A Sequential Regularization Method for Time-Dependent Incompressible Navier--Stokes Equations

The objective of the paper is to present a method, called the sequential regularization method (SRM), for the nonstationary incompressible Navier--Stokes equations from the viewpoint of regularization of differential-algebraic equations (DAEs), and to provide a way to apply a DAE method to partial differential-algebraic equations (PDAEs). The SRM is a functional iterative procedure. It is proved that its convergence rate is O (ᵐ), where m is the number of the SRM iterations and is the regularization parameter. The discretization and implementation issues of the method are considered. In particular, a simple explicit-difference scheme is analyzed and its stability is proved under the usual step-size condition of explicit schemes. It appears that the SRM formulation is new in the Navier--Stokes context. Unlike other regularizations or pseudocompressibility methods in the Navier--Stokes context, the regularization parameter in the SRM need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any. Hence the stability condition is independent of even for explicit time discretization. Numerical experiments are given to verify our theoretical results.