Burgers' equation

File:Inviscid Burgers Equation in Two Dimensions.gif

This is a numerical simulation of the inviscid Burgers Equation in two space variables up until the time of shock formation.

Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895–1981).

For a given velocity u and viscosity coefficient \(\nu \), the general form of Burgers' equation is:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}\].

When \(\nu = 0\), Burgers' equation becomes the inviscid Burgers' equation:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,\]

which is a prototype for equations for which the solution can develop discontinuities (shock waves). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is:

\[\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial}{\partial x}\big(u^2\big) = 0.\]

Solution

Inviscid Burgers' equation

The inviscid Burgers' equation is a first order partial differential equation (PDE). Its solution can be constructed by the method of characteristics. This method yields that if \(X(t)\) is a solution of the ordinary differential equation

\[\frac{dX(t)}{dt} = u[X(t),t]\]

then \(U(t) := u[X(t),t]\) is constant as a function of \(t\). Hence \([X(t),U(t)]\) is a solution of the system of ordinary equations:

\[\frac{dX}{dt}=U,\]

\[\frac{dU}{dt}=0.\]

The solutions of this system are given in terms of the initial values by:

\[\displaystyle X(t)=X(0)+tU(0),\]

\[\displaystyle U(t)=U(0).\]

Substitute \(X(0)= \eta\), then \(U(0)=u[X(0),0]=u(\eta,0)\). Now the system becomes

\[\displaystyle X(t)=\eta+tu(\eta,0)\]

\[\displaystyle U(t)=U(0).\]

Conclusion:

\[\displaystyle u(\eta,0)=U(0)=U(t)=u[X(t),t]=u[\eta+tu(\eta,0),t]. \]

This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist.

Viscous Burgers' equation

The viscous Burgers' equation can be linearized by the Cole–Hopf transformation

\[u=-2\nu \frac{1}{\phi}\frac{\partial\phi}{\partial x},\]

which turns it into the diffusion equation

\[\frac{\partial\phi}{\partial t}=\nu\frac{\partial^2\phi}{\partial x^2}.\]

That allows one to solve an initial value problem:

\[u(x,t)=-2\nu\frac{\partial}{\partial x}\ln\Bigl\{(4\pi\nu t)^{-1/2}\int_{-\infty}^\infty\exp\Bigl[-\frac{(x-x')^2}{4\nu t} -\frac{1}{2\nu}\int_0^{x'}u(x'',0)dx''\Bigr]dx'\Bigr\}.\]

Generalized Burgers' equation

Non-linear kinematic wave for debris flow can be written as a generalized Burgers' equation with complex non-linear coefficients:

\[\frac{\partial h}{\partial t} + C \frac{\partial h}{\partial x} = D \frac{\partial^2 h}{\partial x^2},\]

where \(h\) is the debris flow height, \(t\) is the time, \(x\) is the downstream channel position, \(C\) is the pressure gradient and the depth dependent nonlinear variable wave speed, and \(D\) is a flow height and pressure gradient dependent variable diffusion term. This equation can also be written in the conservative form:

\[\frac{\partial h}{\partial t} + \frac{\partial F}{\partial x} =0,\]

where \(F\) is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For \(F=h^2/2\), this equation reduces to the Burgers' equation.

External links

• Burgers' Equation at EqWorld: The World of Mathematical Equations.
• Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
• Burgers shock-waves and sound in a 2D microfluidic droplets ensemble Phys. Rev. Lett. 103, 114502 (2009).de:Burgersgleichung