Numerical Solution of Transport Equations.

In this dissertation we discuss the numerical solution of systems of hyperbolic partial differential equations with lower order terms and step function initial data. These equations arise in modeling the propagation of a signal with loss, such as a signal in a resistive co-axial cable, or the flow of neutrons in a reactor. Majda and Osher have shown that dissipative finite difference approximations to such problems display a numerical artifact which is not encountered for scalar equations. Namely, noise from an initial discontinuity propagates into a large region behind the discontinuity. Their results do not apply in the vicinity of a discontinuity, and our goal is to discover the detailed behavior in this region. This information will be of use in constructing algorithms that attempt to accurately approximate solutions with discontinuities or shocks.