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

Ping Lin

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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

    Fingerprint

    Sequential Methods
    Incompressible Navier-Stokes Equations
    Regularization Method
    Navier Stokes equations
    Differential equations
    Explicit Scheme
    Partial differential equations
    Algebraic Differential Equations
    Regularization Parameter
    Navier-Stokes
    Regularization
    Partial Differential Algebraic Equations
    Time Discretization
    Iterative Procedure
    Difference Scheme
    Stability Condition
    Experiments
    Convergence Rate
    Discretization
    Numerical Experiment

    Keywords

    • Sequential regularization method
    • Navier-Stokes equations

    Cite this

    Lin, Ping. / A sequential regularization method for time-dependent incompressible Navier--Stokes equations. In: SIAM Journal on Numerical Analysis. 1997 ; Vol. 34, No. 3. pp. 1051-1071.
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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.",
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    Lin, P 1997, 'A sequential regularization method for time-dependent incompressible Navier--Stokes equations', SIAM Journal on Numerical Analysis, vol. 34, no. 3, pp. 1051-1071. https://doi.org/10.1137/S0036142994270521

    A sequential regularization method for time-dependent incompressible Navier--Stokes equations. / Lin, Ping.

    In: SIAM Journal on Numerical Analysis, Vol. 34, No. 3, 1997, p. 1051-1071.

    Research output: Contribution to journal › Article

    TY - JOUR

    T1 - A sequential regularization method for time-dependent incompressible Navier--Stokes equations

    AU - Lin, Ping

    N1 - © 1997 Society for Industrial and Applied Mathematics

    PY - 1997

    Y1 - 1997

    N2 - 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.

    AB - 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.

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    Lin P. A sequential regularization method for time-dependent incompressible Navier--Stokes equations. SIAM Journal on Numerical Analysis. 1997;34(3):1051-1071. https://doi.org/10.1137/S0036142994270521

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