Computers & Fluids 41 (2011) 15–19

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Computers & Fluids

journal homepage: www.elsevier .com/ locate /compfluid

Reflections on the evolution of implicit Navier–Stokes algorithms

W. Roger Briley ⇑, Henry McDonaldSimCenter: National Center for Computational Engineering, University of Tennessee at Chattanooga, 701 East M.L. King Blvd., Chattanooga, TN 37403, United States

a r t i c l e i n f o

Article history:Available online 19 September 2010

Keywords:Implicit algorithmNavier–StokesLinearizationApproximate factorizationADIUpwind

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.09.014

⇑ Corresponding author.E-mail address: roger-briley@utc.edu (W.R. Briley)

a b s t r a c t

This article gives a historical perspective on the evolution of implicit algorithms for the Navier–Stokesequations that utilize time or Newton linearizations and various forms of approximate factorizationincluding alternating-direction, lower/upper, and symmetric relaxation schemes. A theme of the paperis how progress in implicit Navier–Stokes algorithms has been influenced and enabled by the introduc-tion of characteristic-based upwind approximations, unstructured-grid discretizations, parallelization,and by advances in computer performance and architecture. Historical examples of runtime, problemsize, and estimated cost are given for actual and hypothetical Navier–Stokes flow cases from the past40 years.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

We are very grateful for the Editor’s invitation to reflect on thehistory and progress of our joint work on implicit solution of theNavier–Stokes equations. It has given us a better appreciation ofhow far our field has come since the early days of limited computerresources and incomplete understanding of the important and nec-essary advances that would emerge to solve complicated practicalproblems. We will briefly comment here on the development ofour own noniterative time-linearized, block-tridiagonal alternat-ing-direction implicit (ADI) scheme reported during 1973–1977,and then give some personal reflections on the subsequent evolu-tion of these ideas and some of the progress achieved by othersthrough related ideas and innovations. The literature on even thissubset of CFD is very large, with many more contributions than wecould possibly mention here, and so we ask forbearance for theunavoidable omission of important work by our colleagues andcoworkers.

The impetus for implicit viscous flow solvers has been the needto solve complex flows requiring highly nonuniform grids and withmultiple time and length scales. Such problems can present severealgorithmic challenges when resolving disparate local length scalesintroduced by geometry, very thin shear layers, and other localizedflow structures. In addition, differing time scales such as convec-tion, diffusion, sound propagation, and chemical reaction can resultin equation stiffness, a term given to systems of differential equa-tions whose system matrices have a wide range of eigenvalues. Inboth instances, the enhanced stability properties of implicit

ll rights reserved.

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schemes can help by allowing larger time steps, within an objec-tive of either time accuracy or convergence to a steady solution.

The challenge facing our community in the early 1970s is exem-plified by considering that a typical Navier–Stokes solution wastwo-dimensional, steady, often laminar, had a simple geometry,and was limited to perhaps 1000–3000 grid points and 100–1000time steps. However, there were aspirations to solve more realisticproblems, perhaps a three-dimensional flow at Reynolds number108 past a moderately complex geometry. Making rough allow-ances for possible length-scale variations of 10�3 for small geomet-ric features, 10�3 for shear-layer thickness and another 10�3 forviscous sublayer resolution, it could be foreseen that resolutionof geometry and thin shear layers could easily require grids with10 million points and with minimum to maximum cell-volume ra-tios of 10�12 and perhaps much worse. Even assuming that suitablegrids could be generated and more powerful computers were avail-able, such variations in mesh length scales would greatly affect thestability and convergence behavior of then-existing Navier–Stokesalgorithms. These algorithm, grid, and computer challenges ap-peared very formidable or even intractable in the early days ofCFD, and it has taken many years to do this kind of problemroutinely.

