1d-Shallow Water Linear Advection Report

Results: Courant Number Analysis

In this test, the model Courant number is modified to test how the different CFL-criterion valid numbers impact the integration. The values of U and dx are held constant. Thus the variation of the Courant number is achieved through changes to the time spacing dt. As this variable is changed, the value of Nt is also modified to approximately preserve the distance the wave travels during the integration. As the value of dt gets smaller, the value of Nt is increased to compensate.

The following plots show graphically the results of this analysis. The Courant number is progressively lowered from a starting value of 0.99 (just under the stability condition for this differencing scheme), to 0.1. Once again, Gaussian waves are used to ensure clarity in the analysis.

Figure 9: A Gaussian wave with integrated with a Courant number of 0.99 (dt = 9.9s).

Figure 10: A Gaussian wave with integrated with a Courant number of 0.6 (dt = 6s).

Figure 11: A Gaussian wave with integrated with a Courant number of 0.3 (dt = 3s).

Figure 12: A Gaussian wave with integrated with a Courant number of 0.1 (dt = 1s).

As the Courant number (and thus time spacing) is reduced, the solution appears to become more stable, with less low-amplitude noise both interacting with and away from the primary wave. This is likely the result of increasing the number of stable integrations of the model. As more integrations are performed, the unstable initial forward-in-time integration has less of an effect on the resulting solution.

This website designed by Brian Vanderwende, 2011.