Fully iterative ILU preconditioning of the unsteady Navier–Stokes equations for GPGPU

In this work we investigate the numerical difficulties that arise in optimizing the efficiency of Newtonian fluids simulations on a massively parallel computing hardware like a GPU. In particular, we will concentrate on the resulting algebraic problem. We will present an approximate, fully-iterative, ILU preconditioner and we will show that this solution is very efficient on a GPU if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner, which indeed goes faster than more robust techniques, like ILU, despite their strong decrease in the number of iterations. We refer to GMRES as the iterative scheme used to solve the linear system. In particular, we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between Reynolds and Atwood numbers. We will show that the numerical simulation at medium–high Reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner, while at low Reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from GMRES and, decreasing the Reynolds number, unable to let GMRES converge at all. In this situation, we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ILU preconditioning, turns out to be robust and efficient. We will show numerical results for variable-density fluids, discretized with the scheme described in Calgaro et al. (2008), in classical benchmarks and, in particular, in the well-known Rayleigh–Taylor instability.     

Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

A characteristic stabilized finite element method for the non-stationary Navier–Stokes equations

This paper is concerned with the analysis of a new stabilized method based on the local pressure projection. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures. Error estimates of the velocity and the pressure are obtained for both the continuous and the fully discrete versions. Finally, some numerical experiments show that this method is highly efficient for the non-stationary Navier–Stokes equations.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations

Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L ² can be established, as well as one in a broken H ¹-norm for the velocity. Received: January 1999 / Accepted: April 1999     

A comparison of parallel solvers for the incompressible Navier–Stokes equations

The paper compares coupled multigrid methods and pressure Schur complement schemes (operator splitting schemes) for the solution of the steady state and time dependent incompressible Navier–Stokes equations. We consider pressure Schur complement schemes with multigrid as well as single grid methods for the solution of the Schur complement problem for the pressure. The numerical tests have been carried out on benchmark problems using a MIMD parallel computer. They show the superiority of the coupled multigrid methods for the considered class of problems. Received: 14 October 1997 / Accepted: 11 February 1998     

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

Characteristic stabilized finite element method for the transient Navier–Stokes equations

Based on the lowest equal-order conforming finite element subspace (X_h, M_h) (i.e. P₁–P₁ or Q₁–Q₁ elements), a characteristic stabilized finite element method for transient Navier–Stokes problem is proposed. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, and averting the difficulties caused by trilinear terms. Existence,uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method. Finally, a serious of numerical experiments are given to show that this method is highly efficient for transient Navier–Stokes problem.     

Numerical simulations of solutions of a two-level averaged Navier–Stokes system

An averaging procedure for the Navier–Stokes equations has been proposed in an earlier article [I. Moise, R.M. Temam, Renormalization group method. Application to Navier–Stokes Equation, Discrete Contin. Dyn. Syst. 6 (1) (2000) 191–210]. This averaging procedure is based on a two-level decomposition of the solution into low and high frequencies. The aim of the present article is to investigate, with the help of numerical simulations, the behavior of the small scales of the corresponding system. Space-periodic solutions with a non-resonant period are considered. The time evolution of the averaged and standard (non-averaged) small scales are compared at different Reynolds numbers and for different values of the cut-off level used to separate large and small scales of the flow variables. The numerical results illustrate the efficiency of the proposed averaging procedure for the Navier–Stokes equations. The averaged small scales provide an accurate prediction of the time-averaged small scales of the Navier–Stokes solutions. As the computational cost is reduced for the averaged equations, long time integrations on more than 50 eddy-turnover times have been performed for cut-off levels ensuring a proper resolution of the large scales. In these cases, development of instabilities in the averaged small scale equation is observed.     

An accelerated algorithm for Navier–Stokes equations

In this paper, the numerical solution of the Navier–Stokes equations by the Characteristic-Based-Split (CBS) scheme is accelerated with the Minimum Polynomial Extrapolation (MPE) method to obtain the steady state solution for evolution incompressible and compressible problems.The CBS is essentially a fractional time-stepping algorithm based on an original finite difference velocity-projection scheme where the convective terms are treated using the idea of the Characteristic-Galerkin method. In this work, the semi-implicit version of the CBS with global time-stepping is used for incompressible problems whereas the fully-explicit version is used for compressible flows.At the other end, the MPE is a vector extrapolation method that transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator.The developed algorithm, tested on two-dimensional benchmark problems, demonstrates the new computational features arising from the introduction of the extrapolation procedure to the CBS scheme. In particular, the results show a remarkable reduction of the computational cost of the simulation.     

A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.     

Radial basis function meshless method for the steady incompressible Navier–Stokes equations

A meshless collocation method based on radial basis functions is proposed for solving the steady incompressible Navier–Stokes equations. This method has the capability of solving the governing equations using scattered nodes in the domain. We use the streamfunction formulation, and a trust-region method for solving the nonlinear problem. The no-slip boundary conditions are satisfied using a ghost node strategy. The efficiency of this method is demonstrated by solving three model problems: the driven cavity flows in square and rectangular domains and flow over a backward-facing step. The results obtained are in good agreement with benchmark solutions.     

Note on helicity balance of the Galerkin method for the 3D Navier–Stokes equations

We study the helicity balance of the Galerkin method for the 3D Navier–Stokes equations, and show that although it does not appear to correctly balance helicity in the usual sense, it instead admits a slightly altered helicity balance that matches that of the underlying physics, up to boundary conditions.     

A new class of massively parallel direction splitting for the incompressible Navier–Stokes equations

We introduce in this paper a new direction splitting algorithm for solving the incompressible Navier–Stokes equations. The main originality of the method consists of using the operator (I ? ?_xx)(I ? ?_yy)(I ? ?_zz) for approximating the pressure correction instead of the Poisson operator as done in all the contemporary projection methods. The complexity of the proposed algorithm is significantly lower than that of projection methods, and it is shown the have the same stability properties as the Poisson-based pressure-correction techniques, either in standard or rotational form. The first-order (in time) version of the method is proved to have the same convergence properties as the classical first-order projection techniques. Numerical tests reveal that the second-order version of the method has the same convergence rate as its second-order projection counterpart as well. The method is suitable for parallel implementation and preliminary tests show excellent parallel performance on a distributed memory cluster of up to 1024 processors. The method has been validated on the three-dimensional lid-driven cavity flow using grids composed of up to 2 × 10⁹ points.