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Burgers' equation (Rev #9)

Contents

Idea

Burger’s 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 especially of a 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.

In one spatial dimension it is

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

in a spatial region Ω\Omega, positive time t>0t \gt 0 and with a real constant ν>0\nu \gt 0 (it is the kinematic viscosity), with appropriate boundary 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

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

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.

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

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 iktdx=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 t} \; 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)