Contents | Viscosity and initialization | Boundary layer | Amplitude | Conclusion
Effect of the viscosity and initialization
In the physical description of the Burgers'equation, the effect of viscosity was mentionned as a mean to attenuate the solution along the time. To verify this phenomenon, some tests were realized with different values of viscosity and the sinusoidal and straight line initializations.
The tests were performed with three values of viscosity: 0.1, 0.01 and 0.001 m2/s
For nu = 0.1 m2/s (1), we observe that the shock does not appear and that the solution is strongly attenuated by the effect of viscosity.
For nu = 0.01 m2/s (2), the solution tends to form a soften shock. This shock is not complete, because of the effect of viscosity and the solution is gradually attenuated.
For nu = 0.001 m2/s (3), the solution forms a sharper shock. The effect of viscosity is to weak to compense the non linear convective term.
Straight line initialization
The tests were performed with three values of viscosity: 0.5, 0.1 and 0.01 m2/s.
The comportment is not similar to the sinusoidal initialization: the solution tends to reach a limit value and after does not evoluate.
For nu = 0.5 m2/s, we observe that the shock does not appear. The solution is very closed to the straight line. The effect of viscosity are predominent. The limit is reached very fast.
For nu = 0.1 m2/s, the solution tends to its limit less rapidly. This limit is smooth. The effect of viscosity attenuates the shock which was forming.
For nu = 0.01 m2/s, the solution forms a sharper shock. The effect of viscosity is to weak to compense the non linear convective term. The limit value is here the vertical straight line in the middle of the domain (x = 0.5 m)
In conclusion of this study, we can say that the program reproduces well the awaited results. The influence of viscosity is determining the appearance of a shock, and for the straight line initialization, the limit value of the solution.
The pictures (1) to (6) show that a kind of boudary layer can be found in the solution. For the sinusoidal initialization, it represents the distance between the maximum and the minimum of the solution. The evolution of this boudary layer is in the form:
In order to verify this phenomenon, the logarithm of the width epsilon was plotted versus the logarithm of the viscosity (from 0.001 to 0.1), with the time as parameter (from 0.2 to 1.6s, step 0.2).
We can observe that there are two zones in the diagram were the values are around a straight line with a coefficient of 0.5, for the very little and the high values of the viscosity. In the middle values, the coefficient is higher (0.8).
So we can say that the theory is repected for little and high values of the viscosity, that means when a shock occurs or when the viscosity has a dominant effect. The higher coefficient in the middle values is due to the transition between the two phenomenons.
Amplitude of the solution
The other point concerns the study of the evolution of the amplitude versus time. In theory, we should observe an evolution in .
For this study, the sinusoidal initialization is also used with different values of time and with the vicosity as parameter (from 0.001 to 0.1). The total amplitude of the solution is plotted versus time.
The theory was also verified in this case. The curve obtained (in black), has an equation corresponding to the theory: . This curve approximate the solution for the little values of the viscosity and for the times superior to 0.4s, that means when the shock occurs and develops.
The study of the results and the comparison with theory permit to say that the model used is correct to simulate the Burger's equation and that the program reproduces well the awaited results.