Schemes' studies

Contents |

Study realized The different schemes were compared to evaluate their properties and their comportment with the simulation parameters dt, dx and the cinematic viscosity nu. The aim is to determine which scheme fits better to the parameters, especially to test the dispersion of the schemes. The other aim is to determine the range of value for the parameters and to evaluate the values of dt and dx which allow a good range of values for the cinematic viscosity.

Dispersion The test of the dispersion was realized with dx = 0.01 m and dt = 0.01 s, for two values of nu, 0.01 and 0.001, with the sinusoidal initialization. For nu = 0.01, the results provided by the schemes are correct, with no dispersion. (the curves are plotted for t=0.2s). We can see that all the schemes have approximately the same comportment, and that there is no dispersion.  For nu = 0.001, the results provided by the schemes begin to be incorrect. The three first scheme present oscillation in the proximity of the shock. Only the 4points-upwind scheme does not present such an effect: the oscillation is just beginning.  For nu = 0.0005, the results provided by the schemes are less and less correct. The dispersion is enforced by the reduction of the viscosity. The results for the 4 points-upwind scheme are acceptable: the dispersion is not very strong. For the other schemes, it increases.  As conclusion, to enlarge the spectrum of the values of the viscosity, the 4 points-upwind scheme will be used. UP

Numerical parameters

The aim is to determine the ranges of values for the numerical parameters, to increase the precision and decrease the dispersion. The first solution is to decrease the space step. Because of the structure of the output file, the maximum number of space points is 121. This value has been chosen. The other direction is to decrease the time step. After several studies, the value of 0.001s was chosen. It permits to avoid the dispersion and allows a good precision. The total time for the calculation is acceptable at the same time. Finally, with these parameters, the dispersion appears for nu = 0.0001. So the viscosity must be superior than 0.0001. UP

Conclusion

With the scheme comparison and the study of the different parameters, the following ranges of value were chosen for the simulations:

Viscosity

dt

dx

> 0.0001 m2/s

0.001 s

0.083 m UP