Burgers equation

The Burgers equation (after the Dutch physicist Johannes Martinus Burgers ) is a simple nonlinear partial differential equation of the second order for a function of two variables. It occurs in various areas of applied mathematics . ${\displaystyle u}$ ${\displaystyle x,t.}$

In general terms, the equation looks like this (also called the viscous Burgers equation ):

${\displaystyle {\begin{alignedat}{2}&{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}&&=\mu {\frac {\partial ^{2}u}{\partial x^{2}}}\\\Leftrightarrow &{\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial }{\partial x}}(u^{2})&&=\mu {\frac {\partial ^{2}u}{\partial x^{2}}}.\end{alignedat}}}$

The parameter can be interpreted here as a viscosity parameter. ${\displaystyle \mu >0}$

Often, the above equation is the event called Burgers equation, some authors call this special case frictionless Burgers 'equation (ger .: inviscid Burgers' equation ): ${\displaystyle \mu =0}$

${\displaystyle {\begin{alignedat}{2}&{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}&&=0\\\Leftrightarrow &{\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial }{\partial x}}(u^{2})&&=0.\end{alignedat}}}$

Both representations are formally equivalent, but the second, frictionless form is more advantageous for numerical calculations . The reason for this is the conservation form of the differential equation (see finite volume method ).

application

The viscous Burgers equation is a simple example of a nonlinear parabolic differential equation, and so it is often used as a test case for numerical algorithms for this type of equation.

Because of its similarity to the non-linear part of the Navier-Stokes equation , the Burgers equation can also be interpreted as a simple model of a one-dimensional flow . The traffic density in road traffic is often taken as an example, the course of which over time can be modeled with the help of the Burgers equation.

solutions

The viscous Burgers equation can be solved with the help of the Hopf-Cole transformation .

For the non-viscous equation, the method of characteristics leads to the goal. However, the equation does not necessarily have a unique solution. With suitably chosen initial values can shock observed. The viscous equation then motivates the concept of a solution with vanishing viscosity for the Euler equations . This is the solution of the unviscous Burgers equation that corresponds to a solution of the viscous equation with vanishing viscosity.

literature

• JM Burgers: Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Negotiations of the Koninklijke Nederlandse Akademie van Wetenschappen, Afdeeling Natuurkunde, Series 1, ISSN  0373-4668 , Vol. 17, No. 2, 1939, pp. 1-53.
• M. Case, SC Chiu, Burgers' turbulence models. Physics of Fluids, ISSN  0031-9171 , Vol. 12, 1969, pp. 1799-1808.
• Tomasz Dlotko: The one-dimensional Burgers' equation: existence, uniqueness and stability. Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Matematyczne, ISSN  0450-9005 , Vol. 23, 1982, pp. 157-172.
• Samuel S. Shen, A Course on Nonlinear Waves. , Nonlinear Topics in the Mathematical Sciences, Kluwer Academic, Dordrecht 1993, ISBN 0-7923-2292-4
• Christof Obertscheider: Burgers' Equation. (PDF; 412 kB) Accessed on May 22, 2011 (English).