Analysis and application of high order implicit Runge–Kutta schemes to collocated finite volume discretization of the incompressible Navier–Stokes equations

• A temporally high order solution algorithm is presented for incompressible Navier–Stokes.
• High order implicit Runge–Kutta schemes are used for time integration.
• Collocated finite volume method is used for spatial discretization.
• A face-velocity interpolation which preserves temporal design order is presented.

The application of a family of high order implicit Runge–Kutta time integration schemes, namely the explicit first-stage singly diagonally implicit Runge–Kutta schemes (ESDIRK), to cell-centered collocated finite volume discretization of the unsteady incompressible Navier–Stokes is considered. Although achieving computational efficiency relative to commonly used second order implicit schemes has been the motivating factor, this study focuses on temporal order analysis of the high order schemes on the collocated grid. In particular, a face velocity interpolation procedure (Rhie–Chow) which preserves the temporal design order of the ESDIRK schemes is introduced. The details of an iterative pressure-based time advancing algorithm comprising the designed interpolation method are presented (iterated-PISO). The results from solving numerical examples demonstrate the temporal order preservation of the algorithm.