Efficient numerical methods for option pricing is an active field of research. This project has the goal to examine possible ways to improve an established method of numerical pricing. The method is based on an adaptive finite difference method in price and uses the backwards differentiation formula of order 2, BDF2, in time. The project will focus on improvements to the time integration through implementation of discontinuous Galerkin methods, dG. Empirical convergence and accuracy results are obtained for equidistant dG-methods up to order 3 and performance is compared to BDF2. The dG-methods do not succeed in outperforming the BDF2-method when comparing accuracy to time for computation, but they do match the performance. Possible ways for improvements are suggested.