Contents

Idea

Burgers’ 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. The turbulent behaviour of the stochastically forced Burgers’ equation is sometimes dubbed Burgulence.

It is a nonlinear equation for which exact solutions are known and is therefore important as a benchmark problem for numerical methods.

Derivation

We may obtain Burgers’ equation as a simplified version of the Navier-Stokes equations. For a Newtonian incompressible fluid, the Navier-Stokes equations say

$\rho \left(\frac{\partial u}{\partial t} + u \cdot \nabla u \right) = -\nabla p + \mu \nabla^2 u + F$

where $\rho$ is the density, $u$ is the velocity vector field, $p$ is the pressure, $\mu$ is the viscosity, and $F$ is an external force. If we drop the pressure term we get Burgers’ equation:

$\rho \left(\frac{\partial u}{\partial t} + u \cdot \nabla u \right) = \mu \nabla^2 u + F$

The equation simplifies further if we assume the external force is equal to zero and take advantage of the fact that $\rho$ is a constant for an incompressible fluid; this allows us to define a new constant, the kinematic viscosity $\nu = \mu / \rho$, and write Burgers’ equation as

$\frac{\partial u}{\partial t} + u \cdot \nabla u = \nu \nabla^2 u$

An even further simplification arises when we assume the viscosity is zero. Then we obtain the inviscid Burgers’ equation:

$\frac{\partial u}{\partial t} + u \cdot \nabla u = 0$

We may derive the inviscid Burgers’ equation by considering a gas of free (that is, noninteracting) particles where at the point $\mathbf{x} \in \mathbb{R}^n$ at time $t$ the velocity of the particles there is $u(t,\mathbf{x})$. As time passes, the velocity of each particle does not change. However, the particles move, so it is the the convective derivative of $u$ that vanishes:

$\frac{\partial u}{\partial t} + u \cdot \nabla u = 0$

In what follows we will consider Burgers’ equation with nonzero viscosity, but in the special case of one space dimension ($n = 1$). Then the equation becomes

$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0$

We may consider this in some spatial region $\Omega$ for positive times $t \ge 0$ and with a real constant $\nu \gt 0$, with appropriate boundary conditions and initial conditions.

We write for the initial condition:

$u(x, o) = u_0(x)$

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

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

with

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

Details

Exact Solution of Initial Value Problem

It is possible to get an exact solution to the Burgers’ equation with given initial conditions.

Cole-Hopf-Transformation to the Heat Equation

It is possible to transform Burgers’ equation to the heat equation via

$u = -2 \nu \frac{\phi_x}{\phi}$

so that the function $\phi$ satisfies

$\frac{\partial \phi}{\partial t} - \nu \frac{\partial^2 \phi}{\partial x^2} = 0$

iff the function $u$ satisfies Burgers’ equation.

This transformation is usually called Cole-Hopf transformation.

The initial condition for $\phi$ is

$u_0 (x) = -2 \nu \frac{\phi_x (x, 0)}{\phi(x, 0)}$

which can be integrated to yield:

$\phi(x)_0 := \phi(x, 0) = \exp \left(- \frac{1}{2 \nu} \; \int_0^x \; u_0(\alpha) \; d \alpha \right)$

Solution Obtained From The Heat Equation

The initial problem of the heat equation has a well known solution, which is

$\phi(x, t) = \frac{1}{2 \sqrt{\pi \nu t}} \; \int_{- \infty}^{\infty} \; \phi_0(\zeta) \; \exp \left(- \frac{(x- \zeta)^2 }{4 \nu t} \right) \; d \zeta$

We can use our knowledge about the form of the inital value $\phi_0$ in order to rewrite this formula in a form that is a little bit more convenient, by introducing the help function

$f(\zeta, x, t) := \int_0^{\zeta} \; u_0(\alpha) \; d \alpha + \frac{(x- \zeta)^2 }{2 t}$

and get

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

for the solution $\phi$ itself, and thus for its derivative:

