On hybrid residual distribution-Galerkin discretizations for steady and time dependent viscous laminar flows

This paper is concerned with the extension of second order residual distribution (RD) schemes to time dependent viscous flows. We provide a critical analysis of the use of a hybrid RD-Galerkin approach for both steady and time dependent problems. In particular, as in Ricchiuto (2008) and Villedieu et al. (2011), we study the coupling of a Residual Distribution (RD) discretization of the advection operator with a Galerkin approximation for the second order derivatives, with a Peclet dependent modulation of the upwinding introduced by the RD scheme. The final objective is to be able to achieve uniform second order of accuracy with respect to variations of the mesh size or, equivalently, of the local Peclet/Reynolds number. Starting from the scalar formulation given in the second order case in Ricchiuto et al. (2008), we perform an accuracy and stability analysis to extend the approach to time-dependent problems, and provide thorough numerical validation of the theoretical results. The schemes, formally extended to the system of laminar Navier-Stokes equations, are also compared to a finite volume scheme with least-squares linear reconstruction on the solution of standard test problems. While the scalar tests show the potential of this approach in providing uniform accuracy w.r.t. the range of values of the Peclet/Reynolds number, the results for the laminar Navier-Stokes equations show that in practice, the use of Peclet number based corrections does not change dramatically the quality of the solutions obtained. This justifies the quest for new formulations.