DISCRETIZATION ERROR ESTIMATES USING EXACT SOLUTIONS TO NEARBY PROBLEMS (INVITED)

Christopher J. Roy† and Matthew M. Hopkins‡

Sandia National Laboratories*

P. O. Box 5800

Albuquerque, NM 87185

AIAA Paper 2003-0629

AbstractA methodology is presented for generating exact so-

lutions to equations that are “near” the Navier-Stokesequations. First, a highly accurate numerical solution tothe Navier-Stokes equations is computed. Second, ananalytic function is fit to the numerical solution by leastsquares optimization. Next, this analytic solution is op-erated on by the Navier-Stokes equations (includingauxiliary relations) to obtain a small analytic sourceterm. When the Navier-Stokes equations are perturbedby adding this source term, the analytic function is re-covered as the exact solution. Approaches are presentedwhich address the “goodness” of the curve-fitting proce-dure and the “nearness” of the modified set of equationsto the Navier-Stokes equations. Two examples are givenfor compressible fluid flow: fully developed flow in achannel, and lid-driven cavity flow. The channel flow isfully captured by a third-order polynomial fit, while thedriven-cavity solution is not adequately represented bypolynomial curve fits up to fourth order. The generationof an exact solution to a set of equations near the Navi-er-Stokes equations allows for the evaluation of variousdiscretization error estimators, without reverting to sim-plification of the governing equations or use of a highlyrefined “truth” mesh. Preliminary results for a numberof extrapolation-based error estimators are also present-ed.

1

† Senior Member of Technical Staff, MS 0825, E-mail:

cjroy@sandia.gov, Senior Member AIAA‡ Senior Member of Technical Staff, MS 0834, E-mail: mm-

hopki@sandia.gov* Sandia is a multiprogram laboratory operated by Sandia

Corporation, a Lockheed Martin Company, for the UnitedStates Department of Energy under Contract DE-AC04-94AL85000.

This paper is declared a work of the U. S. Government and isnot subject to copyright protection in the United States.

Nomenclaturee energy, m2/s2

et total energy, m2/s2

f source termh measure of grid spacingk thermal conductivity, W/(m·K), or grid levelL reference length, mL2 L2 normp spatial order of accuracy, or pressure, N/m2

Pr Prandtl number (Pr = 1.0)q heat flux vector, N/(m·s)R specific gas constant (R = 287.0 N·m/(kg·K))r grid refinement factort time, sT temperature, Ku, v Cartesian velocity components, m/s x, y Cartesian spatial coordinates, mγ ratio of specific heats (γ = 1.4)µ absolute viscosity, N·s/m2

ρ density, kg/m3

τ shear stress tensor, N/m2

φ general solution variable

Subscriptsexact exact value k mesh level (k = 1, 2, 3, etc., fine to coarse)n flowfield node index

IntroductionSources of error in Computational Fluid Dynamics

(CFD) can be classified into two main categories: mod-eling errors and numerical errors. Modeling errors arisedue to a given model’s inability to reproduce the behav-ior observed in the real world. For example, a turbu-lence model may be calibrated for attached, zero

AIAA 99-xxxx

pressure gradient flow, but fail to predict the correctseparation characteristics in a flow with a strong adversepressure gradient. Numerical errors can be associatedwith a number of sources including mesh resolution,time step, discretization scheme, iterative convergence,round-off, and coding mistakes. For complex CFD sim-ulations (e.g., with turbulence, chemistry, unsteadyflow, shock waves, etc.), it is particularly important tocontrol and understand numerical errors; failure to do socan lead to incorrect conclusions in model validationstudies.

Numerical errors associated with the mesh and dis-cretization scheme (i.e., discretization errors) are impor-tant to assess not only for the purpose of modelvalidation, but also for grid adaptation. Unstructuredgrid methodologies,1,2 and to a lesser extent, structuredgrid methods,3 allow for the control of discretization er-ror through selective refinement/de-refinement of thegrid. Such adaptation criteria are often based on localfeatures (gradients, shock waves, etc.). More advancedstrategies can adapt the grid based on global errors, oreven the local contributions to global errors.4,5

