BURGERS_STEADY_VISCOUS, a MATLAB code 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.

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

u * du/dx = nu * d^2u/dx^2
for 0 < nu, a <= x <= b.

Problem data includes the endpoints a and b, the Dirichlet boundary values u(a) = alpha, u(b) = beta, and the value of the viscosity nu.

We can discretize the problem by specifying a sequence of n (perhaps equally spaced) points x, and applying standard finite difference approximations to the derivatives in the continuous equation. A piecewise linear discretization of the solution can then be computed by bsv().

When alpha and beta have opposite sign, the solution must cross the x-axis (at least once). The location x0 of this crossing is of interest, and can be computed by bsv_crossing().

The crossing location may be quite susceptible to the values of alpha and beta.

The conservative form of the equation is

1/2 * d(u^2)/dx = nu * d^2u/dx^2
and this is the version we discretize. The residual associated with node i is then
f(i) = 1/2 * ( u(i+1)^2 - u(i-1)^2 / ( 2 * dx ) - nu * ( u(i-1) - 2 * u(i) + u(i+1) ) / dx^2
and we can apply Newton's method to seek a solution u for which f is zero.

Languages:

BURGERS_STEADY_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 code which evaluates an exact solution of the time-dependent 1D viscous Burgers equation.

BURGERS_TIME_VISCOUS, a MATLAB code 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 code 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 code 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 code 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,