BURGERS_TIME_VISCOUS
Time-Dependent Viscous Burgers Equation

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, 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

for 0 < nu, a <= x <= b, tmin <= t <= tmax with initial condition

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

and fixed Dirichlet conditions

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

and this is the version we discretize.

Licensing:

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

Languages:

BURGERS_TIME_VISCOUS is available in a MATLAB version.

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_steady_viscous_test

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:

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.