In this paper, a high-order method is used to simulate the viscous fingering fluid instability during the miscible displacement process in porous media, on structured and unstructured grids. The numerical model incorporates decoupling in time, discontinuous Galerkin method of high order, flux reconstruction and parallel implicit solvers to produce an accurate and efficient predictive tool for finger growth. This paper shows that the proposed numerical approach is a competitive method to simulate several viscous fingering problems, such as rectilinear flow, density-driven flow and radial flow. The numerical model does not suffer from grid orientation, and accurately measures finger growth rate. Convergence of the fingering pattern is obtained under mesh refinement and increased polynomial degree.