A sequential regularization method for time-dependent incompressible Navier--Stokes equations

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    Abstract

    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 $O(\epsilon^m)$, where $m$ is the number of the SRM iterations and $\epsilon$ 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 $\epsilon$ in the SRM need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any $\epsilon$. Hence the stability condition is independent of $\epsilon$ even for explicit time discretization. Numerical experiments are given to verify our theoretical results.
    Original languageEnglish
    Pages (from-to)1051-1071
    Number of pages21
    JournalSIAM Journal on Numerical Analysis
    Volume34
    Issue number3
    DOIs
    Publication statusPublished - 1997

    Keywords

    • Sequential regularization method
    • Navier-Stokes equations

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