A novel computational algorithm for the direct numerical simulation of miscible-porous-media flows is developed and applied that offers multiple benefits. It is based on the streamfunction-vorticity formulation of Darcy's law and resolves all physically relevant length scales including diffusion. By employing an implicit discretization based on compact finite differences, it combines ease of implementation and the ability to handle nontrivial geometries with the superior computational accuracy usually reserved for spectral methods. The formal accuracy is O(Αx4,Αt2), although diffusion terms are discretized with O(Αx6). Test calculations are performed that provide a quantitative comparison with linear stability results for strong mobility contrasts in the practically relevant-quarter five-spot configuration. The excellent agreement, both with regard to the algebraic growth rate as well as the preferred wave number, demonstrates the very low levels of numerical diffusion, thereby eliminating grid orientation effects. The capabilities of the numerical method are furthermore demonstrated by means of representative calculations for both homogeneous and heterogeneous displacements. Among other findings, these simulations exhibit a minimal recovery efficiency for intermediate values of the correlation length of the permeability heterogeneities.