BURGERS_TIME_INVISCID
Time-Dependent Inviscid Burgers Equation

BURGERS_TIME_INVISCID is a MATLAB library which solves the time-dependent inviscid Burgers equation with one of six solution methods selected by the user, by Mikal Landajuela.

The function u(x,t) is to be solved for in the equation:

du/dt + u * du/dx = 0

u(x,tmin) = uinit(x);

u(a,t) = alpha, u(b,t) = beta

Problem data includes the spatial endpoints a and b, the Dirichlet boundary values u(a,t) = alpha, u(b,t) = beta, and the final time t_max.

The conservative form of the equation is

du/dt + 1/2 * d(u^2)/dx = 0

There are six solution methods provided in this package:

1. Upwind nonconservative;
2. Upwind conservative;
3. Lax Friedrichs;
4. Lax Wendroff;
5. MacCormack;
6. Godunov.

The problem specification requires an initial condition, to be determined by a MATLAB function. Five such functions are provided, and the user is free to write a new one:

1. ic_expansion.m, an expansion wave;
2. ic_gaussian.m, a gaussian;
3. ic_shock.m, a shock wave;
4. ic_spike.m, a spike;
5. ic_spline.m, a spline through data points.

Licensing:

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

Related Data and Programs:

BURGERS, a dataset directory which contains some solutions to the viscous Burgers equation.

BURGERS_CHARACTERISTICS, a MATHEMATICA program which solves the time dependent inviscid Burgers equation using the method of characteristics, by Mikel Landajuela.

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

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.

BURGERS_TIME_VISCOUS, a MATLAB library which solves the time-dependent viscous Burgers equation using a finite difference discretization of the conservative form of the equation.

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 time-dependent Burgers equation in one spatial dimension.

PCE_BURGERS, 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.

Reference:

1. Daniel Zwillinger,
   Handbook of Differential Equations,
   Academic Press, 1997,
   ISBN: 0127843965,
   LC: QA371.Z88.

Source Code:

• burgers_time_inviscid.m, integrates a discretized form of the time dependent inviscid Burgers equation.
• ic_expansion.m, an initial condition function for an expansion wave.
• ic_gaussian.m, an initial condition function for a Gaussian.
• ic_shock.m, an initial condition function for a shock wave.
• ic_spike.m, an initial condition function for a spike.
• ic_spline.m, an initial condition function for a spline through data.
• timestamp.m, prints the YMDHMS date as a timestamp.

Examples and Tests:

• bti_test.m, runs all the tests.
• bti_test01.m, expansion initial condition, periodic boundary, upwind nonconservative.
• bti_test01.png.
• bti_test02.m, expansion initial condition, periodic boundary, upwind conservative.
• bti_test02.png.
• bti_test03.m, expansion initial condition, periodic boundary, Lax Friedrichs.
• bti_test03.png.
• bti_test04.m, expansion initial condition, periodic boundary, Lax Wendroff.
• bti_test04.png.
• bti_test05.m, expansion initial condition, periodic boundary, MacCormack.
• bti_test05.png.
• bti_test06.m, expansion initial condition, periodic boundary, Godonov.
• bti_test06.png.
• bti_output.txt, the output file.

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