In this work we analyse and develop shock capturing (SC) techniques to improve the behaviour of one dimensional Finite Element methods for nonsmooth solutions. After investigating the state-of-the-art of the current SC techniques., the most interesting ones have been selected to experimentally analyse them. We have organized the method in three groups: SC techniques for continuous Galerkin methods, limiters and SC for high order methods. A Fortran 90 code has been developed in order to implement the methods in the literature. The code is capable to solve the convection-diffusion-reaction equation (and in particular the transport equation) using FEM in space and theta-methods for the integration in time. The method is able to use continuous and discontinuous Galerkin in space as well as any order of approximation desired. The selected SC methods of the literature have been implemented in the code. The objective is to understand the behaviour of these techniques and be able to propose modifications and even new SC schemes. In particular, a new SC method is proposed in the context of SC for CG under the name of gradient jump viscosity method (GJV). . In this project, the student will carry out a detailed state-of-the-art review on numerical methods for the approximation of the linear transport equation (limiters, stabilized methods and artificial viscosity methods). A prototypical 1d solver will be developed and a wide set of these methods implemented. In a next step, a deep comparison of these techniques will be carried out. In particular, the student will evaluate the capability of these techniques to deal with continuous and discontinuous Galerkin schemes and implicit time integration.