Our work in this area began in 1968 with McDonald’s work onimplicit compressible boundary-layer methods [1] and Briley’s dis-sertation [2] on a two-dimensional Peacemen–Rachford [3] ADIscheme for the incompressible vorticity stream-function equa-tions. This ADI scheme for vorticity and stream function was suc-cessful in producing validated, time-accurate incompressiblesolutions for axisymmetric flow in a cylindrical vessel for Re < O(103), using time steps with Courant number over 20 [4]. However,4–5 iterations were needed at each time step to couple the equa-tions and vorticity boundary conditions. We improved this method

16 W.R. Briley, H. McDonald / Computers & Fluids 41 (2011) 15–19

and used it for prediction of laminar and transitional separationbubbles on airfoils using both Navier–Stokes and boundary-layerequations, with a local viscous-inviscid interaction model for free-stream boundary conditions [1,5,6]. However, vorticity-basedmethods were not workable for our primary application objectiveof three-dimensional viscous subsonic internal flows in turboma-chinery, and so we began to explore implicit Navier–Stokes algo-rithms based on primitive variables.

2. The noniterative time-linearized block-tridiagonal ADIscheme

With much appreciated support from Dr. Morton Cooper at theOffice of Naval Research, we began work on our implicit algorithmfor the three-dimensional compressible Navier–Stokes equations[7–11]. This algorithm combined (a) a noniterative implicit timelinearization in primitive variables, (b) a three-dimensionalblock-tridiagonal ADI scheme with consistent intermediate steps,and (c) the use of implicitly coupled, characteristic-compatibleboundary conditions.

Our work with vorticity and stream function had convinced usthat implicit coupling of equations and boundary conditions wasimportant, and we had used block-tridiagonal elimination to solvelinear coupled systems in one-dimension. This suggested thatdeveloping a block-tridiagonal ADI scheme was a promising ap-proach, but two issues had to be addressed. The first was how todevelop a coupled ADI scheme for three dimensions. This was re-solved by recognizing that the 1964 Douglas–Gunn [12] procedurefor generating ADI schemes from linear, scalar, implicit time-marching schemes could be applied formally to coupled systemsof linear parabolic/hyperbolic equations. This gave us a procedureto derive a three-dimensional ADI scheme in which each interme-diate step was constructed to be consistent with the underlyingimplicit equations, which allows straightforward use of physicalboundary conditions for the intermediate steps. The remainingtask was to develop a suitable linearization technique for the non-linear equations and boundary conditions. After considering alter-native approaches, we linearized the implicit equations andboundary conditions by Taylor expansion with respect to the prim-itive variables Q = (q,u,v,w,T)T about the known solution Qn fromthe previous time step. This choice of variables was convenient be-cause the viscous and heat-conduction terms were already linearin velocity and temperature. The time linearization coupled theequations naturally, thus obviating ad hoc decoupling techniquesthat we and others were using at the time, and it was also unam-biguous in guiding the implicit coupling of nonlinear boundaryconditions.

This scheme was first tested for the trivial one-dimensionalproblem of inviscid uniform flow at constant Mach number. Webecame convinced that this noniterative time linearization was akey ingredient when it provided stability for large Courant num-bers over a full range of Mach numbers, using central differencingwith no added dissipation, and if implicit characteristic-compatibleinflow/outflow boundary conditions were used. For other prob-lems, we added a small amount of second-order dissipation. The fi-nal test problem was a three-dimensional viscous subsonicinternal flow in a straight duct, which was stable and rapidly con-vergent for CFL in the range 20–80. This confirmed that the nonit-erative three-dimensional implicit method was both effective andefficient, at least for this problem. A somewhat related two-dimen-sional scheme for magnetohydrodynamics was developed inde-pendently in 1973 by Lindemuth and Killeen [13]. They appliedthe Peaceman–Rachford ADI scheme without linearization, andthen linearized progressively about the two successive solutionsobtained for each of the half-steps.

Over the next few years, our group applied the time-linearizedADI scheme to numerous three-dimensional steady subsonic prob-lems. The scheme was used in 1975 as a spatial marching algo-rithm for a steady three-dimensional viscous supersonic jet flow[9], with the equations in conservation form and using an isoener-getic flow assumption that eliminates the need to solve an energyequation. In 1975, it was extended and successfully demonstratedfor a steady three-dimensional combustor flow with sub-modelsfor turbulence, finite-rate hydrocarbon chemistry, droplet vapori-zation and burning, and radiation transport [10].

