# The Azimuth Project Burgers' equation (Rev #4, changes)

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# 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. 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

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

in a spatial region $\Omega$, positive time $t \gt 0$ and with a real constant $\nu \gt 0$, with appropriate boundary 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

### 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.

### 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:

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

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

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

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

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