The incompressible Navier-Stokes (INS) equations upon discretization on fixed meshes become a system of differential algebraic equations (DAE) of index 2. It is proved in this paper that for the general explicit and implicit Runge-Kutta (RK)methods, the time accuracy of velocity is the same as that for the ordinary differential equations, by taking into consideration of the special form of the resulting DAE; (the time accuracy of pressure can be lower). For the three-stage secondorder explicit RK method, algorithms with less (than three) Poisson solutions of pressure are proposed and verified by numerical experiments. However, in practical computation of complex flows it is found that the method must satisfy the so-called consistency condition for the components of the solution (here the velocity and the pressure) of the DAE for the method to be robust.
An efficient and robust pressure correction projection method with the CNMT1 finitedifference scheme is presented in this paper for the numerical solution of the incompressible Navier-Stokes equations. It is proved that on fixed spatial grids the method is of secondorder global accuracy in time; this is confirmed with numerical experiment on an examplewith an exact solution. Then the method is used for numerical simulation of the drivencavity flow problems; the asymptotic periodic solution for Re=10000 is preseated.
A second order accurate implicit finite difference scheme CNMT2 is proposed in this paper for the unsteady incompressible Navier-Stokes equations. It is proved that the scheme is unconditionally nonlinearly stable on smoothly nonuniform halfstaggered meshes; this stability also holds for this scheme with the pressure correction projection method. However, it is found that the pressure correction projection method may lead to deviation problems in practical simulation of high Re flow;the reason and the cure is given in this paper in terms of differential-algebraic equations.