It was significant that the time linearization error and the errorintroduced by the ADI scheme are both O(Dt2). Thus, halving thetime step significantly reduced the errors due to linearization,ADI, and time differencing, at the same cost as a single Newtoniteration [9]. This Douglas–Gunn ADI scheme also gives identicalresults (except for round-off error) when solved for Qn+1 as in [7–9] or for the time increment DQ = (Qn+1 � Qn) as in [11]. A singlenoniterative time step cost only about twice that of explicit meth-ods, and convergence to steady solutions was much faster. Thisefficiency was needed at a time when computers were inadequatefor many practical CFD problems. However, many subsequentimprovements in implicit algorithms and solution methodologywould be needed to achieve modern high-fidelity simulations, asincreasingly powerful computers enabled them.

In our view, the primary long-term contribution of our Navier–Stokes algorithm was to introduce the noniterative time lineariza-tion for the coupled Navier–Stokes equations and boundary condi-tions, and to recognize the value of the scalar Douglas–Gunnformalism to generate a three-dimensional block-tridiagonal ADIscheme with consistent intermediate steps. This stimulated con-siderable interest and further research in implicit algorithms forthe Navier–Stokes equations.

3. Early work at NASA Ames Research Center

Our 1974 seminar at NASA Ames helped to generate some ofthis interest, and many important contributions were made there.In 1976, Beam and Warming developed a two-dimensional implicitalgorithm for hyperbolic systems in conservation-law form [14].They gave an elegant derivation by expressing the time lineariza-tion in terms of flux Jacobian matrices for the conserved variablesQ = (q,qu,q v,e)T and then using approximate factorization (AF).They then simplified this factorization for homogeneous flux func-tions and expressed their algorithm as a splitting in terms of Qn+1

due to D’Yakonov. They applied this scheme to the Euler equationswith spatially switched central/upwind differencing. In 1977,Warming and Beam used approximate factorization of a delta formof the equations written for DQ = (Qn+1 � Qn), with a three-level,single-step temporal scheme [15]. Steger and Kutler applied thescheme to the three-dimensional incompressible Navier–Stokesequations [16] using Chorin’s [17] artificial compressibility formu-lation, and they also used it for the two-dimensional compressibleEuler equations. In 1978, Beam and Warming developed their DQfactorization for the two-dimensional compressible Navier–Stokesequations [18] and applied it to a shock boundary-layer interactionproblem. Steger and Pulliam [19,20] implemented this time-linear-ized AF scheme for the thin-layer Navier–Stokes equations in twoand three-dimensional curvilinear coordinates.

For a given set of linearized equations, the approximate factor-ization in terms of DQ and the Douglas–Gunn procedure for gener-ating ADI schemes are equivalent [21]; however, approximatefactorization is more direct and soon became the standard deriva-tion. The Beam and Warming scheme differed from ours primarilyin their use of conserved variables, fourth-order dissipation,

W.R. Briley, H. McDonald / Computers & Fluids 41 (2011) 15–19 17

different treatment of boundary conditions, and provision forthree-level time differencing.

Warming and Beam did extensive work on stability of implicitschemes, and one important development was their 1979 stabilityanalysis indicating that the three-dimensional Douglas–Gunn ADIor delta-form AF scheme is unconditionally unstable for the first-order wave equation [22]. In the context of implicit algorithms,the von Neumann stability analysis assumes linear equations withconstant coefficients, a uniform grid, and periodic boundary condi-tions. In 1985, we performed a matrix stability analysis [23] forthis inviscid equation showing that when the periodic conditionsare replaced by characteristic-based inflow/outflow conditions,the algorithm is conditionally stable for inviscid Courant numberless than about 1.2. This may explain why many researchersincluding ourselves were able to obtain numerous steady solutionsin three dimensions. Although it is unlikely that any algorithmwould be unconditionally stable for complex nonlinear systems,the lack of unconditional stability for this model problem properlymotivated the subsequent development of much improved two-factor AF schemes for three dimensions, most notably the lower–upper (LU) factorizations.

4. Some post-1980 developments

4.1. Characteristic-based upwind schemes

A major algorithmic advance came in the early 1980s when vanLeer and others developed high-resolution characteristic-basedupwind numerical fluxes [24–27]. These inviscid flux formulasare implemented as local one-dimensional flux projections andare capable of preserving discontinuities, with consequent higherresolution of flows with discontinuities and/or thin viscous layers.Upwind flux formulations are also widely used for artificial com-pressibility formulations. While upwind schemes were perhapsoriginally motivated by their discontinuity-preserving properties,their local upwind properties later became a key attribute in devel-oping more effective two-factor implicit methods for three-dimen-sional structured and unstructured grids.

