In the relatively new but extremely fast growing field of topological matter research, this thesis should contribute to the understanding of the behavior of the underlying quasiparticle: the Dirac fermion. For this purpose the reader gets a brief introduction to the emergence of relativistic quantum mechanics in solid state systems. The connection to the topological invariant and its indispensability in giving a unified description of the physics at topological defects like surfaces, domain-walls, and vortices is made. The main focus, however, lies in the description of the dynamical behavior of the Dirac fermion quasiparticle in this non-trivial spatial and, possibly, time dependent potential and mass landscapes. To allow an efficient numerical simulation, a new scheme is developed. Along the way, the famous fermion doubling problem is introduced, and avoided by a special time and space staggering of the numerical finite difference grid. Like in the underlying differential equation, this special discretization of the Dirac equation treats time and space on an equal footing. Schemes for (1+1)D, (2+1)D, and (3+1)D are formulated, and their numerical properties are derived. The important topic of open boundary conditions is discussed and, in the one-dimensional case, perfect absorbing boundary conditions (so-called discrete transparent boundary conditions) are derived. In (2+1)D absorbing boundary conditions, using imaginary potential regions, are introduced. On the applied physics side, the utilization of domain-wall fermions for dissipation-less electric circuits is proposed. Using these principles, an interferometer device which can be controlled by an electrical gate is envisioned and its working principle is shown numerically.