Discretization error estimators can be broadly classi-fied into three categories: 1) extrapolation-based, 2) re-sidual-based, and 3) recovery-based error estimators.The extrapolation-based error estimators are based onRichardson extrapolation,6 where the numerical solu-tions on two meshes are extrapolated to zero elementsize to approximate the exact solution. The most popularextrapolation-based approach used today is Roache’sGrid Convergence Index (GCI).7 The residual-based er-ror estimators were initially developed for finite elementformulations and include both standard a posteriori8and adjoint-based4,5 error estimators. The recovery-based error estimators were also initially developed forfinite elements (e.g., the Zienkiewicz-Zhu errorestimator9), and have recently been extended to the fi-nite-volume approach.10

The standard methods for evaluating the efficacy oferror estimators involve the use of either exact solutionsor benchmark numerical solutions. For complex partialdifferential equations (e.g., the Navier-Stokes equa-tions), there are generally only a limited number of ex-act solutions available. Furthermore, these exactsolutions often involve significant simplifications anddo not test the general governing equations. One exam-ple is the flow between moving parallel plates (Couetteflow) where the velocity gradient is linear and thus thediffusion term, a second derivative of velocity, is identi-cally zero and is therefore not exercised. The use of abenchmark numerical solution (or “truth” mesh) is alsoproblematic since the accuracy of the benchmark solu-tion is generally unknown. In addition, assessing theconvergence rate of the numerical method is difficult

when the exact solution is not known.There has been some prior work in the literature deal-

ing with the generation of exact solutions. One exampleis the method of manufactured solutions,11-13 where ananalytic solution is chosen a priori and the governingequations are modified by the addition of analyticsource terms. The purpose of manufactured solutions isfor code verification, that is, to ensure to the highest de-gree possible that a given simulation code is free fromcoding mistakes. The manufactured solutions are gener-ally chosen a priori for their smoothness and for theirability to exercise all terms in the governing equations.However, code verification is a mathematical exercisethat does not attempt to assess the adequacy of the phys-ical models, thus the solutions are generally nonphysi-cal.

Another approach to generating exact solutions wasdeveloped by Lee and Junkins14 for one-dimensionalnonlinear ordinary differential equations (ODEs). Thebasic idea behind their work is summarized in the fol-lowing steps:

Step 1) Compute a numerical solution on a highly re-fined mesh (i.e., small time step).

Step 2) Generate an analytic solution from a global fit to the fine grid numerical solution based on the least squares approach using Chebyshev polyno-mials.

Step 3) Use symbolic manipulation (in their case MACSYMA) to plug the analytic polynomial solu-tion into the original ODEs to generate small source terms.

Step 4) Solve the nearby problem, consisting of the original ODEs plus the small source terms, on a se-ries of different discretizations.

The goal of their work was to determine the optimal nu-merical integration parameters for a given problem,which generally involved determining where round-offerrors started to affect the discretization error. Junkinsand Lee15 later extended their methodology to nonlinearhybrid ODE/PDEs that arise from flexible multi-bodydynamical systems in two dimensions (both time andspace).

The goal of our current research program is to evalu-ate the reliability of various error estimators for use inboth model validation and grid adaptation. This paperdescribes a first effort at a methodology for generatingexact solutions to small perturbations of the Navier-Stokes equations. The methodology extends the work ofLee and Junkins14 for dynamical systems to the couplednonlinear PDEs that make up the Navier-Stokes equa-tions. Two cases are examined herein: fully developedlaminar flow in a narrow channel, and the flow in a lid-driven cavity.

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Numerical Formulation

Navier-Stokes Equations

The perturbed 2D Navier-Stokes equations in conser-vation form are

(1)

(2)

(3)

where generalized source terms are shown on the righthand side. Of course, when all source terms are equal tozero, the (unperturbed) Navier-Stokes equations are re-covered. For a calorically perfect gas, the Navier-Stokesequations are closed with auxiliary relations for energy

. (4)

(5)

and with the ideal gas equation of state

(6)

For the solutions presented herein, the ratio of specificheats is γ = 1.4 and the specific gas constant isR = 287.0 N·m/(kg·K). The shear stress tensor is

(7)

and the heat flux vector is given by:

(8)

The absolute viscosity µ is chosen to be a constant, andthe thermal conductivity k is determined from the vis-cosity by choosing the Prandtl number (here Pr = 1):

(9)

CFD Simulation Codes

Two different Navier-Stokes codes are employed inthe current work. The SACCARA code is used to estab-lish the highly refined initial numerical solutions. Thiscode was developed from a parallel distributed memoryversion16 of the INCA code, originally written byAmtec Engineering. The SACCARA code is used tosolve the Navier-Stokes equations for conservation ofmass, momentum, and energy in two dimensions. Thegoverning equations are discretized using a cell-cen-tered finite-volume approach. The convective fluxes atthe interface are calculated using Yee’s symmetric TVDscheme.17 Second-order reconstructions of the interfacefluxes are obtained via MUSCL extrapolation.18 Theviscous terms are discretized using central differences.

