Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.     

Splitting algorithms as applied to the Navier-Stokes equations

Implicit finite-difference schemes of approximate factorization and predictor-corrector schemes based on a special splitting of operators are proposed for the numerical solution of the Navier-Stokes equations governing a viscous compressible heat-conducting gas. The schemes are based on scalar tridiagonal Gaussian elimination and are unconditionally stable. The accuracy and efficiency of the algorithms are confirmed by computing two-dimensional flows of complex geometry.     

Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in the three space dimensions with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.     

Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations

The paper is devoted to studying the distribution of stationary solutions for 3D Navier-Stokes equations perturbed by a random force. Under a non-degeneracy assumption, we show that the support of such a distribution coincides with the entire phase space, and its finite-dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the Lebesgue measure. Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function. The results of this paper were announced in the short note [A. Shirikyan, Controllability of three-dimensional Navier-Stokes equations and applications, in: Sémin. Équ. Dériv. Partielles, 2005-2006, École Polytech., Palaiseau, 2006].     

Vorticity-velocity formulation for the Navier-Stokes equations: The three-dimensional case

In this article, we propose a mixed method for the vorticity-velocity formulation of the stationary Stokes and Navier-Stokes equations in space dimension three, the unknowns being the vorticity and the velocity of the fluid. We give a similar variational formulation for the nonstationary Stokes equations in the vorticity-velocity variables. © 1997 John Wiley & Sons, Inc.     

Spectral-finite element method for solving three-dimensional unsteady Navier-Stokes equations

This paper is devoted to the combined Fourier spectral and finite element approximations of three-dimensional, semi-periodic, unsteady Navier-Stokes equations. Fourier spectral method and finite element method are employed in the periodic and non-periodic directions respectively. A class of fully discrete schemes are constructed with artificial compression. Strict error estimations are proved. The analysis shows also that the classical two-dimensional velocity-pressure elements can be readily extended to solving such three-dimensional semi-periodic problems, provided they satisfy the two-dimensional “inf-suf” condition.     

Implicit algorithms and their linearization for the transient incompressible Navier-Stokes equations

Email: na.asmith2{at}na-net.ornl.gov Email: na.silvester{at}na-net.ornl.gov The issue of appropriate time discretization methods for theunsteady incompressible Navier-Stokes equations is consideredfrom a practical perspective. Conventional implicit time-steppingalgorithms are not feasible for long-time simulations sincethey inherit the quadratic nonlinearity of the steady-stateequations. As a result, two new linearized versions of the ‘pure’algorithms are analyzed herein. These have similar stabilityproperties and comparable accuracy to the underlying nonlinearmethods.     

State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations

The mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained.     

Projection and projection-difference methods for the solution of the Navier-Stokes equations

The projection and projection-difference methods for the approximate solution of the nonlinear unsteady Navier-Stokes equations in a bounded two-dimensional region are studied. Asymptotic estimates for the convergence rate of the approximate solutions and the time and space derivatives in the uniform topology are obtained.     

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study. In the time-independent context, the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the Galerkin framework. This upwind approach has significant advantage of seeking solutions free from cross-stream diffusion error. Finite element solutions have been found by mixed formulation, implemented in quadratic cubic elements which are characterized as possessing the so-called LBB (Ladyzhenskaya-Babuška-Brezzi) condition. An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows. The developed three-dimensional finite element code is first rectified by solving a problem amenable to analytic solution. A well-known lid-driven cavity flow problem in a cubical cavity is also studied.     

Explicit and implicit projection-difference schemes for the solution of the Navier-Stokes equations

Journal of Mathematical Sciences -     

Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier-Stokes equations in R³ for any specific heat ratio γ>1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.