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<html>
<head>
<title>
BURGERS_TIME_VISCOUS - Time-Dependent Viscous Burgers Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BURGERS_TIME_VISCOUS <br>
Time-Dependent Viscous Burgers Equation
</h1>
<hr>
<p>
<b>BURGERS_TIME_VISCOUS</b> is a MATLAB library which
solves the time-dependent viscous Burgers equation
using a finite difference discretization of the conservative form
of the equation, and then carrying out a simple parabolic integration scheme.
</p>
<p>
The function u(x,t) is to be solved for in the equation:
<blockquote>
du/dt + u * du/dx = nu * d^2u/dx^2
</blockquote>
for 0 < nu, a <= x <= b, tmin <= t <= tmax
with initial condition
<blockquote>
u(x,tmin) = uinit(x);
</blockquote>
and fixed Dirichlet conditions
<blockquote>
u(a,t) = alpha, u(b,t) = beta
</blockquote>
</p>
<p>
Problem data includes the spatial endpoints a and b, the Dirichlet boundary values
u(a,t) = alpha, u(b,t) = beta, the time limits
tmin and tmax, and the (positive) value of the viscosity nu.
</p>
<p>
The conservative form of the equation is
<blockquote>
du/dt + 1/2 * d(u^2)/dx = nu * d^2u/dx^2
</blockquote>
and this is the version we discretize.
<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_TIME_VISCOUS</b> is available in
<a href = "../../m_src/burgers_time_viscous/burgers_time_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_steady_viscous/burgers_steady_viscous.html">
BURGERS_STEADY_VISCOUS</a>,
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>
<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 = "burgers_time_viscous.m">burgers_time_viscous.m</a>,
integrates a discretized form of the time dependent viscous Burgers equation.
</li>
<li>
<a href = "ic_expansion.m">ic_expansion.m</a>,
an initial condition function for an expansion wave.
</li>
<li>
<a href = "ic_gaussian.m">ic_gaussian.m</a>,
an initial condition function for a Gaussian.
</li>
<li>
<a href = "ic_shock.m">ic_shock.m</a>,
an initial condition function for a shock wave.
</li>
<li>
<a href = "ic_spike.m">ic_spike.m</a>,
an initial condition function for a spike.
</li>
<li>
<a href = "ic_spline.m">ic_spline.m</a>,
an initial condition function for a spline through data.
</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 = "btv_test.m">btv_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "btv_test01.m">btv_test01.m</a>,
gaussian initial condition, periodic boundary conditions.
</li>
<li>
<a href = "btv_test01.png">btv_test01.png</a>,
a plot of several successive solutions.
</li>
<li>
<a href = "btv_test02.m">btv_test02.m</a>,
gaussian initial condition, Dirichlet left and right.
</li>
<li>
<a href = "btv_test02.png">btv_test02.png</a>
</li>
<li>
<a href = "btv_test03.m">btv_test03.m</a>,
gaussian initial condition, Dirichlet left, Neumann right.
</li>
<li>
<a href = "btv_test03.png">btv_test03.png</a>,
a plot of several successive solutions.
</li>
<li>
<a href = "btv_test04.m">btv_test04.m</a>,
shock initial condition, periodic boundary condition.
</li>
<li>
<a href = "btv_test04.png">btv_test04.png</a>,
</li>
<li>
<a href = "btv_test05.m">btv_test05.m</a>,
expansion initial condition, periodic boundary condition.
</li>
<li>
<a href = "btv_test05.png">btv_test05.png</a>,
</li>
<li>
<a href = "btv_test06.m">btv_test06.m</a>,
spline initial condition, periodic boundary condition.
</li>
<li>
<a href = "btv_test06.png">btv_test06.png</a>,
</li>
<li>
<a href = "btv_test07.m">btv_test07.m</a>,
gaussian initial condition, plot solutions as a 3D surface.
</li>
<li>
<a href = "btv_test07.png">btv_test07.png</a>,
</li>
<li>
<a href = "btv_test08.m">btv_test08.m</a>,
spline initial condition, draw profile at final time.
</li>
<li>
<a href = "btv_test08.png">btv_test08.png</a>,
</li>
<li>
<a href = "btv_test09.m">btv_test09.m</a>,
spike initial condition, periodic boundary conditions.
</li>
<li>
<a href = "btv_test09.png">btv_test09.png</a>,
</li>
<li>
<a href = "btv_output.txt">btv_output.txt</a>,
the output file.
</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 22 April 2012.
</i>
<!-- John Burkardt -->
</body>
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</html>