Many technologically important problems are coupled in nature. For example, blood flow in deformable arteries, flow past (flexible) tall buildings, coupled deformation-diffusion, degradation, etc. It is, in general, not possible to solve these problems analytically, and one needs to resort to numerical solutions. An important ingredient of a numerical framework for solving these problems is the coupling algorithm, which couples individual solvers of the subsystems that form the coupled system, to obtain the coupled response. A popular coupling algorithm widely employed in numerical simulations of such coupled problems is the conventional staggered scheme (CSS). However, there is no systematic study on the stability characteristics of the CSS. The stability of coupling algorithms is of utmost importance, and assessment of the stability on real problems is not feasible given the computational costs involved. The main aim of this thesis, is to address this issue - assess the accuracy and stability characteristics of CSS using various canonical problems. In this thesis we show that the stability of CSS depends on the relative sizes of the domain, disparity in material properties, and the time step.