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heat_mpi.html
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<html>
<head>
<title>
HEAT_MPI - Solve the 1D Time Dependent Heat Equation using MPI
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HEAT_MPI <br> Solve the 1D Time Dependent Heat Equation using MPI
</h1>
<hr>
<p>
<b>HEAT_MPI</b>
is a FORTRAN90 program which
solves the 1D Time Dependent Heat Equation using MPI.
</p>
<h3 align = "center">
The continuous problem
</h3>
<p>
This program solves
<pre>
dUdT - k * d2UdX2 = F(X,T)
</pre>
over the interval [A,B] with boundary conditions
<pre>
U(A,T) = UA(T),
U(B,T) = UB(T),
</pre>
over the time interval [T0,T1] with initial conditions
<pre>
U(X,T0) = U0(X)
</pre>
</p>
<h3 align = "center">
The finite difference discretization
</h3>
<p>
To apply the finite difference method, we define a grid of
points X(1) through X(N), and a grid of times T(1) through T(M).
In the simplest case, both grids are evenly spaced. We denote
by U(I,J) the approximate solution at spatial point X(I) and
time T(J).
</p>
<p>
A second order finite difference can be used to approximate the
second derivative in space, using the solution at three
points equally separated in space.
</p>
<p>
A forward Euler approximation to the first derivative in time
is used, which relates the value of the solution to its value
at a short interval in the future.
</p>
<p>
Thus, at the spatial point X(I) and time T(J), the discretized differential
equation defines a relationship between U(I-1,J), U(I,J), U(I+1,J)
and the "future" value U(I,J+1). This relationship can be drawn
symbolically as a four node stencil:
<pre>
U(I,J+1)
|
|
U(I-1,J)-----U(I,J)--------U(I+1,J)
</pre>
</p>
<p>
Since we are given the value of the solution at the initial time,
we can use the stencil, plus the boundary condition information,
to advance the solution to the next time step. Repeating this
operation gives us an approximation to the solution at every
point in the space-time grid.
</p>
<h3 align = "center">
Using MPI to compute the solution:
</h3>
<p>
To solve the 1D heat equation using MPI, we use a form of domain
decomposition. Given P processors, we divided the interval [A,B]
into P equal subintervals. Each processor can set up the stencil
equations that define the solution almost independently. The exception
is that every processor needs to receive a copy of the solution
values determined for the nodes on its immediately left and right sides.
</p>
<p>
Thus, each processor uses MPI to send its leftmost solution value to its
left neighbor, and its rightmost solution value to its rightmost neighbor.
Of course, each processor must then also receive the corresponding information
that its neighbors send to it. (However, the first and last processor
only have one neighbor, and use boundary condition information to determine
the behavior of the solution at the node which is not next to another
processor's node.)
</p>
<p>
The naive way of setting up the information exchange works, but can
be inefficient, since each processor sends a message and then waits for
confirmation of receipt, which can't happen until some processor has
moved to the "receive" stage, which only happens because the first or
last processor doesn't have to receive information on a given step.
</p>
<p>
It is worth investigating how to improve the information exchange
(an exercise for the reader!). The odd processors could SEND while the
even processors RECEIVE for instance, guaranteeing that messages would
not have to wait in a buffer.
</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>HEAT_MPI</b> is available in
<a href = "../../c_src/heat_mpi/heat_mpi.html">a C version</a> and
<a href = "../../cpp_src/heat_mpi/heat_mpi.html">a C++ version</a> and
<a href = "../../f77_src/heat_mpi/heat_mpi.html">a FORTRAN77 version</a> and
<a href = "../../f_src/heat_mpi/heat_mpi.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/communicator_mpi/communicator_mpi.html">
COMMUNICATOR_MPI</a>,
a FORTRAN90 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 = "../../f_src/hello_mpi/hello_mpi.html">
HELLO_MPI</a>,
a FORTRAN90 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 = "../../f_src/mpi/mpi.html">
MPI</a>,
FORTRAN90 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 = "../../f_src/mpi_stubs/mpi_stubs.html">
MPI_STUBS</a>,
a FORTRAN90 library which
is a set of "stub" MPI routines, which allows
a user to compile, load, and possibly run an MPI program on a
serial machine.
</p>
<p>
<a href = "../../f_src/multitask_mpi/multitask_mpi.html">
MULTITASK_MPI</a>,
a FORTRAN90 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 = "../../f_src/prime_mpi/prime_mpi.html">
PRIME_MPI</a>,
a FORTRAN90 program which
counts the number of primes between 1 and N, using MPI for parallel execution.
</p>
<p>
<a href = "../../f_src/quad_mpi/quad_mpi.html">
QUAD_MPI</a>,
a FORTRAN90 program which
approximates an integral using a quadrature rule, and carries out the
computation in parallel using MPI.
</p>
<p>
<a href = "../../f_src/random_mpi/random_mpi.html">
RANDOM_MPI</a>,
a FORTRAN90 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 = "../../f_src/ring_mpi/ring_mpi.html">
RING_MPI</a>,
a FORTRAN90 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 = "../../f_src/satisfy_mpi/satisfy_mpi.html">
SATISFY_MPI</a>,
a FORTRAN90 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 = "../../f_src/search_mpi/search_mpi.html">
SEARCH_MPI</a>,
a FORTRAN90 program which
searches integers between A and B for a value J such that F(J) = C,
using MPI for parallel execution.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
William Gropp, Ewing Lusk, Anthony Skjellum,<br>
Using MPI: Portable Parallel Programming with the
Message-Passing Interface,<br>
Second Edition,<br>
MIT Press, 1999,<br>
ISBN: 0262571323,<br>
LC: QA76.642.G76.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "heat_mpi.f90">heat_mpi.f90</a>,
the source code;
</li>
<li>
<a href = "heat2_mpi.f90">heat2_mpi.f90</a>,
a revised version of the source code which tries to
speed up the transfer of information by having odd and even
processors do SENDS and RECEIVES in pairs.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>HEAT_FSU</b> compiles and runs the program on the FSU HPC cluster.
<ul>
<li>
<a href = "heat_fsu.sh">heat_fsu.sh</a>,
the MOAB script.
</li>
<li>
<a href = "heat_fsu_output.txt">heat_fsu_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>HEAT_SYSX</b> compiles and runs the program on Virginia Techs's System X.
<ul>
<li>
<a href = "heat_sysx.sh">heat_sysx.sh</a>,
BASH commands to compile, link and load the source code on System X.
</li>
<li>
<a href = "heat_sysx_output.txt">heat_sysx_output.txt</a>,
the output file.
</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 22 October 2011.
</i>
<!-- John Burkardt -->
</body>
</html>