The Premo code19 is used for the implementation ofthe source terms and generalized boundary conditions.This code is being developed as part of the Departmentof Energy’s Accelerated Strategic Computing Initiative(ASCI) to meet the needs of the Stockpile StewardshipProgram. The Premo code is one of a number of me-chanics and energy transport codes that serves as a mod-ule to the SIERRA multi-mechanics framework.20 TheSIERRA framework provides services for I/O, domaindecomposition, massively parallel processing, meshadaptivity, load balancing, code coupling, and an inter-face to a host of linear and nonlinear solvers.

The spatial discretization employed in the Premocode is a node-centered finite-volume formulation. Thisdiscretization is implemented on unstructured meshesusing an edge-based scheme which allows arbitrary ele-ment topologies, where an element is determined byconnecting nearest-neighbor nodes. The convectivefluxes are evaluated with Roe’s approximate Riemannsolver.21 Second-order spatial accuracy is achieved viaMUSCL extrapolation18 for the primitive variables tothe control volume surface using a Least Squares (LSQ)gradient operator.22 The gradient is also used in theevaluation of the viscous fluxes at the control-volume

ρ( )∂t∂

----------- ρu( )∂x∂

-------------- ρv( )∂y∂

--------------+ + fm=

ρu( )∂t∂

--------------ρu2 p+ τxx–( )∂

x∂--------------------------------------- +

ρuv τxy–( )∂

y∂-------------------------------+ fx=

ρv( )∂t∂

--------------ρvu τxy–( )∂

x∂------------------------------- +

ρv2 p τyy–+( )∂

y∂---------------------------------------+ fy=

ρet( )∂t∂

---------------ρuet pu uτxx– vτxy– qx+ +( )∂

x∂----------------------------------------------------------------------------+

ρvet pv uτxy– vτyy– qy+ +( )∂

y∂----------------------------------------------------------------------------+ fe=

e 1γ 1–-----------RT=

et e u2 v2 w2+ +2

------------------------------+=

p ρRT=

τxx23---µ 2 u∂

x∂----- v∂

y∂-----–

=

τyy23---µ 2 v∂

y∂----- u∂

x∂-----–

=

τxy µ u∂y∂

----- v∂x∂

-----+ =

qx k T∂x∂

------–=

qy k T∂y∂

------–=

k γRγ 1–----------- µ

Pr------=

3

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surface, resulting in a second-order discretization for theviscous terms. For the steady-state simulations dis-cussed herein, the governing equations are integrated intime using a low-storage, four-stage Runge-Kutta meth-od.23 See Ref. 19 for more details on the temporal andspatial discretization of the Premo code.

Exact Solution MethodologyThe proposed method for generating exact solutions

to equations near the Navier-Stokes equations is basedon that of Lee and Junkins14 and is summarized as fol-lows:

Step 1) Establish accurate numerical solutionStep 2) Generate analytic curve fit to #1 aboveStep 3) Generate analytic source termsStep 4) Numerically solve “nearby” problemStep 5) Evaluate error estimators

These five steps are described in detail below.

1. Accurate Numerical Solution

Once the problem of interest is identified, the firststep is to compute a highly refined numerical solution.While this solution will have some associated discreti-zation error, this fact will not pose a problem as will beshown later. All of the initial numerical solutions com-puted were generated using the SACCARA Navier-Stokes code.

2. Analytic Curve Fit

This step is generally the most difficult step and in-volves generating an accurate analytic fit of the numeri-cal solution computed in step 1 above. In order to avoidissues with continuity of properties and derivativesacross zonal boundaries, this paper examines globalcurve fits only. In addition, the boundary conditions arenot enforced at the domain boundaries in order to sim-plify the fit approximations. Once the curve fit has beengenerated, some measure of the “goodness” of the fitwill be quantified to determine how well it satisfies thegiven data (i.e., the original numerical solution).

