  Representing massless Dirac fermions on a spatial lattice poses a potential
challenge known as the Fermion Doubling problem. Addition of a quadratic term
to the Dirac Hamiltonian circumvents this problem. We show that the modified
Hamiltonian with the additional term results in a very small Hamiltonian matrix
when discretized on a real space square lattice. The resulting Hamiltonian
matrix is considerably more efficient for numerical simulations without
sacrificing on accuracy and is several orders of magnitude faster than the
atomistic tight binding model. Using this Hamiltonian and the Non-Equilibrium
Green's Function (NEGF) formalism, we show several transport phenomena in
graphene, such as magnetic focusing, chiral tunneling in the ballistic limit
and conductivity in the diffusive limit in micron sized graphene devices. The
modified Hamiltonian can be used for any system with massless Dirac fermions
such as Topological Insulators, opening up a simulation domain that is not
readily accessible otherwise.
