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We introduce a generalization of the artificial compressibility method for approximation of the incompressible Navier-Stokes equations.It allows for the construction of schemes of any order in time that require the solution of a fixed number of vectorial parabolic problems,depending only on the desired order of the scheme.These problems have a condition number that scales like O(δth ?2),with δt being the time step and h being the spatial grid size.This approach has several advantages in comparison to the traditional projection schemes widely used for the unsteady Navier-Stokes equations.First,it allows for the construction of schemes of any order for both,the velocity and pressure,while the best proven accuracy achievable by a projection scheme is second order on the velocity and 3/2 order on the pressure.Second,the projection schemes require the solution of an elliptic scalar problem for the pressure that has a condition number O(h ?2),in addition to a vectorial parabolic problem for the velocity.This makes them slower if iterative methods are used to solve the linear systems.Third,the artificial compressibility schemes of second or higher order based on a defect correction approach presented below have superior stability properties as compared to a second order projection scheme.Finally,the ability to construct higher order schemes allows to incorporate time step control techniques which greatly improves their efficiency.