4.2. Reducing factorization error in time-linearized schemes

The low computational cost of the noniterative implicit algo-rithms was attained at the expense of introducing time-lineariza-tion and factorization errors. Both of these errors vanish inconverged steady solutions, which was their primary use at thetime, especially in three-dimensions. However, as more complexproblems were addressed, it became evident that in some casesfactorization errors were degrading stability and convergence. Ofparticular concern were the three-factor stability limitation andthe influence of increased stiffness at very low Mach numbers.

4.2.1. Lower–upper approximate factorization schemesThe 1981 introduction of implicit lower–upper approximate

factorization (LU/AF) schemes for hyperbolic systems was a veryimportant development [28,25]. Jameson and Turkel [28] derivedimplicit LU/AF schemes for centered spatial differences, with twostable diagonally dominant factors for any number of spatialdimensions. A finite-volume adaptation of this scheme for the Eu-ler equations was later applied to two-dimensional cascades byBuratynski and Caughey [29]. Steger and Warming [25] developedtheir implicit two-factor scheme for the Euler equations using flux-vector splitting, time-linearized flux Jacobian matrices, and up-wind differencing. Requiring only two factors in three-dimensions,the LU approach to factorization improves stability, reduces factor-ization error, and does not require block tridiagonal solutions. This

approach eventually became a key ingredient in developing effec-tive implicit schemes for unstructured grids. Riemann-basednumerical fluxes such as Roe’s 1981 flux-difference scheme arenow widely used in conjunction with LU/AF schemes. It is of someinterest that LU/AF schemes fall within the Douglas–Gunn formal-ism for generating ADI schemes, which was based on the conceptthat each ADI step be easily solved but not neccessarily associatedwith a coordinate direction.

4.2.2. Preconditioning for low Mach number flowsAnother means for reducing factorization error is the use of pre-

conditioning techniques. In 1983, we developed a formulation forlow Mach number simulations using a constant diagonal precondi-tioning matrix that was constructed to eliminate ill-conditioning(stiffness) in the factorization error when computing steady isoen-ergetic flows [30]. This greatly improved the convergence rate of anADI scheme applied to the two-dimensional isoenergetic Navier–Stokes equations at low Mach number. A similar constant precon-ditioner for the Euler equations was developed in 1985 by Choi andMerkle [31]. Preconditioning techniques designed to alter the sys-tem eigenvalues locally to remove large disparities in wave speedswere introduced in 1984 by Turkel [32]. These basic ideas haveevolved into a number of other well known preconditioners suchas those described in [33,34].

4.2.3. Iterative time-linearized schemes for structured andunstructured grids

More progress came with the use of iteration at each time step toeliminate the factorization error altogether. Chakravarthy and Mac-Cormack began using iterative symmetric line relaxation methods[35,36] that eliminated factorization errors in the time-linearizedscheme. Later, iterative schemes also became the basis for success-ful implicit algorithms that are applicable to unstructured grids.Useful unstructured spatial discretizations incorporating upwindschemes and an efficient edge-based data structure were given,for example, by Barth and Jespersen [37,38]. Although line-orientedfactorizations are not applicable to unstructured grids, the time-lin-earized implicit equations for unstructured grids can be solved bypoint iterative methods and by LU relaxation or factorization. Itera-tive time-linearized implicit schemes for unstructured meshes wereintroduced by Batina using point-implicit, two-sweep Gauss-Seidel,and two-sweep point-Jacobi relaxation [39], by Venkatakrishnanand Mavriplis using preconditioned generalized minimal residual(GMRES) iteration [40], and by Anderson using multicolor vectoriz-able Gauss-Seidel relaxation [41].

