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pce_burgers.html
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<html>
<head>
<title>
PCE_BURGERS
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
PCE_BURGERS <br> Polynomial Chaos Expansion for Burgers Equation
</h1>
<hr>
<p>
<b>PCE_BURGERS</b>
is a C++ library which
defines and solves a version of the time-dependent viscous Burgers equation,
with uncertain viscosity, using a polynomial chaos expansion,
in terms of Hermite polynomials,
by Gianluca Iaccarino.
</p>
<p>
The time-dependent viscous Burgers equation to be solved is:
<pre>
du/dt = - d ( u*(1/2-u)) /dx + nu d2u/dx2 for -3.0 <= x <= 3.0
</pre>
with boundary conditions
<pre>
u(-3.0) = 0.0, u(+3.0) = 1.0.
</pre>
</p>
<p>
The viscosity nu is assumed to be an uncertain quantity with
normal distribution of known mean and variance.
</p>
<p>
A polynomial chaos expansion is to be used, with Hermite polynomial
basis functions h(i,x), 0 <= i <= n.
</p>
<p>
Because the first two Hermite polynomials are simply 1 and x,
we have that
<pre>
nu = nu_mean * h(0,x) + nu_variance * h(1,x).
</pre>
We replace the time derivative by an explicit Euler approximation,
so that the equation now describes the value of U(x,t+dt) in terms
of known data at time t.
</p>
<p>
Now assume that the solution U(x,t) can be approximated
by the truncated expansion:
<pre>
U(x,t) = sum ( 0 <= i <= n ) c(i,t) * h(i,x)
</pre>
In the equation, we replace U by its expansion, and then multiply
successively by each of the basis functions h(*,x) to get a set of
n+1 equations that can be used to determine the values of c(i,t+dt).
</p>
<p>
This process is repeated until the desired final time is reached.
</p>
<p>
At any time, the coefficients c(0,t) contain information definining
the expected value of u(x,t) at that time, while the higher order coefficients
can be used to determine higher moments.
</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>PCE_BURGERS</b> is available in
<a href = "../../c_src/pce_burgers/pce_burgers.html">a C version</a> and
<a href = "../../cpp_src/pce_burgers/pce_burgers.html">a C++ version</a> and
<a href = "../../f77_src/pce_burgers/pce_burgers.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pce_burgers/pce_burgers.html">a FORTRAN90 version</a> and
<a href = "../../m_src/pce_burgers/pce_burgers.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../cpp_src/pce_ode_hermite/pce_ode_hermite.html">
PCE_ODE_HERMITE</a>,
a C++ program which
sets up a simple scalar ODE for exponential decay with an uncertain
decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.
</p>
<p>
<a href = "../../cpp_src/sde/sde.html">
SDE</a>,
a C++ library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis,
by Desmond Higham;
</p>
<h3 align = "center">
Author:
</h3>
<p>
The original FORTRAN90 version of this program was written by Gianluca Iaccarino.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Ghanem, Pol Spanos,<br>
Stochastic Finite Elements: A Spectral Approach,<br>
Revised Edition,<br>
Dover, 2003,<br>
ISBN: 0486428184,<br>
LC: TA347.F5.G56.
</li>
<li>
Dongbin Xiu,<br>
Numerical Methods for Stochastic Computations: A Spectral Method Approach,<br>
Princeton, 2010,<br>
ISBN13: 978-0-691-14212-8,<br>
LC: QA274.23.X58.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "pce_burgers.cpp">pce_burgers.cpp</a>, the source code.
</li>
<li>
<a href = "pce_burgers.sh">pce_burgers.sh</a>,
BASH commands to compile and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "pce_burgers_output.txt">pce_burgers_output.txt</a>,
the output file.
</li>
<li>
<a href = "burgers.history.txt">burgers.history.txt</a>,
the value of the solution (expansion coefficients) at selected times.
</li>
<li>
<a href = "burgers.modes.txt">burgers.modes.txt</a>,
the modes of the solution at the final time.
</li>
<li>
<a href = "burgers.moments.txt">burgers.moments.txt</a>,
the mean and variance of the coefficients at each point,
at the final time.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for PCE_BURGERS.
</li>
<li>
<b>HE_DOUBLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*e^(-x^2/2).
</li>
<li>
<b>HE_TRIPLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</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 modified on 16 March 2012.
</i>
<!-- John Burkardt -->
</body>
</html>