For the present study, polynomial functions of x and yare examined up to degree four. A least squares optimi-zation is performed to fit the numerical solution of eachprimitive variable (ρ, u, v, and p). The form and deriva-tion of the polynomial approximations are given inAppendix A.

3. Generation of Analytic Source Terms

The analytic curve fit from step 2 now becomes theexact solution to a set of equations “near” the Navier-Stokes equations. In fact, these neighboring equationsdiffer from the original Navier-Stokes equations only by

a (hopefully) small source term. These source termscome from operating the Navier-Stokes equations(Eqs. 1-3), along with the auxiliary equations (Eqs. 4-9),onto the curve fit solution from step 2. As the size ofthese source terms approach zero, solutions to the per-turbed equations approach the solution of the originalNavier-Stokes equations (with possible smoothness con-straints).

How small do the source terms need to be? Relatingthe “solution distance” to the “equation distance” for anonzero source term is difficult. Theory may tell us howto measure these distances as a function of source termsize for very simple cases (e.g., linear PDEs), but the au-thors are not aware of any work addressing this issue forgeneral coupled nonlinear PDEs. The resulting per-turbed equations would still be valuable as a verificationproblem, but would possibly not be as close to the start-ing equations as one would like (i.e., the “physics re-gime” is too different). We nonetheless expect that theresulting methodology presented here to be of greatpractical value.

The closeness of these neighboring equations to theNavier-Stokes equations is determined by examiningthe size of the associated source terms. Recall the defini-tion of the L2 norm of a function f on a domain ,

. (10)

Because the domain of integration should be obvious,and we are only using the L2 norm, we make the simpli-fication that .

We often create a continuous function from a discretenumerical solution by interpolating solution nodal val-ues located at mesh nodes to other spatial locations. Inthe present context this is done by bilinear interpolationfrom the corners of a quadrilateral mesh element to inte-rior points. However, one would expect that other nu-merical methods would yield their own interpolationmethods. For example, in finite elements there is thenatural finite element basis. In any case, Eq. (10) suffic-es for comparing source term sizes.

In the present case, the L2 norm of the source term foreach governing equation is calculated by integratingover the domain of interest . In theory this integrationcould be performed analytically. Unfortunately, as thegoverning equations or curve fit functions become morecomplex, analytic integration (using a symbolic mathpackage) becomes less efficient. We instead numerical-ly integrate the source term on successively finer mesh-es until the norm converges.

Ω

f L2 Ω( )2 f 2 ωd

Ω∫=

f f L2 Ω( )2=

Ω

4

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4. Numerical Solution to Nearby Problem

The neighboring problem is then discretized andcomputed on a series of meshes, including the sourceterm from the last step and the perturbed boundary con-ditions. For consistent numerical schemes and suffi-ciently refined meshes, the formal order of accuracy ofthe numerical scheme should be observed, even on ourperturbed equations. In general, the discretization errorshould drop as 1/r p, where in the current case the gridrefinement factor is r = 2 and the nominal order of accu-racy is p = 2; thus the error should drop by a factor offour on each successively refined mesh level. In order toexamine the global discretization error behavior, we de-fine the discrete error function

(11)

where k refers to the discrete mesh level and n variesover all N mesh nodes in space (including both interiorand boundary nodes) with the exception of the Dirichletboundary nodes for which the discretization error isidentically zero. Here, refers to the exact solu-tion evaluated at node n.

5. Evaluation of Error Estimators

While the focus of this paper is on the generation ofexact solutions, a key aspect of future efforts will be theevaluation of error estimators. Initial error estimators tobe examined include: Richardson Extrapolation,11

Roache’s Grid Convergence Index (GCI),7 and Roy’smixed-order error estimator.24

Results: Channel FlowThe first case to be examined is that of fully-devel-

oped laminar flow in a narrow channel. The flow is driv-en by a pressure gradient, and the bulk Mach number ischosen as 0.1, giving a maximum Mach number of 0.15.The channel height L is chosen as 0.01 m, and the vis-cosity is chosen as 4.086×10-4 N·s/m2. The Reynoldsnumber based on the channel height is thus ReL = 1000.The channel length is chosen as 3L, and the longitudinalpressure gradient is -1702.3 N/m3. Standard no-slipboundary conditions and a fixed wall temperature of300.6 K are applied at the walls. At the inflow bound-ary, the incompressible fully-developed velocity profileis specified

. (12)

The isentropic relation and the definition of the speed ofsound are used to specify the inflow temperature as

(13)

and the inflow pressure is set to 101,376.07 N/m2. At theoutflow boundary, the back pressure is set to atmospher-ic (101,325.0 N/m2), and the other boundary propertiesare found via extrapolation from the interior. While anexact solution exists in the limit as the Mach number ap-proaches zero, there is no general solution to the com-pressible flow problem.

