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<html>
<head>
<title>
FD1D_ADVECTION_LAX - Finite Difference Method, 1D Advection Equation, Lax Method
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_ADVECTION_LAX <br>
Finite Difference Method<br>
1D Advection Equation<br>
Lax Method
</h1>
<hr>
<p>
<b>FD1D_ADVECTION_LAX</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 Lax method for the time derivative,
writing graphics files for processing by gnuplot.
</p>
<p>
The Lax method is an improvement to the FTCS method. The FTCS method is
always unstable; nonphysical oscillations appear and grow. The Lax method
is stable, if the time step is small enough; however, it does cause the
wave to gradually spread out and flatten.
</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>
The Lax method modifies the FTCS approximation to the time derivative:
<ul>
<li>
du/dt = (u(t+dt,x)-0.5*u(t,x-dx)-0.5*u(t,x+dx))/dt
</li>
<li>
du/dx = (u(t,x+dx)-u(t,x-dx))/2/dx
</li>
</ul>
</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. Unlike in the case of the FTCS approach, we do not
see unstable oscillations. However, instead, we definitely find the
wave gradually flattening out.
</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_LAX</b> is available in
<a href = "../../c_src/fd1d_advection_lax/fd1d_advection_lax.html">a C version</a> and
<a href = "../../cpp_src/fd1d_advection_lax/fd1d_advection_lax.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_advection_lax/fd1d_advection_lax.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_advection_lax/fd1d_advection_lax.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_advection_lax/fd1d_advection_lax.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_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_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_lax.cpp">fd1d_advection_lax.cpp</a>, the source code.
</li>
<li>
<a href = "fd1d_advection_lax.sh">fd1d_advection_lax.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_lax_output.txt">fd1d_advection_lax_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_lax.png">advection_lax.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_LAX</b> solves the advection equation using the Lax method.
</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 27 January 2013.
</i>
<!-- John Burkardt -->
</body>
</html>