4.3. Newton-linearized iterative unsteady schemes

Another significant advance came in 1986 when Rai and Cha-kravarthy [42,43] introduced an (approximate) Newton lineariza-tion with iterative ADI or line relaxation that converges to asolution of the nonlinear unsteady equations. These Newton-line-arized unsteady methods are distinct from time-linearized meth-ods, which become a Newton linearization of the steadyequations as the time step becomes infinite. A converged Newtoniteration eliminates both time-linearization and factorization er-rors at each time step. This introduction of Newton-iterativeschemes became feasible with increases in computer processingspeed and available memory, which allowed both larger problemsizes and algorithm improvements. Newton-linearized iterativeschemes were also developed for the incompressible artificial com-pressibility equations by Pan and Chakravarthy, and Rogers andKwak [44,45].

In 1991, we began our current long-term collaboration withDave Whitfield and his group, then at the National Science Founda-tion Engineering Research Center at Mississippi State University.

18 W.R. Briley, H. McDonald / Computers & Fluids 41 (2011) 15–19

Whitfield and Taylor had developed an iterative Newton-linearizedunsteady upwind scheme for incompressible flow that incorpo-rated symmetric Gauss-Seidel (LU/SGS) relaxation to solve theNewton linearization [46]. This discretized Newton relaxationmethod used numerical differentiation to compute accurate Roe-flux Jacobian matrices instead of using approximate Jacobians.Their implementation exploited CPU/memory trade-offs enabledby much larger memory resources that had become available, inparticular Cray SSD. By organizing the computation into sequentialstructured-grid blocks and storing the residuals and large numeri-cal Jacobian matrices, the linear LU/SGS iteration sweeps becameextremely inexpensive, and the factorization error could be elimi-nated at very little cost. For unsteady flows, this algorithm wasused with multiple Newton iteration cycles, using LU/SGS sub-iter-ation to solve each linear Newton iterate (e.g., [47]).

Another important advance came in 1998, when Newman III,Anderson and Whitfield recognized the utility of using complex-variable numerical differentiation, a 1967 technique of Lynessand Molar for differentiating real functions [48], to compute partialderivatives arising in computational fluid dynamics. They devel-oped this technique to compute highly accurate flux Jacobiansand sensitivity derivatives for computational design optimization[49–51].

There is some algorithmic unity in the fact that symmetricGauss-Seidel relaxation and two-factor LU versions of DQ/AF orDouglas–Gunn ADI are equivalent linear iteration schemes,although they are generally written in different forms. The New-ton-linearized schemes work well for steady solutions and can beused without the added cost of Newton iterations, but they areespecially beneficial for time-accurate simulations, since they per-mit fully nonlinear implicit time-integration schemes. Accordingly,the Newton linearization of the unsteady equations is now com-monly used. A unified treatment of iterative Newton-linearizedschemes and both iterative and noniterative time-linearizedschemes is given in [52] using an operator notation.

4.4. Scalable parallel algorithms for structured and unstructured grids

In the early 1990s, we began exploring a number of alternativeapproaches for transitioning implicit algorithms to parallel com-puters. In 1995, Pankajakshan and Briley modified Taylor andWhitfield’s iterative Newton-linearized Navier–Stokes algorithmto enable parallel solution with minimal effect on its algorithmicperformance [53]. This parallel algorithm used a modified block-Ja-cobi symmetric Gauss-Seidel (BJ-SGS) sub-iteration scheme withcoarse-grained domain decomposition for mapping to distrib-

Table 1Historical examples of actual and hypothetical Navier–Stokes flow cases.

Year Computer Case Grid Step

1968 CDC-6600 Ac 26 � 101 1001973 Univac 1108 Bd 11 � 11 � 11 601985 Cray X-MP Ce 29 � 29 � 29 601999 Cray T-3E Df 4.5 M 14,02005 SimCenter Cluster 1 Eg 18 M 54002008 SimCenter Cluster 2 Fh 20.5 M 48,0

a The last two columns give estimated runtime and cost for a hypothetical test case Gb The memory available would have been insufficient to run this case.c [2] University of Texas Austin 1968 – Time-accurate, incompressible, axisymmetricd [7] United Technologies Research Center 1973 – Steady flow in a straight square due [23] Scientific Research Associates, Inc. 1985 – Steady horseshoe vortex flow: Nonitf [55] NSF ERC Mississippi State University 1999 – Time-accurate 6-DOF maneuver of

propeller revolutions: Parallel, structured Newton-iterative LU/BJ-SGS scheme (3D).g [58] University of Tennessee SimCenter 2005 – Time-accurate release of contaminan