The channel flow is solved on the half-domain fromthe bottom wall (y/L = 0) to the centerline (y/L = 1/2),with symmetry applied at the channel half-height. Acontour plot of the channel is shown in Fig. 1 with lon-gitudinal velocity contours shown along the lower half,and pressure contours shown along the upper half. It isclear from the figure that the pressure is nearly linear inx, and constant in y.

1. Accurate Numerical Solution

The velocity profiles across the channel half-heightare presented in Fig. 2 for the SACCARA solution using256×256 cells. A number of longitudinal stations areshown in the figure from . Each of thecurves for the longitudinal velocity u are identical, thus

E φk( )

φk n, φexact,n– 2

n 1=

N

∑N

--------------------------------------------------

1 2⁄

=

φexact n,

u∞ y( ) L2

2µ------dp

dx------ y

L ----- 2 y

L ----- –=

v∞ 0=

T∞ y( ) T0γ 1–

2-----------

u∞2

γR------–=

x/L

y/L

0 1 2 30

0.5

1

1.5

2

2.5101375101370101365101360101355101350101345101340101335101330101325

p (N/m2)

x/L

y/L

0 1 2 30

0.5

1

1.5

2

2.550454035302520151050

Channel FlowReL = 1000256x256 Cells

u (m/s)

p (N/m2)

u (m/s)

Fig. 1 Contours of axial velocity (bottom) and pressure (top) for the channel flow (SACCARA solution).

0.5 x L 2.5≤⁄≤

5

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showing that there are negligible u velocity gradients inthe x direction. While there is some variation in the ver-tical velocity v, these variations are small (approximate-ly 0.001 m/s) and approach zero as the grid is refined.

2. Analytic Curve Fit

The above SACCARA solution on the 256×256 cellmesh was used to generate polynomial approximationswith order ranging from zero to four. A truncated do-main of was used in order to minimizeerrors occurring at the boundaries. The L2 norms of thedifference between the polynomial approximations andthe fine grid SACCARA solution are given in Fig. 3plotted versus polynomial order. The vertical velocity isapproximated well by a constant value, the density andpressure by a linear value, and the axial velocity by asecond-order polynomial. The velocity profiles acrossthe channel half-height for the third order-polynomialare given in Fig. 4. The longitudinal velocity maintainsits quadratic behavior across the channel, while the ver-tical velocities are small and near (but not identically)zero at the lower wall and the channel half-height. Forthe remainder of this study of the channel flow, third-or-der polynomial fits for each of the primitive variables(ρ, u, v, and p) are employed.

3. Generation of Analytic Source Terms

Analytic source terms were generated by operatingthe Navier-Stokes equations (and auxiliary equations)on the chosen third-order polynomial solutions. Thesesource terms were generated using Mathematica, whichhas options for outputting the results in either the FOR-TAN or C programming languages. The resulting source

term for the mass conservation equation is given inAppendix B. Source terms for the other three governingequations are significantly more complex, and are avail-able upon request from the first author.

The initial SACCARA solutions of the Navier-Stokesequations were obtained on a series of grids ranging insize from 256×256 cells (Mesh 1) to 8×8 cells (Mesh 6).Each of these initial solutions were fitted to a third-orderpolynomial using least squares optimization. The L2norms of the source terms were integrated over the do-main of interest ( , ) fol-lowing Eq. (10). While this integration was performednumerically, the discretization size for the integrationwas varied until the source term norms did not change.

v (m/s)

u (m/s)

y/L

-0.001 -0.0005 0

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5Channel FlowReL = 1000256x256 Cells

v (m/s)

u (m/s)

Fig. 2 Velocity profiles across the channel half-height (SACCARA solution).

