This thesis proposes and examines various algorithms for analysis of steady ideal fluid capillary flows with free/moving boundary.;For this class of problems, the leading parameter is the capillary number C which for an established flow and fixed geometry of the solution domain is proportional to the ratio of a velocity scale and surface tension. When C {dollar}\ll{dollar} 1 the problem can be simplified and is solved using Small Deformation Theory (SDT). Conditions for validity of SDT are identified.;When C = 0(1) one has to seek a simultaneous solution for two dependent variables, i.e. a flow field and a free surface shape. Depending on the order of linearization of the governing equations one arrives at the Picard Algorithm (linearization of the first order) and the 1-Step Algorithm (of the second order). The latter one provides significantly faster convergence.;All algorithms are based on a finite-difference approximation and the Alternate Direction Implicite (ADI) scheme has been chosen as a method of solution. A thorough study of an algebraic stability of equations of the flow field and the free surface, has been carried out. This is supplemented by an analytical and numerical analysis of existence and uniqueness of the solutions to the free surface equation. The Wachspress optimization of relaxation parameters has been used in order to accelerate convergence of the ADI.;Finite differences discretization implemented in the thesis is based on the Hermitian equations, which generated compact difference schemes of the second and higher order accuracy. In the thesis one can find a rigorous study of the interrelationship between the order of differencing scheme and the rate of convergence of the computed results with grid refining. This rate, called \u27grid-convergence order\u27 has been used as the criterion for identification of the minimum dimensionality of the computational grid. It was found that higher order methods require much finer grids than second order methods, to provide desired grid-convergence order.;For higher order method, the new fourth order compact difference estimate (independent of coordinate direction) of a mixed derivative was found. Its application significantly improved the grid-convergence order.;The discussion of the algorithms is supplemented by a physical interpretation of the results obtained for a number of particular cases
