An overview of projection methods for incompressible flows

Abstract

Abstract. We introduce and study a new class of projection methods-namely, the velocitycorrection methods in standard form and in rotational form-for solving the unsteady incompressible Navier-Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1 -norm for the velocity and of the L 2 -norm for the pressure. We also show that the class of fractional-step methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986) Phys., 97 (1991), pp. 414-443] can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.

Interactions between vortex tubes and magnetic-flux rings at high kinetic and magnetic Reynolds numbers

“…The (strongly nonlinear) mathematical model is solved via a staggered grid, fractional step, projection, finite volume method [37,38]. A book-length discussion of the method is available in [39].…”

Group-wise construction of reduced models for understanding and characterization of pulmonary blood flows from medical images

“…Problem (2) is discretized in time with a Chorin-Temam projection scheme (see, e.g., (Guermond et al, 2006;Chorin, 1968;Temam, 1968)) in which velocity and pressure are solved separately in two substeps. Namely, let us denote with τ the time-step size, setting t n def = nτ for 1 ≤ n ≤ N .…”

An embedded approach for immiscible multi‐fluid problems

“…Alternatives to the coupled scheme such as e.g. incremental pressurecorrection scheme in rotational form that provides the consistent boundary condition for the pressure [21], [35] must be investigated. Possibility of using this approach in both domains and its impact on the computational efficiency will be investigated in the future.…”

“…Interesting alternatives include the use of non-singular approximations of the Laplacian matrix in the pressure Poisson's equation or the schemes based on consistent pressure boundary condition (e.g. incremental pressure-correction scheme in rotational form, see [21]). …”

“…Fractional step or pressure projection approach (see [19], [20] or [21] ) uncouples the velocity and the pressure. Instead of one large and poorly conditioned system of equations two smaller and better conditioned systems are solved.…”