$\phi_x(x, t) = - \frac{1}{4 \nu \sqrt{\pi \nu t} } \; \int_{- \infty}^{\infty} \; \frac{x - \zeta}{t} \; \exp \left(- \frac{f}{2 \nu} \right) \; d \zeta$

Now plugging both formulas into the Cole-Hopf transformation gets us

$u(x, t) = \frac{\int_{- \infty}^{\infty} \; \frac{x - \zeta}{t} \; \exp \left(- \frac{f}{2 \nu} \right) \; d \zeta}{\; \int_{- \infty}^{\infty} \; \exp \left(- \frac{f}{2 \nu} \right) \; d \zeta}$

The involved integrals may of course turn out to be hard to evaluate, both analytically and numerically, depending, of course, on the initial condition, but this exact formula is still very useful to crosscheck numerical schemes.

An Exact Solution of a Boundary Value Problem

We will state the exact solution of the combined initial and boundary value problem, where the periodic boundary condition is for $0 \lt x \lt l$:

$u(0, t) = u(l, t) = 0$

and we state a specific initial condition:

$u(x, 0) = u_0 \sin(\frac{\pi x}{l})$

From the Cole-Hopf-transformation we get as corresponding initial condition of the heat equation:

$\phi(x, 0) = \exp \left[ - \frac{u_0 l}{2 \pi \nu} \left(1 - \cos(\frac{\pi x}{l}) \right) \right]$

The solution of the wave equation is:

$\phi(x, t) = a_0 + \sum_{n=1}^{\infty} a_n \exp \left(- \frac{n^2 \pi^2 \nu t}{l^2} \right) \cos \left( \frac{n \pi x}{l} \right)$

with the coefficients:

$a_0 = \frac{1}{l} \int_0^l \; \exp \left[- \frac{u_0 l}{2 \pi \nu} \left( 1 - \cos(\frac{\pi x}{l}) \right) \right] \; d x = \exp \left( - \frac{u_0 l}{2 \pi \nu} \right) \; I_0 \left( \frac{u_0 l}{2 \pi \nu} \right)$

and

$a_n = \frac{2}{l} \int_0^l \; \exp \left[- \frac{u_0 l}{2 \pi \nu} \; \left( 1 - \cos(\frac{\pi x}{l}) \right) \right] \; \cos \left( \frac{n \pi x}{l} \right) \; d x = 2 \exp \left(- \frac{u_0 l}{2 \pi \nu} \right) \; I_n \left( \frac{u_0 l}{2 \pi \nu} \right)$

where $I_n$ are the modified Bessel functions of the first kind.

Therefore, the solution of Burgers’ equation is:

$u(x, t) = \frac{4 \pi \nu}{l} \frac{ \sum_{n = 1}^{\infty} \; n \; I_n \left( \frac{u_0 l}{2 \pi \nu} \right) \; \exp \left( - \frac{n^2 \pi^2 \nu t}{l^2} \right) \; \; \sin \left( \frac{n \pi x}{l} \right)} {I_0 \left( \frac{u_0 l}{2 \pi \nu} \right) + 2 \sum_{n = 1}^{\infty} n I_n \left( \frac{u_0 l}{2 \pi \nu} \right) \; \exp \left( - \frac{n^2 \pi^2 \nu t}{l^2} \right) \; \cos \left( \frac{n \pi x}{l} \right)}$

Examples

N-Wave Solution

From the exact formula for the initial value problem, we can calculate the solution of the Burgers’ equation $u$ from the solution of the heat equation

$\phi(x, t) = 1 + \sqrt{ \frac{\tau}{t}} \exp \left( \frac{- x^2}{4 \nu t} \right)$

with a positive constant $\tau$.

Plugging this into the formula for the exact solution yields:

$u(x, t, \tau, \nu) = \frac{x}{t} \left( 1 + \sqrt{ \frac{t}{\tau}} \exp \left( \frac{x^2}{4 \nu t} \right) \right)^{-1}$

This solution is commonly called the N-solution, because for fixed values of $t, \tau, \nu$ it has a N-shaped graph: Approximation via Fourier-Galerkin Spectral Method

We will use a spectral method for an approximate solution of Burgers’ equation on the domain $\Omega = (0, 2 \pi)$. “Fourier” means that we will use an approximation via a Fourier series, and “Galerkin” means that we will use the approximation functions also as test functions. As usual, we will use the spectral approximation for the spatial dimension only, not for the temporal. This results in an ansatz that is usually called “separation of variables”. One could indeed call “separation of variables” a special case of spectral methods.

