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

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

U2 - 10.1137/S0036142994270521

DO - 10.1137/S0036142994270521

M3 - Article

SN - 0036-1429

VL - 34

SP - 1051

EP - 1071

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 3

ER -