Keith Rudd and Dr. Jeffery Humpherys, Mathematics

Traveling waves are, mathematically speaking, partial differential equations which can be expresses as U(x-ct), where x is the spatial variable, t is time, and c is the speed of the wave. Traveling waves have applications in fluid dynamics, optics, and many other areas.

In our research we continued what we had already started, taking a close look at the Kortweg-DeVries equation. We were able to produce Matlab movies to observe the known point of instability which occurs at p =4. After this we moved on to high amplitude shock waves in fluid, or viscous dispersive shocks, modeled by the following equations:

Or if we differentiate the last two terms (1) and (2) can be written as

To do the numerical model of this system we employed a Crank-Nicolson scheme because it is second order accurate and allows for large time steps. Implementing this we arrive at the following difference equation:

Notice that the last term is non-linear, as the n+1 terms cannot be separated. To deal with this non-linearity we used a Newton solver. This gave us the sequence

and DFv, DGu, and DGv are similar. Iterating this Newton solver gives us a good approximation of the next time step. Below are some of the figures generated by said process. The parameters used were gamma = 1.4 and v+ = 9e-6. The solid lines represent the waves evolving over time, the dashed the actual solution.