One dimensionnal Burgers' equation

Contents | Introduction | Physical behaviour

Introduction

The one-dimensionnal Burger's equation is a typical example for non linear equations. It permits to simulate flows with severe gradients or shocks.

This equation can be written in the following form:

(1)

It is similar to the transport equation except that the convective term is non linear.

The equation can also be written in the conservative form:

(2)

This form is often used in computational algorithms (see section “model”).

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Physical behaviour

If the viscous term is dropped from (1), the result is the inviscid equation. The non linearity allows discontinuous solutions to develop.

A wave is convecting and the points with larger value of u convect faster and consequently overtake parts of the wave with smaller values of u. To have a unique and physically sensible result, it is necessary to introduce a shock accross which u changes discontinuously as it is shown on the diagram below:

The effect of the viscous term is twofold: first it reduces the amplitude of the wave for increasing t and secondly it prevent multivalued solutions.

Another point is that the non linear convective term allows aliasing to occur. The shorter wavelength which can be solved on a grid with a spacing dx is 2dx. The energey associated with wavelengths shorter than 2dx reappears, associated with long wavelengths. The aliasing distorts the true longwave solution and may cause instability where very long time integrations are made.

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