On nonlinear preconditioners in Newton–Krylov methods for unsteady flows

The application of nonlinear schemes like dual time stepping as preconditioners in matrix‐free Newton–Krylov‐solvers is considered and analyzed, with a special emphasis on unsteady viscous flows. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix‐free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes, which is demonstrated through numerical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations

We study different variants of the augmented Lagrangian (AL)‐based block‐triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput. 2006; 28 : 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual method (GMRES) applied to various finite element and Marker‐and‐Cell discretizations of the Oseen problem in two and three space dimensions. Both steady and unsteady problems are considered. Numerical experiments show the effectiveness of the proposed preconditioners for a wide range of problem parameters. Implementation on parallel architectures is also considered. The AL‐based approach is further generalized to deal with linear systems from stabilized finite element discretizations. Copyright © 2010 John Wiley & Sons, Ltd.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Improvements to compressible Euler methods for low‐Mach number flows

In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43 : 357–372] for non‐structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low‐Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order ILU preconditioners for CFD problems

This paper tests a number of incomplete lower–upper (ILU)‐type preconditioners for solving indefinite linear systems, which arise from complex applications such as computational fluid dynamics (CFD). Both point and block preconditioners are considered. The paper focuses on ILU factorization that can be computed with high accuracy by allowing liberal amounts of fill‐in. A number of strategies for enhancing the stability of the factorizations are examined. Copyright © 2000 John Wiley & Sons, Ltd.     

Two preconditioners for saddle point problems in fluid flows

In this paper two preconditioners for the saddle point problem are analysed: one based on the augmented Lagrangian approach and another involving artificial compressibility. Eigenvalue analysis shows that with these preconditioners small condition numbers can be achieved for the preconditioned saddle point matrix. The preconditioners are compared with commonly used preconditioners from literature for the Stokes and Oseen equation and an ocean flow problem. The numerical results confirm the analysis: the preconditioners are a good alternative to existing ones in fluid flow problems. Copyright © 2006 John Wiley & Sons, Ltd.     

An efficient shock-capturing algorithm for compressible flows in a duct of variable cross-section

A numerical scheme is presented for the solution of the Euler equations of compressible flow of a gas in a single spatial co-ordinate. This includes flow in a duct of variable cross-section as well as flow with slab, cylindrical or spherical symmetry and can prove useful when testing codes for the two-dimensional equations governing compressible flow of a gas. The resulting scheme requires an average of the flow variables across the interface between cells and for computational efficiency this average is chosen to be the arithmetic mean, which is in contrast to the usual ‘square root’ averages found in this type of scheme. The scheme is applied with success to five problems with either slab or cylindrical symmetry and a comparison is made in the cylindrical case with results from a two-dimensional problem with no sources.     

An efficient algorithm for compressible flows with real gases

An efficient algorithm is presented for the solution of the Euler equations of gas dynamics with a general convex equation of state. The scheme is based on solving linearized Riemann problems approximately, and in more than one dimension incorporates operator splitting. In particular, only one function evaluation in each computational cell is required by using a local parametrization of the equation of state. The scheme is applied to two standard test problems in gas dynamics for some specimen equations of state.     

Ainv and bilum preconditioning techniques

It was proposed that a robust and efficient parallelizable preconditioner for solving general sparse linear systems of equations, in which the use of sparse approximate inverse (AINV) techniques in a multi-level block ILU (BILUM) preconditioner wereinvestigated. The resulting preconditioner retains robustness of BILUM preconditioner and has two advantages over the standard BILUM preconditioner : the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner.     

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.     

Level Set方法和多介质可压缩流

多介质可压缩流问题计算的关键是如何精确的捕获不同时刻物质界面的位置,从而将多介质问题分解成多个单介质问题去处理.Level Set方法的优点是不用显示的追踪物质界面,而用距离函数就能精确定位界面.同时,用Level Set方法追踪界面运动易于处理界面拓扑结构的变化、易于处理大变形问题.本文成功地将Level Set方法应用在二维多介质可压缩流计算.     

A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.     

