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<html>
<head>
<title>
FD1D_WAVE - Finite Difference Method, 1D Wave Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_WAVE <br>
Finite Difference Method<br>
1D Wave Equation
</h1>
<hr>
<p>
<b>FD1D_WAVE</b>
is a FORTRAN90 program which
applies the finite difference method to solve
a version of the wave equation in one spatial dimension.
</p>
<p>
This program solves the 1D wave equation of the form:
<pre>
Utt = c^2 Uxx
</pre>
over the spatial interval [X1,X2] and time interval [T1,T2],
with initial conditions:
<pre>
U(T1,X) = U_T1(X),
Ut(T1,X) = UT_T1(X),
</pre>
and boundary conditions of Dirichlet type:
<pre>
U(T,X1) = U_X1(T),<br>
U(T,X2) = U_X2(T).
</pre>
The value <b>C</b> represents the propagation speed of waves.
</p>
<p>
The program uses the finite difference method, and marches
forward in time, solving for all the values of U at the next
time step by using the values known at the previous two time steps.
</p>
<p>
Central differences may be used to approximate both the time
and space derivatives in the original differential equation.
</p>
<p>
Thus, assuming we have available the approximated values of U
at the current and previous times, we may write a discretized
version of the wave equation as follows:
<pre>
Uxx(T,X) = ( U(T, X+dX) - 2 U(T,X) + U(T, X-dX) ) / dX^2
Utt(T,X) = ( U(T+dt,X ) - 2 U(T,X) + U(T-dt,X ) ) / dT^2
</pre>
If we multiply the first term by C^2 and solve for the single
unknown value U(T+dt,X), we have:
<pre>
U(T+dT,X) = ( C^2 * dT^2 / dX^2 ) * U(T, X+dX)
+ 2 * ( 1 - C^2 * dT^2 / dX^2 ) * U(T, X )
+ ( C^2 * dT^2 / dX^2 ) * U(T, X-dX)
- U(T-dT,X )
</pre>
(Equation to advance from time T to time T+dT, except for FIRST step!)
</p>
<p>
However, on the very first step, we only have the values of U
for the initial time, but not for the previous time step.
In that case, we use the initial condition information for dUdT
which can be approximated by a central difference that involves
U(T+dT,X) and U(T-dT,X):
<pre>
dU/dT(T,X) = ( U(T+dT,X) - U(T-dT,X) ) / ( 2 * dT )
</pre>
and so we can estimate U(T-dT,X) as
<pre>
U(T-dT,X) = U(T+dT,X) - 2 * dT * dU/dT(T,X)
</pre>
If we replace the "missing" value of U(T-dT,X) by the known values
on the right hand side, we now have U(T+dT,X) on both sides of the
equation, so we have to rearrange to get the formula we use
for just the first time step:
<pre>
U(T+dT,X) = 1/2 * ( C^2 * dT^2 / dX^2 ) * U(T, X+dX)
+ ( 1 - C^2 * dT^2 / dX^2 ) * U(T, X )
+ 1/2 * ( C^2 * dT^2 / dX^2 ) * U(T, X-dX)
+ dT * dU/dT(T, X )
</pre>
(Equation to advance from time T to time T+dT for FIRST step.)
</p>
<p>
It should be clear now that the quantity ALPHA = C * DT / DX will affect
the stability of the calculation. If it is greater than 1, then
the middle coefficient 1-C^2 DT^2 / DX^2 is negative, and the
sum of the magnitudes of the three coefficients becomes unbounded.
</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_WAVE</b> is available in
<a href = "../../c_src/fd1d_wave/fd1d_wave.html">a C version</a> and
<a href = "../../cpp_src/fd1d_wave/fd1d_wave.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_wave/fd1d_wave.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_wave/fd1d_wave.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_wave/fd1d_wave.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
FD1D_BURGERS_LAX</a>,
a FORTRAN90 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 = "../../f_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
FD1D_BURGERS_LEAP</a>,
a FORTRAN90 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 = "../../f_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a FORTRAN90 program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fd1d_heat_explicit/fd1d_heat_explicit.html">
FD1D_HEAT_EXPLICIT</a>,
a FORTRAN90 program which
uses the finite difference method and explicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fd1d_heat_implicit/fd1d_heat_implicit.html">
FD1D_HEAT_IMPLICIT</a>,
a FORTRAN90 program which
uses the finite difference method and implicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a FORTRAN90 program which
uses the finite difference method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fd1d_predator_prey/fd1d_predator_prey.html">
FD1D_PREDATOR_PREY</a>,
a FORTRAN90 program which
implements a finite difference algorithm for predator-prey system
with spatial variation in 1D.
</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_wave.f90">fd1d_wave.f90</a>, the source code.
</li>
<li>
<a href = "fd1d_wave.sh">fd1d_wave.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_wave_prb.f90">fd1d_wave_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "fd1d_wave_prb.sh">fd1d_wave_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fd1d_wave_prb_output.txt">fd1d_wave_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>test01_plot</b> sets up the "shark wave".
<ul>
<li>
<a href = "test01_plot.txt">test01_plot.txt</a>,
the data.
</li>
<li>
<a href = "test01_plot.png">test01_plot.png</a>,
a plot of the data created by MATLAB's <b>surf</b> command.
</li>
</ul>
</p>
<p>
<b>test02_plot</b> sets up a sine wave.
<ul>
<li>
<a href = "test02_plot.txt">test02_plot.txt</a>,
the data.
</li>
<li>
<a href = "test02_plot.png">test02_plot.png</a>,
a plot of the data created by MATLAB's <b>surf</b> command.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FD1D_WAVE_ALPHA</b> computes ALPHA for the 1D wave equation.
</li>
<li>
<b>FD1D_WAVE_START</b> takes the first step for the wave equation.
</li>
<li>
<b>FD1D_WAVE_STEP</b> computes a step of the 1D wave equation.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 24 January 2012.
</i>
<!-- John Burkardt -->
</body>
</html>