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Burgers' equation

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

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

where ρ\rho is the density, uu is the velocity vector field, pp is the pressure, μ\mu is the viscosity, and FF is an external force. If we drop the pressure term we get Burgers’ equation:

ρ(ut+uu)=μ 2u+F \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 ρ\rho is a constant for an incompressible fluid; this allows us to define a new constant, the kinematic viscosity ν=μ/ρ\nu = \mu / \rho, and write Burgers’ equation as

ut+uu=ν 2u \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:

ut+uu=0 \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 x n\mathbf{x} \in \mathbb{R}^n at time tt the velocity of the particles there is u(t,x)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 uu that vanishes:

ut+uu=0 \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=1n = 1). Then the equation becomes

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

We may consider this in some spatial region Ω\Omega for positive times t0t \ge 0 and with a real constant ν>0\nu \gt 0, with appropriate boundary conditions and initial conditions.

We write for the initial condition:

u(x,o)=u 0(x) u(x, o) = u_0(x)

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}

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=2νϕ xϕ u = -2 \nu \frac{\phi_x}{\phi}

so that the function ϕ\phi satisfies

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

iff the function uu satisfies Burgers’ equation.

This transformation is usually called Cole-Hopf transformation.

The initial condition for ϕ\phi is

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

which can be integrated to yield:

ϕ(x) 0:=ϕ(x,0)=exp(12ν 0 xu 0(α)dα) \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

ϕ(x,t)=12πνt ϕ 0(ζ)exp((xζ) 24νt)dζ \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 ϕ 0\phi_0 in order to rewrite this formula in a form that is a little bit more convenient, by introducing the help function

f(ζ,x,t):= 0 ζu 0(α)dα+(xζ) 22t f(\zeta, x, t) := \int_0^{\zeta} \; u_0(\alpha) \; d \alpha + \frac{(x- \zeta)^2 }{2 t}

and get

ϕ(x,t)=12πνt exp(f2ν)dζ \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 ϕ\phi itself, and thus for its derivative:

ϕ x(x,t)=14νπνt xζtexp(f2ν)dζ \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)= xζtexp(f2ν)dζ exp(f2ν)dζ 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<x<l0 \lt x \lt l:

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

and we state a specific initial condition:

u(x,0)=u 0sin(πxl) 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:

ϕ(x,0)=exp[u 0l2πν(1cos(πxl))] \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:

ϕ(x,t)=a 0+ n=1 a nexp(n 2π 2νtl 2)cos(nπxl) \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=1l 0 lexp[u 0l2πν(1cos(πxl))]dx=exp(u 0l2πν)I 0(u 0l2πν) 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=2l 0 lexp[u 0l2πν(1cos(πxl))]cos(nπxl)dx=2exp(u 0l2πν)I n(u 0l2πν) 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 nI_n are the modified Bessel functions of the first kind.

Therefore, the solution of Burgers’ equation is:

u(x,t)=4πνl n=1 nI n(u 0l2πν)exp(n 2π 2νtl 2)sin(nπxl)I 0(u 0l2πν)+2 n=1 I n(u 0l2πν)exp(n 2π 2νtl 2)cos(nπxl) 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} 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)}

The formula simplifies somewhat if we make the obvious choice l=2π l = 2 \pi :

u(x,t)=2ν n=1 nI n(u 0ν)exp(n 2νt4)sin(12nx)I 0(u 0ν)+2 n=1 I n(u 0ν)exp(n 2νt4)cos(12nx) u(x, t) = 2 \nu \frac{ \sum_{n = 1}^{\infty} \; n \; I_n \left( \frac{u_0}{\nu} \right) \; \exp \left( - \frac{n^2 \nu \; t}{4} \right) \; \; \sin \left( \frac{1}{2} \; n \; x \right)} {I_0 \left( \frac{u_0}{\nu} \right) + 2 \sum_{n = 1}^{\infty} I_n \left( \frac{u_0}{\nu} \right) \; \exp \left( - \frac{n^2 \nu \; t}{4} \right) \; \cos \left( \frac{1}{2} \; n \; x \right)}

Setting u 0=0u_0 = 0 and ν=0.5\nu = 0.5, a numerical calculation (TODO: add reference to repository project) shows that this solution describes exponential decay:

Burgers equation with 0.5 viscosity

When you consider a high viscosity or big times in the exact solution, you will see that the first summand in the nominator dominates, which describes exponential decay. So the plot seems to be plausible.

Examples

N-Wave Solution

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

ϕ(x,t)=1+τtexp(x 24νt) \phi(x, t) = 1 + \sqrt{ \frac{\tau}{t}} \exp \left( \frac{- x^2}{4 \nu t} \right)

with a positive constant τ\tau.

Plugging this into the formula for the exact solution yields:

u(x,t,τ,ν)=xt(1+tτexp(x 24νt)) 1 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,τ,νt, \tau, \nu it has a N-shaped graph:

N-Wave solution to the Burgers' equation

Approximation via Fourier-Galerkin Spectral Method

We will use a spectral method for an approximate solution of Burgers’ equation on the domain Ω=(0,2π)\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)= k=N2 k=N21u^ n(t)e ikx 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 M(f α),h i=0\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

0 2π(u Nt+u Nu Nxν 2u Nx 2)e ikxdx=0 \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=N2...N21k = - \frac{N}{2}...\frac{N}{2} - 1 .

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

du^ kdt+(u Nu Nx) k+k 2νu^ k=0 \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

(u Nu Nx) k:=12π 0 2πu Nu Nxe ikxdx \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:

u^ k(0)=12π 0 2πu(x,0)e ikxdx \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

(uv)=12π 0 2πuve ikxdx \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:

k(uv)= i+j=ku^ iv^ j \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.

References

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 nn-dimensional Riemannian manifold MM:

tu+uu=0 \partial_t u + u \cdot \nabla u = 0

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

η t:MM\eta_t : M \to M

If a particle starts as pMp \in M at time zero, at time tt it’s at η t(p)\eta_t(p). And they show the inviscid Burgers equation is equivalent to this 1-parameter family η t\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.