(3D).h [59] University of Tennessee SimCenter 2008 – Flow past Class 8 trucks with drag r

uted-memory processors. In his 1997 dissertation, Pankajakshandeveloped a parallel iteration hierarchy for time-accurate solutionsthat encompassed multigrid, Newton and BJ-SGS iterations, andevaluated both algorithmic and software performance [54,55]. Asemi-empirical parallel performance model for his MPICH softwareimplementation indicated constrained-memory scalability to atleast 400 processors using contemporaneous distributed-memorycomputers such as the IBM-SP2 and Cray-T3E. In 1997, Shengand Whitfield began working with Anderson’s implicit unstruc-tured time-linearized flow solver and developed a multi-blockimplementation to reduce memory requirements on single-proces-sor machines [56]. Building on this work, Hyams’s 2000 disserta-tion developed a scalable parallel implicit viscous flow solver forhighly nonuniform multi-element unstructured grids [57] usingdistributed-memory processors. His work addressed issues of par-allel iteration hierarchy for the Newton-linearized method and ofthe treatment of connectivity and coupling of the sub-domaininterfaces. Finally, after our current group was formed in 2002, Sre-enivas et al. developed a new code (now called Tenasi) based onthis same algorithm but with many refinements, extensions andnew capabilities for complex computational engineering analysisand design applications.

Various implicit methods that have evolved from our early workcontinue to be applied to a wide range of problems. Despite theremarkable progress in our field during recent decades, further ad-vances in algorithms and computational methodology are stillneeded for effective solution of many practical problems that areattractive for computational simulation. Our own work in recentyears has focused on various application problems and relatedalgorithm developments. These interests have included multiphaseand chemically reacting flows, the shallow-water and Maxwell’sequations, preconditioning, and treatment of source terms and dis-continuities other than shocks.

5. Historical examples of runtime, problem size, and estimatedcost

We conclude by giving in Table 1 some historical examples offlow cases our group has run on different computers over the years.These six cases (A–F) illustrate the evolution over time in problemsize and runtime. The last two columns give estimated runtimeand cost for a hypothetical test case G having 106 grid points and103 time steps, assuming it would run on each computing platformat the same rate of runtime per point per step as in the correspond-ing cases A–F. The estimated runtime is given for the indicatednumber of processors and assumes sufficient memory is available.The estimated cost assumes typical published hardware-only

s Runtime Hrs (Proc) Runtime Hrs (Proc) a Costa

0.38 (1) 1447 (1)b $328,0000.22 (1) 2713 (1)b $124,0000.13 (1) 89 (1)b $5,000

00 320 (50) 5 (50) $6844 (100) 0.45 (100) $0.80

00 91 (200) 0.09 (200) $0.23

having 106 grid points and 103 time steps.

rotating flow: ADI scheme, vorticity/stream-function method (2D).ct: Noniterative Time-linearized ADI/AF scheme (3D).erative time-linearized ADI/AF scheme (3D).submarine with rotating propulsor and moving control surfaces, 2 hull lengths, 40

t in urban environment: Parallel unstructured Newton-iterative LU/BJ-SGS scheme

eduction devices: Parallel unstructured Newton-iterative LU/BJ-SGS scheme (3D).

W.R. Briley, H. McDonald / Computers & Fluids 41 (2011) 15–19 19

prices without discount, operating continuously without downtime, depreciated over 3 years, and given in 2007 dollars.

It is remarkable that the hardware cost for running the stan-dardized hypothetical case G has decreased from a theoretical esti-mate of $328,000 for the 1968 CDC-6600 to just $0.23 on the 2008UTC SimCenter diskless Linux cluster of in-house design. Althoughthe runtime varies with the number of processors used (con-strained by available memory), the cost is more or less indepen-dent of the number of processors used within the range of linearscalability for a given problem size. Our current benchmarks showa near-linear scalability up to 1200 processors on a problem size of96 M points.

Although not evident from Table 1, the Navier–Stokes algo-rithms have also improved greatly in regard to geometric andproblem complexity, solution accuracy, and general robustness.Although these algorithms will continue to evolve, CFD as a spe-cialty is now having a tremendous impact on practical engineeringanalysis and design problems, and it appears that this impact willcontinue and accelerate for the foreseeable future.

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