Our ansatz for the approximate solution is:

$u_N (x, t) = \sum_{k = - \frac{N}{2}}^{k = \frac{N}{2} - 1} \hat{u}_n (t) e^{i k x}$

Since we choose as test functions our approximation functions, the conditions resulting from the requirement that $\langle \; M(f_{\alpha}), \; h_i \; \rangle = 0$ (see spectral methods for an explanation of the nomenclature) results in our case in the system of equations

$\int_0^{2 \pi} (\frac{\partial u_N}{\partial t} + u_N \frac{\partial u_N}{\partial x} - \nu \frac{\partial^2 u_N}{\partial x^2} ) \; e^{i k x} \; d x = 0$

for $k = - \frac{N}{2}...\frac{N}{2} - 1$ .

This results in a system of ordinary differential equations for the Fourier components $\hat{u}_n$:

$\frac{d \hat{u}_k}{d t} + \mathcal{F}(u_N \frac{\partial u_N}{\partial x})_k + k^2 \; \nu \; \hat{u}_k = 0$

with

$\mathcal{F}(u_N \frac{\partial u_N}{\partial x})_k := \frac{1}{2 \pi} \; \int_0^{2 \pi} \; u_N \frac{\partial u_N}{\partial x} \; e^{- i k x} d x$

The initial conditions are of course:

$\hat{u}_k (0) = \frac{1}{2 \pi} \int_0^{2 \pi} \; u(x, 0) \; e^{- i k x} d x$

and we have left the boundary conditions unspecified.

When we consider the Fourier transformation of a product of functions

$\mathcal{F} (u v) = \frac{1}{2 \pi} \int_0^{2 \pi} \; u v \; e^{- i k x} d x$

and assume that these functions are trigonometric polynomials, we get the k-th component in form of a convolution sum:

$\mathcal{F}_k (u v) = \sum_{i + j = k} \hat{u}_i \; \hat{v}_j$

Convolution sums can be efficiently evaluated by using a Fast Fourier Transform, for example.

Stochastic Forcing

The Burgers’ equation with stochastic forcing is, besides many other applications, also a sandbox problem to study turbulence. Due to the stochastic forcing term, the Burgers’ equation becomes a stochastic partial differential equation.

A lot of information including the derivation of the solution on closed form can be found here:

• Lokenath Debnath: Nonlinear Partial Differential Equations for Scientists and Engineers. (ZMATH)

Some analytical descriptions of solutions to the Burgers equation will be collected on the page analytical hydrodynamics.

These authors study shock waves for the inviscid Burgers equation using differential geometry:

They consider the inviscid Burgers equation on an $n$-dimensional Riemannian manifold $M$:

$\partial_t u + u \cdot \nabla u = 0$

where $u$ is the velocity vector field, a time-dependent vector field on $M$. They show how it’s related to freely moving noninteracting particles on $M$: if you have one such particle at each point of $M$, moving with velocity $u$, tracing out a geodesic, the vector field $u$ obeys the above equation. More abstractly, they think of the motion of these particles as defining a 1-parameter family of diffeomorphisms

$\eta_t : M \to M$

If a particle starts as $p \in M$ at time zero, at time $t$ it’s at $\eta_t(p)$. And they show the inviscid Burgers equation is equivalent to this 1-parameter family $\eta_t$ being a geodesic with respect to a certain Riemannian metric on the diffeomorphism group.

This resembles Arnol’d’s study of the Euler equation for an inviscid incompressible fluid: indeed, the paper is dedicated to Arnol’d. The difference is that Arnol’d got geodesics in the group of volume-preserving diffeomorphisms.

Then they use this idea to study shock waves in the inviscid Burgers equation.