commands to plot the data.
</li>
<li>
<a href = "advection.png">advection.png</a>,
a (not very satisfactory) image of the solution.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FD1D_ADVECTION_FTCS</b> solves the advection equation using the FTCS method.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>INITIAL_CONDITION</b> sets the initial condition.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</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 26 December 2012.
</i>
<!-- John Burkardt -->
</body>
</html>