# Burgers Equation

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Nonlinear PDE's Course
Current Topic Burgers Equation
Next Topic
Previous Topic Reaction-Diffusion Systems

## Introduction

We have already met the conservation law for the traffic equations

$\partial _{t}\rho +c\left( \rho \right) \partial _{x}\rho =0$

and seen how this leads to shocks. We can smooth this equation by adding dispersion to the equation to give us

$\partial _{t}\rho +c\left( \rho \right) \partial _{x}\rho =\nu \partial _{x}^{2}\rho$

where $\nu \gt 0.$

The simplest equation of this type is to write

$\partial _{t}u+u\partial _{x}u=\nu \partial _{x}^{2}u$

(changing variables to $u$ and this equation is known as Burgers equation.

## Travelling Wave Solution

We can find a travelling wave solution by assuming that

$u\left( x,t\right) =u\left( x-ct\right) =u\left( \xi \right)$

$-cu^{\prime }+u^{\prime }u-\nu u^{\prime \prime }=0$

We begin by looking at the phase plane for this system, writing $w=u^{\prime }$ so that

$\begin{matrix} \dfrac{\mathrm{d}u}{\mathrm{d}\xi } &=&w \\ \dfrac{\mathrm{d}w}{\mathrm{d}\xi } &=&\frac{1}{\nu }\left( w\left( u-c\right) \right) \end{matrix}$

This is a degenerate system with the entire $u$ axis being equilibria.

We can also solve this equation exactly as follows.

$-cu^{\prime }+u^{\prime }u-\nu u^{\prime \prime }=0$

can be integrated to give

$-cu+\frac{1}{2}\left( u\right) ^{2}-\nu u^{\prime} =c_{1}$

which can be rearranged to give

$u^{\prime }=\frac{1}{2\nu }\left( \left( u\right) ^{2}-2cu-2c_{1}\right)$

We define the two roots of the quadratic $\left( u\right) ^{2}-2\nu u-2c_{1}=0$ by $u_{1}$ and $u_{2}$ and we assume that $u_{2} \lt u_{1}$. Note that there is only a bounded solution if we have two real roots and for the bounded solution $u_{2} \lt u \lt u_{1}$. We note that the wave speed is

$c=\frac{1}{2}\left( u_{1}+u_{2}\right)$

The equation can therefore be written as

$2\nu u^{\prime }=\left( u-u_{1}\right) \left( u-u_{2}\right)$

which has solution

$u\left( \xi \right) =\frac{1}{2}\left( u_{1}+u_{2}\right) -\frac{1}{2}\left( u_{1}-u_{2}\right) \tanh \left[ \left( \frac{\xi }{4\nu }\right) \left( u_{1}-u_{2}\right) \right]$

## Numerical Solution of Burgers equation

We can solve the equation using our split step spectral method. The equation can be written as

$\partial _{t}u=-\frac{1}{2}\partial _{x}\left( u^{2}\right) +\nu \partial _{x}^{2}u$

We solve this by solving in Fourier space to give

$\partial _{t}\hat{u}=-\frac{1}{2}ik \widehat{\left( u^{2}\right)} -\nu k^{2}\hat{u}$

Then we solve each of the steps in turn for a small time interval to give

$\begin{matrix} \tilde{u}\left( k,t+\Delta t\right) &=&\hat{u}\left( k,t\right) -\frac{ \Delta t}{2}ik\mathcal{F}\left( \left[ \mathcal{F}^{-1}\hat{u}\left( k,t\right) \right] ^{2}\right) \\ \hat{u}\left( k,t+\Delta t\right) &=&\tilde{u}\left( k,t+\Delta t\right) \exp \left( -\nu k^{2}\Delta t\right) \end{matrix}$
Phase plane for a travelling wave solution Numerical solution of Burgers equation
Phase plane for a travelling wave solution of Burgers equation
Numerical solution of Burgers equation

## Exact Solution of Burgers equations

We can find an exact solution to Burgers equation. We want to solve

$\begin{matrix} \partial _{t}u+u\partial _{x}u &=&\nu \partial _{x}^{2}u \\ u\left( x,0\right) &=&F\left( x\right) \end{matrix}$

Frist we write the equation as

$\partial _{t}u+\partial _{x}\left( \frac{u^{2}}{2}-\nu \partial _{x}u\right) =0$

