The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.