The double-diffusive convection (DDC) phenomenon has seen enormous interest among researchers worldwide due to its pragmatic significance. The application range of DDC can easily be spotted in several fields, from geology to oceanography and over astrophysics to metallurgy. Magnetohydrodynamics (MHD) has long been a subject of substantial scientific investigation due to its relevance in some natural-occurring activities, like solar corona and flares. DDC, under the effect of the external magnetic field, has the potency to damp the movement of fluids with applications in many instances, like electromagnetic casting and other metal manufacturing processes, cooling of magnetic storage media or electronic devices under magnetic field, geothermal reservoirs, crystal growth, electromagnetic stirring, etc. The lattice Boltzmann simulation method (LBM) has attracted numerous research findings. The thesis aims to use an in-house developed (C++ programming language) LBM solver for convection heat transfer with combined temperature and mass diffusion subjected to the external magnetic field. In particular, entropy generation study due to MHD double-diffusive natural convection in the rectangular cavity with built-in rectangular blockage, Effect of the aspect ratio of the shallow enclosure, and built-in rectangular blockage on MHD double-diffusive free convection subjugated to non-uniform boundary conditions, and Effect of near-wall blockage on the MHD based double-diffusive convection within the rectangular cavities. The influence of a various range of pertinent parameters, such as Rayleigh number (103 £ Ra £105), Lewis number (2 £ Le £ 10), buoyancy ratio (-2 £ N £ 2), Hartmann number (0 £ Ha £ 300), separation distance (H/4 £ Sd£ H/8), the aspect ratio of the cavity (1 £ AR £ 4),  and blockage (1 £ ar £ 4) on DDC characteristics has been explored. The validation has shown excellent likeness with the open literature. The results show that the heat and mass transfer rate diminishes with the enhancement in Ha for fixed Ra. The Lewis number encourages the mass transfer rate to augment and the heat transfer rate to diminish. The sinusoidal boundary treatment has higher heat and mass transfer than other boundary treatments. The total Nu and total Sh get reduced by decreasing the N value until it reaches a critical value. The enhancement in separation distance shows higher heat and mass transfer rate.