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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 MATLAB 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.
</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 = "../../m_src/fd1d_advection_diffusion_steady/fd1d_advection_diffusion_steady.html">
FD1D_ADVECTION_DIFFUSION_STEADY</a>,
a MATLAB 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.
</p>
<p>
<a href = "../../m_src/fd1d_advection_lax/fd1d_advection_lax.html">
FD1D_ADVECTION_LAX</a>,
a MATLAB 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 = "../../m_src/fd1d_advection_lax_wendroff/fd1d_advection_lax_wendroff.html">
FD1D_ADVECTION_LAX_WENDROFF</a>,
a MATLAB 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>
<p>
<a href = "../../m_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
FD1D_BURGERS_LAX</a>,
a MATLAB 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 = "../../m_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
FD1D_BURGERS_LEAP</a>,
a MATLAB 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_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a MATLAB program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fd1d_heat_explicit/fd1d_heat_explicit.html">
FD1D_HEAT_EXPLICIT</a>,
a MATLAB program which
uses the finite difference method and explicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_heat_implicit/fd1d_heat_implicit.html">
FD1D_HEAT_IMPLICIT</a>,
a MATLAB program which
uses the finite difference method and implicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a MATLAB program which
uses the finite difference method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_predator_prey/fd1d_predator_prey.html">
FD1D_PREDATOR_PREY</a>,
a MATLAB program which
implements a finite difference algorithm for predator-prey system
with spatial variation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_wave/fd1d_wave.html">
FD1D_WAVE</a>,
a MATLAB program which
applies the finite difference method to solve the time-dependent
wave equation utt = c * uxx in one spatial dimension.
</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.m">fd1d_advection_ftcs.m</a>,
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>
<li>
<a href = "fd1d_advection_ftcs.png">fd1d_advection_ftcs.png</a>,
an image of the solution.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 25 December 2012.
</i>
<!-- John Burkardt -->
</body>
</html>