Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods

We consider numerical solution methods for the incompressible Navier-Stokes equations discretized by a finite volume method on staggered grids in general coordinates. We use Krylov subspace and multigrid methods as well as their combinations. Numerical experiments are carried out on a scalar and a vector computer. Robustness and efficiency of these methods are studied. It appears that good methods result from suitable combinations of GCR and multigrid methods.     

Numerical solution of the incompressible Navier-Stokes equations with explicit integration of the pressure term

We propose a formulation of the incompressible Navier-Stokes equations considering a Poisson equation with Neumann boundary conditions for the pressure, and innovative boundary conditions for the velocity. Numerical tests show that the proposed formulation ensures solenoidality of the velocity field. If the initial condition is not divergence-free, exponential decay is observed in time for the error in the fulfillment of the continuity equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of the time-dependent incompressible Navier-Stokes equations in the stream function and vorticity formulation

In this work we study the time-dependent incompressible Navier-Stokes problem. We introduce a suitable technique based on the splitting of the vorticity into two components. Then we discretize in space the resulting uncoupled system by means of continuous Lagrange finite elements. This is achieved by first performing the semi-discretization in time of these equations by a classical characteristics method for the advective term. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.     

Stochastic nonhomogeneous incompressible Navier-Stokes equations

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.     

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.     

A locally conservative LDG method for the incompressible Navier-Stokes equations

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

     

TWO-LEVEL METHOD FOR UNSTEADY NAVIER-STOKES EQUATIONS IN STREAM FUNCTION FORM

Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied. This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid. Moreover,the scaling between these two grid sizes is super-linear. Approximation,stability and convergence aspects of a fully discrete scheme are analyzed. At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.     

Analysis of a sequential regularization method for the unsteady Navier-Stokes equations

The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

     

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.     

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.     

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

The velocity-vorticity-pressure formulation of the steady-state incompressible Navier-Stokes equations in two dimensions is cast as a nonlinear least squares problem in which the functional is a weighted sum of squared residuals. A finite element discretization of the functional is minimized by a trust-region method in which the trustregion radius is defined by a Sobolev norm and the trust-region subproblems are solved by a dogleg method. Numerical test results show the method to be effective.     

On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the incompressible Navier-Stokes equations with damping α|u|^β−1u(α>0) has global strong solution for any β>3 and the strong solution is unique when 3<β?5. This improves earlier results.     

Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations

In this paper the extension of the Legendre least-squares spectral element formulation to Chebyshev polynomials will be explained. The new method will be applied to the incompressible Navier-Stokes equations and numerical results, obtained for the lid-driven cavity flow at Reynolds numbers varying between 1000 and 7500, will be compared with the commonly used benchmark results. The new results reveal that the least-squares spectral element formulations based on the Legendre and Chebyshev Gauss-Lobatto Lagrange interpolating polynomials are equally accurate.     

On a potential-velocity formulation of incompressible Navier-Stokes equations

Computational fluid dynamics has emerged as an essential investigative tool in nearly every field of technology. Despite a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging, especially due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two-dimensional incompressible Navier-Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original equations [1]. A useful application to free surface simulation was found in [2]. Moreover the new formulation is naturally extendible to three dimensions via tensor calculus, involving a non-unique symmetric tensor potential. The corresponding degrees of freedom can be used in order to achieve a numerically convenient representation. Finally an efficient staggered-grid finite difference scheme is applied to a Stokes flow problem in a 3D lid-driven cavity to demonstrate the capabilities of the new approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

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Résumé Dans ce travail nous étudions l'écoulement plan d'un fluide visqueux incompressible dans lequel la rotation locale est proportioneile à la fonction de courant perturbée par un courant uniforme. Conformément aux travaux de Taylor et Kovasznay les équations de Navier-Stokes pour cet écoulement deviennent linéaires. Par conséquent nous utilisons la solution générale pour démontrer que seulement deux catégories d'écoulement stationnaire peuvent exister: l'écoulement de Kovasznay en aval d'une grille plane, et l'écoulement inversé de Lin et Tobak pour une plaque plane avec aspiration. Nous étudions aussi l'écoulement non stationnaire et nous découvrons des classes nouvelles de solutions exactes qui contiennent, en particulier, le réseau de tourbillons de Taylor. Enfin nous démontrons que ces écoulements sont pseudo-stationnaires dans un système de coordonnées en mouvement avec le courant uniforme non perturbé; ce qui signifie que l'amplitude de l'écoulement stationnaire croit ou décroit exponentiellment dans le temps. Toutes ces solutions sont valides pour tous les nombres de Reynolds.

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On leave from University of Waterloo, Ontario, Canada.     

A stochastic representation for backward incompressible Navier-Stokes equations

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.     

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.     

A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations

Summary. We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious domain and the boundary mesh are independent, up to a mesh-length ratio. Received February 24, 1999 / Revised version received January 30, 2000 / Published online October 16, 2000     

On a relaxation approximation of the incompressible Navier-Stokes equations

We consider a hyperbolic singular perturbation of the incompressible Navier Stokes equations in two space dimensions. The approximating system under consideration arises as a diffusive rescaled version of a standard relaxation approximation for the incompressible Euler equations. The aim of this work is to give a rigorous justification of its asymptotic limit toward the Navier Stokes equations using the modulated energy method.