Presentation of

Burgers' equation

Burgers' equation:

for discretization purposes we will use the more convenient form:

Burgers’ equation was initially proposed as a turbulence model until further studies including the work of Hopf and Cole showed that it did not ave the adequate properties. Even if there is no physical behavior to be modelized by this equation it is still a interresting study because of its ligh approach to non linear equations. Non linearity still represent a great challenge for analytical as well as for numerical solving.

Burgers’ equation leads a low viscosity and with the proper initial conditions (a decreasing speed with x) to the formation of shocks (when the fatest fluid catches up the slowest). In case of and inviscid fluid multivalued solutions appear. After the schock formation the solution decays wlile the maximm moves away from he schock postion as a result of th be viscosity.

To caraterize Burgers’ behavior it is quit clear that the Reynolds number
Re is appropriate. For u=1m/s and a domain length l=1m (the values used
in this study) Re=1/n. So it is quite clear that viscosity is the main
parameter od this study. If Re=1 viscosity and advection at from the same
order. For Re>>1 advection is dominant and will possibly lead to the shock’s
formation.