BURGERS_TIME_VISCOUS
Time-Dependent Viscous Burgers Equation

BURGERS_TIME_VISCOUS 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.

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

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

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, the time limits tmin and tmax, and the (positive) value of the viscosity nu.

The conservative form of the equation is

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

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.

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_viscous.m, integrates a discretized form of the time dependent viscous 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:

• btv_test.m, runs all the tests.
• btv_test01.m, gaussian initial condition, periodic boundary conditions.
• btv_test01.png, a plot of several successive solutions.
• btv_test02.m, gaussian initial condition, Dirichlet left and right.
• btv_test02.png
• btv_test03.m, gaussian initial condition, Dirichlet left, Neumann right.
• btv_test03.png, a plot of several successive solutions.
• btv_test04.m, shock initial condition, periodic boundary condition.
• btv_test04.png,
• btv_test05.m, expansion initial condition, periodic boundary condition.
• btv_test05.png,
• btv_test06.m, spline initial condition, periodic boundary condition.
• btv_test06.png,
• btv_test07.m, gaussian initial condition, plot solutions as a 3D surface.
• btv_test07.png,
• btv_test08.m, spline initial condition, draw profile at final time.
• btv_test08.png,
• btv_test09.m, spike initial condition, periodic boundary conditions.
• btv_test09.png,
• btv_output.txt, the output file.

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