A segregated implicit solution algorithm for 2D and 3D laminar incompressible flows

A segregated algorithm for the solution of laminar incompressible, two- and three-dimensional flow problems is presented. This algorithm employs the successive solution of the momentum and continuity equations by means of a decoupled implicit solution method. The inversion of the coefficient matrix which is common for all momentum equations is carried out through an approximate factorization in upper and lower triangular matrices. The divergence-free velocity constraint is satisfied by formulating and solving a pressure correction equation. For the latter a combined application of a preconditioning technique and a Krylov subspace method is employed and proved more effecient than the approximate factorization method. The method exhibits a monotonic convergence, it is not costly in CPU time per iteration and provides accurate solutions which are independent of the underrelaxation parameter used in the momentum equations. Results are presented in two- and three-dimensional flow problems.     

On iterative methods for the incompressible Stokes problem

In this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite‐element method. The proposed solution methods, based on a suitable approximation of the Schur‐complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative methods. Two of these approaches use the pressure mass matrix as preconditioner (or an approximation) to the Schur complement, whereas the third uses an approximation based on the ideas of least‐squares commutators (LSC). We observe that the approximation based on the pressure mass matrix gives h‐independent convergence, for both constant and variable viscosity. Copyright © 2010 John Wiley & Sons, Ltd.     

A segregated method for compressible flow computation Part I: isothermal compressible flows

Traditionally, coupled methods have been employed for the computation of compressible flows, whereas segregated methods have been preferred for the computation of incompressible flows. Compared to coupled methods, segregated solvers present the advantage of reduced computer memory and CPU time requirements, although at the cost of an inferior robustness. Therefore, in a series of papers we present unified computational techniques to compute compressible and incompressible flows with segregated stabilized methods. The proposed algorithms have an increased robustness compared to existing techniques, while possessing additional benefits such as employing standard pressure boundary conditions. In this first part, the thermodynamics of isothermal, thermally perfect compressible flows is set up in the framework of symmetric systems and the corresponding segregated algorithms are introduced. Copyright © 2005 John Wiley & Sons, Ltd.     

Third‐order Cartesian overset mesh adaptation method for solving steady compressible flows

A third‐order mesh generation and adaptation method is presented for solving the steady compressible Euler equations. For interior points, a third‐order scheme is used on Cartesian and curvilinear meshes. Concerning the mesh adaptation, the method of Meakin is also extended to third order. The accuracy of the new overset mesh adaptation method is demonstrated by a grid convergence study for 2‐D inviscid model problems and results are compared with a second‐order method. Finally, the method is applied to the computation of an inviscid 3‐D flow around a hovering blade of the ONERA 7A helicopter rotor exhibiting an improvement in the wake capture. With a 7 million point mesh, the tip vortex can be followed for more than three rotor revolutions with the third‐order method. The CPU time needed for this calculation is only 3% higher than with a conventional second‐order method. Copyright © 2007 John Wiley & Sons, Ltd.     

Development and Study of Newton-Krylov-Schwarz Algorithms

In this paper we develop and study a combination of Newton-Krylov matrix-free methodology with Schwarz-based domain decomposition methods. To illustrate the performance of the resulting algorithms we present numerical applications to the steady solution of compressible Euler flow together with some comparisons with the more standard Schwarz-based methods, namely, Schwarz methods combined with defect-correction procedures.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

Some special purpose preconditioners for conjugate gradient-like methods applied to CFD

Standard preconditioners such as incomplete LU decomposition perform well when used with conjugate gradient-like iterative solvers such as GMRES for the solution of elliptic problems. However, efficient computation of convection-dominated problems requires, in general, the use of preconditioners tuned to the particular class of fluid-flow problems at hand. This paper presents three such preconditioners. The first is applied to the finite element computation of inviscid (Euler equations) transonic and supersonic flows with shocks and uses incomplete LU decomposition applied to a matrix with extra artificial dissipation. The second preconditioner is applied to the finite difference computation of unsteady incompressible viscous flow; it uses incomplete LU decomposition applied to a matrix to which a pseudo-compressible term has been added. The third method and application are similar to the second, only the LU decomposition is replaced by Beam-warming approximate factorization. In all cases, the results are in very good agreement with other published results and the new algorithms are found to be competitive with others; it is anticipated that the efficiency and robustness of conjugate-gradient-like methods will render them the method of choice as the difficulty of the problems that they are applied to is increased.