burgers_solution/burgers_solution.html

<html>

  <head>
    <title>
      BURGERS_SOLUTION - Exact Solution of Time Dependent 1D Viscous Burgers Equation
    </title>
  </head>

  <body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">

    <h1 align = "center">
      BURGERS_SOLUTION <br> Exact Solution of Time Dependent 1D Viscous Burgers Equation
    </h1>

    <hr>

    <p>
      <b>BURGERS_SOLUTION</b>
      is a FORTRAN90 library which
      evaluates an exact solution of the time-dependent 1D viscous Burgers equation.
    </p>

    <p>
      The form of the Burgers equation considered here is:
      <pre>
        du       du        d^2 u
        -- + u * -- = nu * -----
        dt       dx        dx^2
      </pre>
      for -1.0 < x < +1.0, and 0.0 < t.
    </p>

    <p>
      Initial conditions are u(x,0) = - sin(pi*x).  Boundary conditions
      are u(-1,t) = u(+1,t) = 0.  The viscosity parameter nu is taken
      to be 0.01 / pi, although this is not essential.
    </p>

    <p>
      The authors note an integral representation for the solution u(x,t),
      and present a better version of the formula that is amenable to
      approximation using Hermite quadrature.  
    </p>

    <p>
      This program library does little more than evaluate the exact solution
      at a user-specified set of points, using the quadrature rule.
      Internally, the order of this quadrature rule is set to 8, but the
      user can easily modify this value if greater accuracy is desired.
    </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_SOLUTION</b> is available in
      <a href = "../../c_src/burgers_solution/burgers_solution.html">a C version</a> and
      <a href = "../../cpp_src/burgers_solution/burgers_solution.html">a C++ version</a> and
      <a href = "../../f77_src/burgers_solution/burgers_solution.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/burgers_solution/burgers_solution.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/burgers_solution/burgers_solution.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 40 solutions of the Burgers equation in one space dimension and time,
      at equally spaced times from 0 to 1, with values
      at 41 equally spaced nodes in [0,1];
    </p>

    <p>
      <a href = "../../f_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
      FD1D_BURGERS_LAX</a>,
      a FORTRAN90 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 = "../../f_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
      FD1D_BURGERS_LEAP</a>,
      a FORTRAN90 program which
      applies the finite difference method and the leapfrog approach
      to solve the non-viscous Burgers equation in one spatial dimension and time.
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      <ol>
        <li>
          Claude Basdevant, Michel Deville, Pierre Haldenwang, J Lacroix, 
          J Ouazzani, Roger Peyret, Paolo Orlandi, Anthony Patera,<br>
          Spectral and finite difference solutions of the Burgers equation,<br>
          Computers and Fluids,<br>
          Volume 14, Number 1, 1986, pages 23-41.
        </li>
      </ol>
    </p>

    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "burgers_solution.f90">burgers_solution.f90</a>, the source code.
        </li>
        <li>
          <a href = "burgers_solution.sh">burgers_solution.sh</a>,
          BASH commands to compile the source code.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "burgers_solution_prb.f90">burgers_solution_prb.f90</a>,
          a sample calling program.
        </li>
        <li>
          <a href = "burgers_solution_prb.sh">burgers_solution_prb.sh</a>,
          BASH commands to compile and run the sample program.
        </li>
        <li>
          <a href = "burgers_solution_prb_output.txt">burgers_solution_prb_output.txt</a>,
          the output file.
        </li>
        <li>
          <a href = "burgers_test01.txt">burgers_test01.txt</a>,
          a data file of solution values for -1 <= x <= +1, 
          0 <= t <= 3/pi, using 11 grid points in x and in t.
        </li>
        <li>
          <a href = "burgers_test02.txt">burgers_test02.txt</a>,
          a data file of solution values for -1 <= x <= +1, 
          0 <= t <= 3/pi, using 41 grid points in x and in t.
        </li>
        <li>
          <a href = "burgers_test02.png">burgers_test01.png</a>,
          an image of U(X,T) for the burgers_test02 data,
          produced by MATLAB's surf() command.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>BURGERS_SOLUTION</b> evaluates a solution to the Burgers equation.
        </li>
        <li>
          <b>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>HERMITE_EK_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
        </li>
        <li>
          <b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
        </li>
        <li>
          <b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
        </li>
        <li>
          <b>R8MAT_PRINT</b> prints an R8MAT.
        </li>
        <li>
          <b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
        </li>
        <li>
          <b>R8MAT_WRITE</b> writes an R8MAT file.
        </li>
        <li>
          <b>R8VEC_EVEN</b> returns an R8VEC of evenly spaced values.
        </li>
        <li>
          <b>R8VEC_PRINT</b> prints an R8VEC.
        </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 = "../f_src.html">
      the FORTRAN90 source codes</a>.
    </p>

    <hr>

    <i>
      Last revised on 16 November 2011.
    </i>

    <!-- John Burkardt -->

  </body>

  <!-- Initial HTML skeleton created by HTMLINDEX. -->

</html>