Numerical Schemes for Linear Hyperbolic Problems with Interfaces

In this thesis, we investigate the numerical techniques for solving interface problem of linear hyperbolic system of equations with piecewise constant coefficients and jump conditions across the interface. Such problem arises naturally in practical physics, for example, electromagnetic waves propagating from one material to the other. Standard numerical techniques for solving hyperbolic systems fail near the interface, and special treatments must be offered. Two Cartesian-based methods, ""ghost fluid method"" and ""immersed interface method"", are introduced to catch the jump discontinuity.