  The non-modal linear stability of miscible viscous fingering in a two
dimensional homogeneous porous medium has been investigated. The linearized
perturbed equations for Darcy's law coupled with a convection-diffusion
equation is discretized using finite difference method. The resultant initial
value problem is solved by fourth order Runge-Kutta method, followed by a
singular value decomposition of the propagator matrix. Particular attention is
given to the transient behavior rather than the long-time behavior of
eigenmodes predicted by the traditional modal analysis. The transient behaviors
of the response to external excitations and the response to initial conditions
are studied by examining the $\epsilon-$pseudospectra structures and the
largest energy growth function. With the help of non-modal stability analysis
we demonstrate that at early times the displacement flow is dominated by
diffusion and the perturbations decay. At later times, when convection
dominates diffusion, perturbations grow. Furthermore, we show that the dominant
perturbation that experiences the maximum amplification within the linear
regime lead to the transient growth. These two important features were
previously unattainable in the existing linear stability methods for miscible
viscous fingering. To explore the relevance of the optimal perturbation
obtained from non-modal analysis, we performed direct numerical simulations
using a highly accurate pseudo-spectral method. Further, a comparison of the
present stability analysis with existing modal and initial value approach is
also presented. It is shown that the non-modal stability results are in better
agreement, than the other existing stability analyses, with those obtained from
direct numerical simulations.
