The Azimuth Project
Burgers' equation (Rev #1)

Contents

Idea

Burger’s equation is a partial differential equation that was originally proposed as a simplified model of turbulence as exhibited by the full-fledged Navier-Stokes equations.

In one spatial dimension it is

ut+uuxν 2ux 2=0 \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0

in a spatial region Ω\Omega, with a real constant ν>0\nu \gt 0, with appropriate boundary and initial conditions.

It is possible to write Burgers’s equation in a conservation form (with a flow that is conserved):

ut+F(u)x=0 \frac{\partial u}{\partial t} + \frac{\partial F(u)}{\partial x} = 0

with

F(u):=12u 2νux F(u) := \frac{1}{2} u^2 - \nu \frac{\partial u}{\partial x}

References