This paper is devoted to the derivation of absorbing boundary conditions for the Klein-Gordon and Dirac equations modeling quantum and relativistic particles subject to classical electromagnetic fields. Microlocal analysis is the main ingredient in the derivation of these boundary conditions, which are obtained in the form of pseudodifferential equations. Numerical schemes are derived and analyzed to illustrate the accuracy of the derived boundary conditions.

S ubm itted to the School o f M athem atics and S tatistics in p a rtia l fu lfillm e n t o f the requirem ents fo r the degree of M aster of Science in A p p lie d M athem atics A bstract T he th eory o f A rtific ia l B oundary C onditions described by A n to in e et al. [2,4-6] for the Schrodinger equation is applied to the K lein-G ordon (K G) in two-dim ensions (2-D) for spinless particles subject to electrom agnetic fields. We begin b y p rovid ing definitions for a basic understanding o f the th eory of operators, d iffe re n tia l geom etry and wave fro n t sets needed to discuss the fa cto riza tio n theorem thanks to N irenberg and H orm ander [14,16]. The laser-free K lein-G ordon equation in 1-D is th e n dis cussed, followed by the case in clu d in g electrodynam ics potentials, concluding w ith the K G equation in 2-D space w ith electrodynam ics potentials. We then consider nu m erical sim ulations of the laser-particle K G equation, w hich includes a b rie f analysis o f a fin ite difference scheme. T he conclusion integrates a discussion o f the num eri cal results, the successful com pletion of the objective set fo rth , a declaration o f the unanswered encountered questions and a suggestion of subjects for fu rth e r research.