This dissertation mainly focuses on the development of new numerical models to simulate transport phenomena and predict the occurrence of macrosegregation defects known as freckles in directional solidification processes. Macrosegregation models that include double diffusive convection are very complex and require the simultaneous solution of the conservation equations of mass, momentum, energy and solute concentration. The penalty method and Galerkin Least Squares (GLS) method are the most commonly employed methods for predicting the interdendritic flow of the liquid melt during the solidification processes. The solidification models employing these methods are computationally inefficient since they are based on the formulations that require the coupled solution to velocity components in the momentum equation Motivated by the inefficiency of the previous solidification models, this work presents three different numerical algorithms for the solution of the volume averaged conservation equations. First, a semi explicit formulation of the projection method that allows the decoupled solution of the velocity components while maintaining the coupling between body force and pressure gradient is presented. This method has been implemented with a standard Galerkin finite element formulation based on bi-linear elements in two dimensions and tri-linear elements in three dimensions. This formulation is shown to be robust and very efficient in terms of both the memory and the computational time required for the macrosegregation computations. The second area addressed in this work is the use of adaptive meshing with linear triangular elements together with the Galerkin finite element method and the projection formulation. An unstructured triangular mesh generator is integrated with the solidification model to produce the solution adapted meshes. Strategies to tackle the different length scales involved in macrosegregation modeling are presented. Meshless element free Galerkin method has been investigated to simulate the solidification processes to alleviate the difficulties associated with the dependence on the mesh. This method is combined with the fractional step method to predict macrosegregation. The performance of these three numerical algorithms has been analyzed and two and three dimensional simulations showing the directional solidification of binary Pb-Sn and multicomponent Ni base alloys are presented.