High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows (Communications in Computational Physics, 2011)
Within the projection schemes for the incompressible Navier-Stokes equations(namely "pressure-correction" method), we consider the simplest method (of orderone in time) which takes into account the pressure in both steps of the splittingscheme. For this scheme, we construct, analyze and implement a new high order compactspatial approximation on nonstaggered grids. This approach yields a fourth orderaccuracy in space with an optimal treatment of the boundary conditions (without erroron the velocity) which could be extended to more general splitting. We prove theunconditional stability of the associated Cauchy problem via von Neumann analysis.Then we carry out a normal mode analysis so as to obtain more precise results aboutthe behavior of the numerical solutions. Finally we present detailed numerical tests forthe Stokes and the Navier-Stokes equations (including the driven cavity benchmark)to illustrate the theoretical results.