Simulations of the Richtmyer-Meshkov instability (RMI), including inviscid and viscous effects are conducted using an improved localized artificial fluid diffusivity method which is used to treat discontinuities in the form of shocks and material interfaces in the flowfield. On the 2-D inviscid shock-cylinder interaction problem the scheme captures unsteady shocks and material discontinuities without spurious oscillations and shows good agreement with experimental data and previous numerical calculations when initial start-up errors are avoided. Comparison to previous numerical investigation done using high-order WENO schemes shows that the present scheme has lesser numerical dissipation, indicating the suitability of the method for resolving a broad range of scales of turbulence. The scheme is used to simulate the Richtmyer-Meshkov instability in a viscous cylindrical (2- D) and planar (3-D) geometry. Simulations are designed to match initial conditions from previous experiments and good agreement is found between the experimental data and the numerical results. The study of the mixing mechanism shows that a large amount of mixing is associated with the primary instability through the large gradient intensification in particular regions of the flow ('bridge') though secondary instabilities which occur at intermediate times are efficient in mixing the multi-component flow. On the 3-D planar RMI problem results of the mixing zone growth rate shows good agreement between the present numerical study and the available experimental data.