A numerical method is developed for the simulation of solidification of solutions/alloys. The heat and species transport equations are solved with appropriate interface conditions. The interface shape and thermal and solutal fields are calculated in a fully coupled manner. The effects of capillarity are included in the interfacial dynamics. The present mixed Eulerian-Lagrangian framework treats the immersed phase boundary as a sharp solid-fluid interface and a conservative finite-volume formulation allows boundary conditions at the moving surface to be exactly applied. We first compare the planar growth results with published one-dimensional numerical results. We then show that the method can compute the breakdown of the solid-liquid interface due to the Mullins-Sekerka instability. The dendritic growth of the crystals under various growth parameters is computed.