A unified shock physics code, ExLO, in which Lagrangian, ALE and Eulerian solvers are incorporated into a single framework, has recently been developed in Korea. It is based on the three-dimensional explicit finite element method and written in C++. ExLO is mainly designed for the calculation of structural responses to highly transient loading conditions, such as high-speed impacts, high-speed machining and explosions. In this paper the numerical schemes are described. Some improvements of the material interface and advection scheme are included. Details and issues of the momentum advection scheme are provided. Numerical predictions are in good agreement with the existing experimental data. Specific applications of the code are discussed in a separate paper in this journal. Eventually, ExLO will provide an optimum simulation environment to engineering problems including the fluid-structure interaction problems, since it allows regions of a problem to be modeled with Lagrangian, ALE or Eulerian schemes in a single framework.