An improved divergence-free-condition compensated method for solving incompressible flows on collocated grids

A collocated incompressible Navier-Stokes equations solver, which is based on an improved version of the divergence-free-condition compensated (DFC) method, is proposed in this paper. In order to avoid the difficulty and complexity of calculating the DFC source terms, an alternative procedure is proposed. In the present paper, the newly proposed IDFC source terms are derived through the following two steps: (1) solve the momentum equation on the cell-face in an approximated sense; (2) solve the momentum equation on the cell-center using the corresponding equations obtained from step (1). By this consistent treatment, the present method can avoid the odd-even decoupling problem, as well as the time-step-size dependent problem. For the sake of calculating a dispersively accurate spatial solution, the advection terms are approximated by the finite-volume-based dispersion-relation-preserving (DRP) advection scheme. Several benchmark problems with analytic solutions are investigated and compared with the present results, to show the applicability and accuracy of the present solver.