The Evolution of Computational Methods in Aerodynamics

Jameson, A.

doi:10.1115/1.3167188

Cited by 75 publications

(48 citation statements)

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“…Frequently this explicit scheme is augmented with implicit residual smoothing 7 to extend stability, allowing the use of larger time steps. This combination proved to be quite effective in solving inviscid flow problems.…”

Section: Introductionmentioning

confidence: 99%

Convergence Acceleration for Multistage Time-Stepping Schemes

Swanson

Turkel

Rossow

et al. 2006

36th AIAA Fluid Dynamics Conference and Exhibit

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The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 could be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. Numerical dissipation operators (based on the Roe scheme, a matrix formulation, and the CUSP scheme) as well as the number of RK stages are considered in evaluating the RK/implicit scheme. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. In two dimensions, turbulent flows over an airfoil at subsonic and transonic conditions are computed. The effects of mesh cell aspect ratio on convergence are investigated for Reynolds numbers between 5.7 × 10 6 and 100.0 × 10 6 . Results are also obtained for a transonic wing flow. For both 2-D and 3-D problems, it is demonstrated that the computational time of a well-tuned standard RK scheme can be reduced at least a factor of four.

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Section: Introductionmentioning

confidence: 99%

Convergence Acceleration for Multistage Time-Stepping Schemes

Swanson

Turkel

Rossow

et al. 2006

36th AIAA Fluid Dynamics Conference and Exhibit

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“…This is the same as (6), except that the artificial viscosity R in w c variables is replaced by R Q based on Q variables. Hence, equations (6) and (12) have different numerical pseudo-steady states but equations (12) and (13) have the same numerical pseudo-steady state.…”

Section: Choice Of Variablesmentioning

confidence: 99%

“…Hence, equations (6) and (12) have different numerical pseudo-steady states but equations (12) and (13) have the same numerical pseudo-steady state.…”

Section: Choice Of Variablesmentioning

confidence: 99%

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Local preconditioners for steady and unsteady flow applications

Turkel

Vatsa

2005

ESAIM: M2AN

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Abstract. Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi and low speed preconditioning. We can express the algorithm in several sets of variables while using only the conservation variables for the flux terms. We compare the effect of these various variable sets on the efficiency and accuracy of the scheme.Mathematics Subject Classification. 65M06, 76M12.

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“…Special cases are the celebrated Rosenbrock methods [14] and the Liniger-Willoughby methods [13]. In this paper, we consider generalized RKN methods obtained by replacing in the RKN method all righthand side evaluations f by Sf (see also [11] where related right-hand side smoothings are discussed). The preconditioning matrix Sis required to be such that Sf converges to f as h tends to 0.…”

Section: A-stable Composite Methods Withmentioning

confidence: 99%

Stability of collocation-based Runge-Kutta-Nyström methods

Houwen¹,

Sommeijer²,

Công

1991

BIT

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Abstract.We analyse the attainable order and the stability of Runge-Kutta-Nystri:im (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order of s-stage methods can be raised to 2s for all s. The attainable stage order is one higher and equals s + 1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order. AMS Subject classification: 65M10, 65M20.l. Introduction.In this paper we shall be concerned with the analysis of implicit Runge-KuttaNystrom (RKN methods) based on collocation for integrating the initial-value problem (IVP) for systems of special second-order, ordinary differential equations (ODEs) of dimension d, i.e. the problem,y: IR--. !Rd, f: IR x !Rd--. !Rd, t 0 :::; t :::; T Our motivation for studying implicit RKN methods is the arrival of parallel computers which enables us to solve the implicit relations occurring in the stage vector equation quite efficiently, so that, what is so far considered as the main disadvantage of fully implicit RKN methods, is reduced a great deal. We consider two types of collocation methods for second-order equations: methods based on direct collocation and on indirect collocation (that is, methods obtained by writing the special second-order equation in first-order form and by applying collocation methods for first-order equations [6]). The theory of indirect collocation methods *) These investigations were supported by the University of Amsterdam who provided the third author with a research grant for spending a total of two years in Amsterdam.Received [1]). The main object of the present paper is to extend the work of Kramarz and to derive order and stability results for direct collocation methods. It will be shown that the attainable step point order is similar to that of indirect collocation methods, but the stage order can be raised to s + 1 leaving all but one collocation parameters free.High stage orders are attractive in the case of stiff problems, provided that the method is A or P-stable. However, it seems that the increased-stage-order methods all have finite stability boundaries. If the stage order is decreased to s, then infinite stability boundaries can be obtained. We found A-stable methods with k = s = 2, k = s = 3 and with k = s -1 = 4 implicit stages.We also investigated two stabilizing techniques for achieving A-stability. The first stabilizing technique is based on the preconditioning of the right-hand side in (1.1), that is, stiff components in the right-hand side are damped. In this way, it is possible to transform conditionally stable RKN methods into unconditionally stable preconditioned RKN methods (P RKN methods) at the cost of a slightly more complicated relation for the stage vector. The second stabilizing technique is based on the combination of different, conditionally stable RKN methods. We will give examples of A-stable, composite methods (CRKN methods) with stag...

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The Evolution of Computational Methods in Aerodynamics

Cited by 75 publications

References 0 publications

Convergence Acceleration for Multistage Time-Stepping Schemes

Convergence Acceleration for Multistage Time-Stepping Schemes

Local preconditioners for steady and unsteady flow applications

Stability of collocation-based Runge-Kutta-Nyström methods

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