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burgers_steady_viscous.html
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<html>
<head>
<title>
BURGERS_STEADY_VISCOUS - Steady Viscous Burgers Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BURGERS_STEADY_VISCOUS <br>
Steady Viscous Burgers Equation
</h1>
<hr>
<p>
<b>BURGERS_STEADY_VISCOUS</b> is a MATLAB library which
solves the steady (time-independent) viscous Burgers equation
using a finite difference discretization of the conservative form
of the equation, and then applying Newton's method to solve the
resulting nonlinear system.
</p>
<p>
The function u(x) is to be solved for in the equation:
<blockquote>
u * du/dx = nu * d^2u/dx^2
</blockquote>
for 0 < nu, a <= x <= b.
</p>
<p>
Problem data includes the endpoints a and b, the Dirichlet boundary values
u(a) = alpha, u(b) = beta, and the value of the viscosity nu.
</p>
<p>
We can discretize the problem by specifying a sequence of n (perhaps equally
spaced) points x, and applying standard finite difference approximations
to the derivatives in the continuous equation. A piecewise linear discretization
of the solution can then be computed by bsv().
</p>
<p>
When alpha and beta have opposite sign, the solution must cross the
x-axis (at least once). The location x0 of this crossing is of interest,
and can be computed by bsv_crossing().
</p>
<p>
The crossing location may be quite susceptible to the values of alpha and beta.
</p>
<p>
The conservative form of the equation is
<blockquote>
1/2 * d(u^2)/dx = nu * d^2u/dx^2
</blockquote>
and this is the version we discretize. The residual associated with node i
is then
<blockquote>
f(i) = 1/2 * ( u(i+1)^2 - u(i-1)^2 / ( 2 * dx )
- nu * ( u(i-1) - 2 * u(i) + u(i+1) ) / dx^2
</blockquote>
and we can apply Newton's method to seek a solution u for which f is zero.
<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>BURGERS_STEADY_VISCOUS</b> is available in
<a href = "../../m_src/burgers_steady_viscous/burgers_steady_viscous.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/burgers/burgers.html">
BURGERS</a>,
a dataset directory which
contains some solutions to the viscous Burgers equation.
</p>
<p>
<a href = "../../math_src/burgers_characteristics/burgers_characteristics.html">
BURGERS_CHARACTERISTICS</a>,
a MATHEMATICA program which
solves the time dependent inviscid Burgers equation using the method of characteristics,
by Mikel Landajuela.
</p>
<p>
<a href = "../../m_src/burgers_solution/burgers_solution.html">
BURGERS_SOLUTION</a>,
a MATLAB library which
evaluates an exact solution of the time-dependent 1D viscous Burgers equation.
</p>
<p>
<a href = "../../m_src/burgers_time_viscous/burgers_time_viscous.html">
BURGERS_TIME_VISCOUS</a>,
a MATLAB library which
solves the time-dependent viscous Burgers equation
using a finite difference discretization of the conservative form
of the equation.
</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 Burgers equation
in one spatial dimension and time.
</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/pce_burgers/pce_burgers.html">
PCE_BURGERS</a>,
a MATLAB program 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>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Daniel Zwillinger,<br>
Handbook of Differential Equations,<br>
Academic Press, 1997,<br>
ISBN: 0127843965,<br>
LC: QA371.Z88.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "bsv.m">bsv.m</a>,
applies Newton's method to a discretized steady viscous Burgers equation.
</li>
<li>
<a href = "bsv_crossing.m">bsv_crossing.m</a>,
determines a point X0 where a discretized solution satisfies U(X0) = 0.
</li>
<li>
<a href = "bsv_upwind.m">bsv_upwind.m</a>,
applies Newton's method to a discretized steady viscous Burgers equation,
using upwinding on the flux term.
</li>
<li>
<a href = "r8_sign.m">r8_sign.m</a>,
returns the sign of an R8.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "bsv_test.m">bsv_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "bsv_test01.m">bsv_test01.m</a>,
runs a simple test with
A = -1, ALPHA = +1, B = +1, BETA = -1, NU = 0.1.
</li>
<li>
<a href = "bsv_test02.m">bsv_test02.m</a>,
runs a simple test with
A = -1, ALPHA = +1, B = +1, BETA = -1,
and a range of NU values.
</li>
<li>
<a href = "bsv_test03.m">bsv_test03.m</a>,
runs a simple test with
A = -1, B = +1, BETA = -1, NU = 0.1.
and a range of ALPHA values.
</li>
<li>
<a href = "bsv_test04.m">bsv_test04.m</a>,
runs a simple test with
ALPHA = +1, B = +1, BETA = -1, NU = 0.1.
and a range of A values.
</li>
<li>
<a href = "bsv_test05.m">bsv_test05.m</a>,
examines the location of the zero-crossing of the solution
as ALPHA is varied.
</li>
<li>
<a href = "bsv_test06.m">bsv_test06.m</a>,
estimates the expected value of the zero crossing
assuming ALPHA is a Gaussian variable with mean 1 and standard deviation 0.05.
</li>
<li>
<a href = "bsv_test07.m">bsv_test07.m</a>,
estimates the variance of the zero crossing
assuming ALPHA is a Gaussian variable with mean 1 and standard deviation 0.05.
</li>
<li>
<a href = "bsv_test08.m">bsv_test08.m</a>,
compares BSV and BSV_UPWIND for NU = 0.1 and NU = 0.01.
</li>
<li>
<a href = "bsv_test01.png">bsv_test01.png</a>,
a plot of the solution as displayed by bsv_test01.
</li>
<li>
<a href = "bsv_test02.png">bsv_test02.png</a>,
a plot of the solutions as displayed by bsv_test02.
</li>
<li>
<a href = "bsv_test03.png">bsv_test03.png</a>,
a plot of the solutions as displayed by bsv_test03.
</li>
<li>
<a href = "bsv_test04.png">bsv_test04.png</a>,
a plot of the solutions as displayed by bsv_test04.
</li>
<li>
<a href = "bsv_test05.png">bsv_test05.png</a>,
a plot of the relation between ALPHA and the zero crossing.
</li>
<li>
<a href = "bsv_test08.png">bsv_test08.png</a>,
a plot comparing two runs of BSV and BSV_UPWIND.
</li>
<li>
<a href = "bsv_output.txt">bsv_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "tanh_plot.m">tanh_plot.m</a>,
plots a sequence of functions u = tanh(2^j*x/2),
scaled to be +1 at x=-1, suggesting the behavior of
solutions of Burgers equation.
</li>
<li>
<a href = "tanh_plot.png">tanh_plot.png</a>,
the plot created by tanh_plot.m.
</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 15 April 2012.
</i>
<!-- John Burkardt -->
</body>
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</html>