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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 MATLAB library which
applies the finite difference method to solve
a version of the wave equation in one spatial dimension.
</p>
<p>
The wave equation considered here is an extremely simplified model
of the physics of waves. Many facts about waves are not modeled by this
simple system, including that wave motion in water can depend
on the depth of the medium, that waves tend to disperse, and that waves
of different frequency may travel at different speeds. However, as
a first model of wave motion, the equation is useful because it captures
a most interesting feature of waves, that is, their usefulness in
transmitting signals.
</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 = "../../m_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">
FD1D_ADVECTION_FTCS</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 FTCS method, forward time difference,
centered space difference.
</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>
<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_alpha.m">fd1d_wave_alpha.m</a>,
computes the value of <b>alpha</b> needed for the computation.
</li>
<li>
<a href = "fd1d_wave_start.m">fd1d_wave_start.m</a>,
takes the first time step.
</li>
<li>
<a href = "fd1d_wave_step.m">fd1d_wave_step.m</a>,
takes all subsequent time steps.
</li>
<li>
<a href = "piecewise_linear.m">piecewise_linear.m</a>,
evaluates a piecewise linear spline.
</li>
<li>
<a href = "r8mat_write.m">r8mat_write.m</a>,
writes an R8MAT to a file.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_wave_test.m">fd1d_wave_test.m</a>,
a program which runs several test cases.
</li>
<li>
<a href = "fd1d_wave_test01.m">fd1d_wave_test01.m</a>,
uses a sort of shark fin as the initial condition.
</li>
<li>
<a href = "fd1d_wave_test02.m">fd1d_wave_test02.m</a>,
uses a sine curve as the initial condition.
</li>
<li>
<a href = "fd1d_test01_step06.png">fd1d_test01_step06.png</a>,
a plot of step 6 for test #1.
</li>
<li>
<a href = "test01_plot.txt">test01_plot.txt</a>,
the data.
</li>
<li>
<a href = "fd1d_test02_step13.png">fd1d_test02_step13.png</a>,
a plot of step 13 for test #2.
</li>
<li>
<a href = "test02_plot.txt">test02_plot.txt</a>,
the data.
</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 24 January 2012.
</i>
<!-- John Burkardt -->
</body>
</html>