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<html>
<head>
<title>
POISSON_SERIAL - A Program for the Poisson Equation in a Rectangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POISSON_SERIAL <br> A Program for the Poisson Equation in a Rectangle
</h1>
<hr>
<p>
<b>POISSON_SERIAL</b>
is a C++ program which
computes an approximate solution to the Poisson equation in a rectangular region.
</p>
<p>
The version of Poisson's equation being solved here is
<pre>
- ( d/dx d/dx + d/dy d/dy ) U(x,y) = F(x,y)
</pre>
over the rectangle 0 <= X <= 1, 0 <= Y <= 1, with exact solution
<pre>
U(x,y) = sin ( pi * x * y )
</pre>
so that
<pre>
F(x,y) = pi^2 * ( x^2 + y^2 ) * sin ( pi * x * y )
</pre>
and with Dirichlet boundary conditions along the lines x = 0, x = 1,
y = 0 and y = 1. (The boundary conditions will actually be zero in
this case, but we write up the problem as though we didn't know that,
which makes it easy to change the problem later.)
</p>
<p>
We compute an approximate solution by discretizing the geometry,
assuming that DX = DY, and approximating the Poisson operator by
<pre>
( U(i-1,j) + U(i+1,j) + U(i,j-1) + U(i,j+1) - 4*U(i,j) ) / dx /dy
</pre>
Along with the boundary conditions at the boundary nodes, we have
a linear system for U. We can apply the Jacobi iteration to estimate
the solution to the linear system.
</p>
<p>
<b>POISSON_SERIAL</b> is intended as a starting point for the implementation of a
parallel version, using, for instance, MPI.
</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>POISSON_SERIAL</b> is available in
<a href = "../../c_src/poisson_serial/poisson_serial.html">a C version</a> and
<a href = "../../cpp_src/poisson_serial/poisson_serial.html">a C++ version</a> and
<a href = "../../f77_src/poisson_serial/poisson_serial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/poisson_serial/poisson_serial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/poisson_serial/poisson_serial.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">
FEM2D_POISSON_RECTANGLE</a>,
a C++ program which
solves the 2D Poisson equation on a rectangle,
using the finite element method,
and piecewise quadratic triangular elements.
</p>
<p>
<a href = "../../cpp_src/fft_serial/fft_serial.html">
FFT_SERIAL</a>,
a C++ program which
demonstrates the computation of a Fast Fourier Transform,
and is intended as a starting point for implementing a parallel version.
</p>
<p>
<a href = "../../cpp_src/fire_serial/fire_serial.html">
FIRE_SERIAL</a>,
a C++ program which
simulates a forest fire over a rectangular array of trees,
starting at a single random location. It is intended as a starting
point for the development of a parallel version.
</p>
<p>
<a href = "../../cpp_src/heated_plate/heated_plate.html">
HEATED_PLATE</a>,
a C++ program which
solves the steady (time independent) heat equation in a 2D
rectangular region, and is intended as
a starting point for implementing a parallel version.
</p>
<p>
<a href = "../../cpp_src/md/md.html">
MD</a>,
a C++ program which
carries out a molecular dynamics simulation, and is intended as
a starting point for implementing a parallel version.
</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/mxm_serial/mxm_serial.html">
MXM_SERIAL</a>,
a C++ program which
sets up a matrix multiplication problem A=B*C,
intended as a starting point for implementing a parallel version.
</p>
<p>
<a href = "../../cpp_src/poisson_openmp/poisson_openmp.html">
POISSON_OPENMP</a>,
a C++ program which
computes an approximate solution to the Poisson equation in a rectangle,
using the Jacobi iteration to solve the linear system, and OpenMP to
carry out the Jacobi iteration in parallel.
</p>
<p>
<a href = "../../cpp_src/quad_serial/quad_serial.html">
QUAD_SERIAL</a>,
a C++ program which
approximates an integral using a quadrature rule,
and is intended as a starting point for parallelization exercises.
</p>
<p>
<a href = "../../cpp_src/search_serial/search_serial.html">
SEARCH_SERIAL</a>,
a C++ program which
searches the integers from A to B for a value J such that F(J) = C.
this version of the program is intended as a starting point for
a parallel approach.
</p>
<p>
<a href = "../../cpp_src/subset_sum_serial/subset_sum_serial.html">
SUBSET_SUM_SERIAL</a>,
a C++ program which
seeks solutions of the subset sum problem, in which it is desired
to find a subset of a set of integers which has a given sum;
this version of the program is intended as a starting point for
a parallel approach.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "poisson_serial.cpp">poisson_serial.cpp</a>, the source code.
</li>
<li>
<a href = "poisson_serial.sh">poisson_serial.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "poisson_serial_output.txt">poisson_serial_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for POISSON_SERIAL.
</li>
<li>
<b>INIT_SERIAL</b> initializes the arrays.
</li>
<li>
<b>SWEEP_SERIAL</b> carries out one step of the Jacobi iteration.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 23 October 2011.
</i>
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