# 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 language English 1051-1071 21 SIAM Journal on Numerical Analysis 34 3 https://doi.org/10.1137/S0036142994270521 Published - 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

@article{0c520e7208ed4c80abde0ffce8111774,
title = "A sequential regularization method for time-dependent incompressible Navier--Stokes equations",
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.",
keywords = "Sequential regularization method, Navier-Stokes equations",
author = "Ping Lin",
note = "{\circledC} 1997 Society for Industrial and Applied Mathematics",
year = "1997",
doi = "10.1137/S0036142994270521",
language = "English",
volume = "34",
pages = "1051--1071",
journal = "SIAM Journal on Numerical Analysis",
issn = "0036-1429",
publisher = "Society for Industrial and Applied Mathematics",
number = "3",

}

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

Research output: Contribution to journalArticle

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.

KW - Sequential regularization method

KW - Navier-Stokes equations

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VL - 34

SP - 1051

EP - 1071

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

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ER -