High order accurate solution of the incompressible Navier–Stokes equations

Abstract

High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier-Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.

A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics

“…The BD/BDE2 scheme has been successfully combined with various spatial discretizations in semi-discrete type formulations for the Navier-Stokes equations (see, for example, [29,30]). However, it is not as popular as the semi-implicit scheme that combines Trapezoidal rule (or Crank-Nicholson) and second order Adams-Bashforth (CN/AB2).…”

“…The final step of the CTU scheme is to again use upwinding to choose the appropriate approximate edge values θ k+1/2 i±1/2,j and θ k+1/2 i,j±1/2 for the fluxes in (28) and (29). The use of limiting in (32) and (33) eliminates spurious oscillations in the CTU scheme (27), but it also reduces the order of accuracy to first order where the magnitude of the gradient of θ is large.…”

Animating smoke with dynamic balance

“…For that reason, developing an adaptive method for reducing numerical errors is a significant problem. 13,14 We found a lot of work related to this problem in computer graphics. Fattal et al 6 forced the fluid to form a user-defined shape, and they use a density error term to reverse the diffusion process so as to match a userdefined shape.…”

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

“…See [45,33] for a review on fundamental formulations of incompressible Navier-Stokes equations. The appearance and growing popularity of "compact schemes" brought a renewed interest in the aforementioned methods ( [26,17,18,43,42,19,50,35,1,13]). The purestreamfunction formulation for the time-dependent Navier-Stokes system in planar domains has been used in [31,32,30] some twenty years ago.…”