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wave_mpi.html
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<html>
<head>
<title>
WAVE_MPI - Parallel Solution of the 1D Wave Equation using MPI
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
WAVE_MPI <br> Parallel Solution of the 1D Wave Equation using MPI
</h1>
<hr>
<p>
<b>WAVE_MPI</b>
is a C++ program which
solves the 1D wave equation in parallel, using MPI.
</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(X,T1) = U_T1(X),
Ut(X,T1) = UT_T1(X),
</pre>
and boundary conditions of Dirichlet type:
<pre>
U(X1,T) = U_X1(T),<br>
U(X2,T) = 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(X,T) = ( U(X+dX,T ) - 2 U(X,T) + U(X-dX,T, ) ) / dX^2
Utt(X,T) = ( U(X, T+dT) - 2 U(X,T) + U(X, T-dt) ) / dT^2
</pre>
If we multiply the first term by C^2 and solve for the single
unknown value U(X,T+dt), we have:
<pre>
U(X,T+dT) = ( C^2 * dT^2 / dX^2 ) * U(X+dX,T)
+ 2 * ( 1 - C^2 * dT^2 / dX^2 ) * U(X, T)
+ ( C^2 * dT^2 / dX^2 ) * U(X-dX,T)
- U(X, T-dT)
</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(X,T+dT) and U(X,T-dT):
<pre>
dU/dT(X,T) = ( U(X,T+dT) - U(X,T-dT) ) / ( 2 * dT )
</pre>
and so we can estimate U(X,T-dT) as
<pre>
U(X,T-dT) = U(X,T+dT) - 2 * dT * dU/dT(X,T)
</pre>
If we replace the "missing" value of U(X,T-dT) by the known values
on the right hand side, we now have U(X,T+dT) 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(X,T+dT) = 1/2 * ( C^2 * dT^2 / dX^2 ) * U(X+dX,T)
+ ( 1 - C^2 * dT^2 / dX^2 ) * U(X, T)
+ 1/2 * ( C^2 * dT^2 / dX^2 ) * U(X-dX,T)
+ dT * dU/dT(X, T )
</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>
<p>
We wish to use MPI in order to accelerate this computation. We use the
method of domain decomposition - that is, we assume we have P MPI processes,
we divide the original interval into P subintervals, and we expect
each process to update the data associated with its subinterval.
</p>
<p>
However, to compute the estimated solution U(X,T+dT) at the next time
step requires information about U(X-dX,T) and U(X+dX,T). When process
ID tries to make these estimates, it will need one value from process
ID-1, and one from process ID+1, before it can make all the updates.
MPI allows the processes to communicate this information using messages.
</p>
<p>
At the end of the complete calculation, we wish to print a table of the
solution at the final time. To do this in an organized fashion, we want
each process to send its final result to the master process (with ID = 0).
Once all the data has been collected, the master process prints it.
</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>WAVE_MPI</b> is available in
<a href = "../../c_src/wave_mpi/wave_mpi.html">a C version</a> and
<a href = "../../cpp_src/wave_mpi/wave_mpi.html">a C++ version</a> and
<a href = "../../f77_src/wave_mpi/wave_mpi.html">a FORTRAN77 version</a> and
<a href = "../../f_src/wave_mpi/wave_mpi.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/communicator_mpi/communicator_mpi.html">
COMMUNICATOR_MPI</a>,
a C++ program which
creates new communicators involving a subset of initial
set of MPI processes in the default communicator MPI_COMM_WORLD.
</p>
<p>
<a href = "../../cpp_src/heat_mpi/heat_mpi.html">
HEAT_MPI</a>,
a C++ program which
solves the 1D Time Dependent Heat Equation using MPI.
</p>
<p>
<a href = "../../cpp_src/hello_mpi/hello_mpi.html">
HELLO_MPI</a>,
a C++ program which
prints out "Hello, world!" using the MPI parallel programming environment.
</p>
<p>
<a href = "../../examples/moab/moab.html">
MOAB</a>,
examples which
illustrate the use of the MOAB job scheduler for a computer cluster.
</p>
<p>
<a href = "../../cpp_src/mpi/mpi.html">
MPI</a>,
C++ programs which
illustrate the use of the MPI application program interface
for carrying out parallel computations in a distributed memory environment.
</p>
<p>
<a href = "../../cpp_src/mpi_stubs/mpi_stubs.html">
MPI_STUBS</a>,
a C++ library which
contains "stub" MPI routines, allowing a user to compile, load, and possibly
run an MPI program on a serial machine.
</p>
<p>
<a href = "../../cpp_src/multitask_mpi/multitask_mpi.html">
MULTITASK_MPI</a>,
a C++ program which
demonstrates how to "multitask", that is, to execute several unrelated
and distinct tasks simultaneously, using MPI for parallel execution.
</p>
<p>
<a href = "../../cpp_src/prime_mpi/prime_mpi.html">
PRIME_MPI</a>,
a C++ program which
counts the number of primes between 1 and N, using MPI for parallel execution.
</p>
<p>
<a href = "../../cpp_src/quad_mpi/quad_mpi.html">
QUAD_MPI</a>,
a C++ program which
approximates an integral using a quadrature rule, and carries out the
computation in parallel using MPI.
</p>
<p>
<a href = "../../cpp_src/random_mpi/random_mpi.html">
RANDOM_MPI</a>,
a C++ program which
demonstrates one way to generate the same sequence of random numbers
for both sequential execution and parallel execution under MPI.
</p>
<p>
<a href = "../../cpp_src/ring_mpi/ring_mpi.html">
RING_MPI</a>,
a C++ program which
uses the MPI parallel programming environment, and measures the time
necessary to copy a set of data around a ring of processes.
</p>
<p>
<a href = "../../cpp_src/satisfy_mpi/satisfy_mpi.html">
SATISFY_MPI</a>,
a C++ program which
demonstrates, for a particular circuit, an exhaustive search
for solutions of the circuit satisfiability problem, using MPI to
carry out the calculation in parallel.
</p>
<p>
<a href = "../../cpp_src/search_mpi/search_mpi.html">
SEARCH_MPI</a>,
a C++ program which
searches integers between A and B for a value J such that F(J) = C,
using MPI.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Geoffrey Fox, Mark Johnson, Gregory Lyzenga, Steve Otto, John Salmon,
David Walker,<br>
Solving problems on concurrent Processors,<br>
Volume 1: General Techniques and Regular Problems, <br>
Prentice Hall, 1988,<br>
ISBN: 0-13-8230226,<br>
LC: QA76.5.F627.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "wave_mpi.cpp">wave_mpi.cpp</a>, the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>WAVE_FSU</b> compiles and runs the program on the FSU HPC cluster.
<ul>
<li>
<a href = "wave_fsu.sh">wave_fsu.sh</a>,
the MOAB script.
</li>
<li>
<a href = "wave_fsu_output.txt">wave_fsu_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>WAVE_LOCAL</b> compiles and runs the program on a local machine
with multiple processors.
<ul>
<li>
<a href = "wave_local.sh">wave_local.sh</a>,
the BASH script.
</li>
<li>
<a href = "wave_local_output.txt">wave_local_output.txt</a>,
the output file.
</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 17 November 2013.
</i>
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