BURGERS_SOLUTION
Exact Solution of Time Dependent 1D Viscous Burgers Equation

BURGERS_SOLUTION is a MATLAB library which evaluates an exact solution of the time-dependent 1D viscous Burgers equation.

The form of the Burgers equation considered here is:

        du       du        d^2 u
        -- + u * -- = nu * -----
        dt       dx        dx^2
      

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.

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.

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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

BURGERS_SOLUTION is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BURGERS, 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];

BURGERS_STEADY_VISCOUS, 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.

FD1D_BURGERS_LAX, 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.

FD1D_BURGERS_LEAP, a MATLAB program which applies the finite difference method and the leapfrog approach to solve the non-viscous Burgers equation in one spatial dimension and time.

Reference:

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

Source Code:

• burgers_solution.m, evaluates a solution to the Burgers equation.
• hermite_ek_compute.m, computes a Gauss-Hermite quadrature rule.
• imtqlx.m, diagonalizes a symmetric tridiagonal matrix.
• r8_sign.m, returns the sign of an R8.
• r8mat_print.m, prints an R8MAT.
• r8mat_print_some.m, prints some of an R8MAT.
• r8mat_write.m, writes an R8MAT file.
• r8vec_print.m, prints an R8VEC.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• burgers_solution_test.m, calls all the tests.
• burgers_solution_test01.m, generates solution values for -1 <= x <= +1, 0 <= t <= 3/pi, using 11 grid points in x and in t.
• burgers_solution_test02.m, generates solution values for -1 <= x <= +1, 0 <= t <= 3/pi, using 41 grid points in x and in t.
• burgers_solution_test_output.txt, the output file.
• burgers_test01.txt, a data file of solution values for -1 <= x <= +1, 0 <= t <= 3/pi, using 11 grid points in x and in t.
• burgers_test02.txt, a data file of solution values for -1 <= x <= +1, 0 <= t <= 3/pi, using 41 grid points in x and in t.
• burgers_test01.png, an image of U(X,T) for the burgers_test02 data, produced by MATLAB's surf() command.

You can go up one level to the MATLAB source codes.