1-D confinement

The addition of a nonlinear "confinement" term to one dimensional advection and wave equations was studied. This confinement allowed pulses concentrated over only two or three grid cells to be convected over long distances without spreading, even on a coarse grid, and with only first order finite difference schemes. The confinement term was found to conserve the total integral of the convected quantity if it was of the form of the derivative of a function. If the confinement term was of the form of the second derivative of a function it was also found not to alter the convection velocity. Forms of the confinement term with these properties were found; even the form that did not have the second property was found to alter the velocity by only a negligible amount. Through a simple mathematic transformation, the confinement term could be altered to allow step discontinuities to propagate with behavior similar to the pulses. Travelling waves created by adding a confinement term to the wave equation had the same propogation characteristics as the advection case. Further, they were found to remain unchanged after interacting with each other, as solutions to the continuum wave equation do. By understanding how confinement functioned in these situations, the results of application to other problems such as vorticity confinement in two or three dimensions should be better understood.