We want to find a function $\psi \left( x,t\right)$ such that

$\partial _{x}\psi =u,\ \ \partial _{t}\psi =\nu \partial _{x}u-\frac{u^{2}}{2 }$

Note that because $\partial _{x}\partial _{t}\psi =\partial _{t}\partial _{x}\psi$ we will satisfy Burgers equation. This gives us the following equation for $\psi$

$\partial _{t}\psi =\nu \partial _{x}^{2}\psi -\frac{1}{2}\left( \partial _{x}\psi \right) ^{2}$

We introduce the Cole-Hopf transformation

$\psi =-2\nu \log \left( \phi \right)$

From this we can obtain the three results:

\begin{align} \partial _{x}\psi &=-2\nu \frac{\partial _{x}\phi }{\phi } \\ \partial _{x}^{2}\psi &=2\nu \left( \frac{\partial _{x}\phi }{\phi }\right) ^{2}-\frac{2\nu }{\phi }\partial _{x}^{2}\phi \\ \partial _{t}\psi &=-2\nu \frac{\partial _{t}\phi }{\phi } \end{align}

Therefore

$\partial _{t}\psi =\nu \partial _{x}^{2}\psi -\frac{1}{2}\left( \partial _{x}\psi \right) ^{2}$

becomes

$-2\nu \frac{\partial _{t}\phi }{\phi }=2\nu ^{2}\left( \frac{\partial _{x}\phi }{\phi }\right) ^{2} -2\nu^2 \frac{\partial_x^2\phi}{\phi} -\frac{1}{2}\left( 2\nu \frac{\partial _{x}\phi }{\phi }\right) ^{2}$

or

$\partial _{t}\phi =\nu \partial _{x}^{2}\phi$

which is just the diffusion equation. Note that we also have to transform the boundary conditions. We have

$F\left( x\right) =u\left( x,0\right) =-2\nu \frac{\partial _{x}\phi \left( x,0\right) }{\phi \left( x,0\right) }$

We can write this as

$\frac{\mathrm{d}}{\mathrm{d}x}\left( \log \left( \phi \right) \right) =-\frac{1}{2\nu }F\left( x\right)$

which has solution

$\phi \left( x,0\right) =\Phi \left( x\right) =\exp \left( -\frac{1}{2\nu } \int_{0}^{x}F\left( s\right) \mathrm{d}s\right)$

We need to solve

$\begin{matrix} \partial _{t}\phi &=&\nu \partial _{x}^{2}\phi \\ \phi \left( x,0\right) &=&\Phi \left( x\right) \end{matrix}$

We take the Fourier transform and obtain

$\begin{matrix} \partial _{t}\hat{\phi} &=&-k^{2}\nu \hat{\phi} \\ \hat{\phi}\left( k,0\right) &=&\hat{\Phi}\left( k\right) \end{matrix}$

which has solution

$\hat{\phi}\left( k,t\right) =\hat{\Phi}\left( k\right) e^{-k^{2}\nu t}$

We can then use the convolution theorem to write

$\begin{matrix} \phi \left( x,t\right) &=&\Phi \left( x\right) * \mathcal{F}^{-1}\left[ e^{-k^{2}\nu t}\right] \\ &=&\frac{1}{2\sqrt{\pi \nu t}}\int_{-\infty }^{\infty }\Phi \left( y\right) \exp \left[ -\frac{\left( x-y\right) ^{2}}{4\nu t}\right] \mathrm{d}y \end{matrix}$

Which can be expressed as

$\phi \left( x,t\right) =\frac{1}{2\sqrt{\pi \nu t}}\int_{-\infty }^{\infty }\exp \left[ -\frac{f}{2\nu }\right] \mathrm{d}y$

where

$f\left( x,y,t\right) =\frac{1}{2\nu }\int_{0}^{y}F\left( s\right) \mathrm{d}s+\frac{ \left( x-y\right) ^{2}}{2t}$

To find $u$ we recall that

$\begin{matrix} u\left( x,t\right) &=&-2\nu \dfrac{\partial _{x}\phi \left( x,t\right) }{\phi \left( x,t\right) } \\ &=&\dfrac{\int_{-\infty }^{\infty }\left( \frac{x-y}{t}\right) \exp \left[ - \dfrac{f}{2\nu }\right] \mathrm{d}y}{\int_{-\infty }^{\infty }\exp \left[ -\frac{f}{ 2\nu }\right] \mathrm{d}y} \end{matrix}$