0.5 x L 2.5≤⁄≤

Polynomial Order

L2Norm

0 1 2 3 4

10-6

10-5

10-4

10-3

10-2

10-1

u (m/s)v (m/s)p (N/m2)ρ (kg/m3)

Channel FlowReL = 1000Baseline Solution:

256x256 Cells

Fig. 3 Error in the polynomial fits relative to the SACCARA solution.

v (m/s)

u (m/s)y/L

-0.001 -0.0005 0

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5Channel FlowReL = 10003rd Order LSQ Fit

v (m/s)

u (m/s)

Fig. 4 Velocity profiles across the channel half-height for the third-order polynomial fit.

0.5 x L 2.5≤⁄≤ 0 y L 0.5≤⁄≤

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The results are plotted versus representative cell size ofthe initial SACCARA solution on a log-log plot inFig. 5. The parameter h is the cell size on mesh level knormalized by the cell size on Mesh 1 (256×256 cells),i.e.,

(14)

Thus, h = 1 corresponds to the finest initial mesh, andh = 32 to the coarsest initial mesh. As the baseline SAC-CARA solution used for the curve fits is refined, thecorresponding size of the source term is reduced. Fur-thermore, the reduction in the size of the source term oc-curs at an approximately second-order rate, at least untila mesh size of 64×64 cells (h = 4). In general, basing thepolynomial fit on the finest initial numerical solutionprovided the smallest source terms.

4. Numerical Solution to Nearby Problem

The “nearby” channel flow problem was examinedbased on the 256×256 cell channel flow solution usingthe third-order polynomial fit. These numerical solu-tions were computed using the Premo code on the fourdifferent mesh levels, from Mesh 1 (65×65 nodes) toMesh 4 (9×9 nodes), shown in Table 1. This nearbyproblem includes the analytic source terms discussed inthe last section, and the boundary conditions are set tothe Dirichlet values determined by the analytic solution.The discrete error functions (see Eq. (11)) are presentedin Fig. 6 versus the normalized mesh spacing (again,h = 1 is the fine mesh). The errors for u, v, and p appearto converge at the expected rate of second order on the

finer meshes, while the norms for the density do notconverge at the expected rate. The reason for the lowerthan expected convergence rate for the density couldcome from two sources. First, all boundary conditionswere specified as Dirichlet for this case, thus leading toan over-specification of the boundary conditions (i.e.,some boundary values should be determined from theinterior solution). Second, since this problem is essen-tially incompressible, it is possible that the small varia-tions in density are captured on the coarser meshes, andadditional mesh resolution does not further resolve thesefeatures.

5. Evaluation of Error Estimators

In order to evaluate the various discretization errorestimators, the estimates of the exact solutions are com-pared to the exact analytical solution. Four different er-ror estimators are employed. The first error estimator isbased on second-order Richardson extrapolation (2ndOrder RE) and requires two solutions. This error estima-tor is expected to be the most accurate since both theformal and the observed order (see Fig. 6) are secondorder. The second error estimator is based on Roy’s

hk∆xk∆x1---------

∆yk∆y1---------= =

h

L2Norm

100 101 10210-4

10-3

10-2

10-1

100

101

102

massxmtmymtmenergy2nd Order Slope

Channel FlowReL = 10003rd Order LSQ Fit

Fig. 5 L2 norm of the source terms for the third-order polynomial fit versus element size of the initial mesh.

Table 1 Meshes for Channel Flow Computations

Mesh MeshNodes

GridSpacing, h

Mesh 1 65×65 1Mesh 2 33×33 2Mesh 3 17×17 4Mesh 4 9×9 8

h

L 2NormError

5 10 15 2010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Pressureu Velocityv VelocityDensity2nd Order Slope

Channel FlowReL = 10003rd Order LSQ Fit

1

Fig. 6 Discrete error function for the computed solution variables relative to the third-order polynomial (initial

mesh is 256×256 cells).

7

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mixed-order extrapolation (Mixed Order1) which re-quires three solutions (see Ref. 24 for details). A variantof this estimator (Mixed Order 2) is also examinedwhere second-order Richardson extrapolation is usedwhen the three solutions are monotonically convergingas the mesh is refined, and the mixed-order extrapola-tion is used when the solutions are non-monotone. Thethird error estimator is based on the locally observed or-der of accuracy, or pth order extrapolation (ObservedOrder). This method requires three solutions, and the or-der is limited to be between first and second order. Fi-nally, an error estimator based on a “Truth” mesh isexamined, where the fine grid is used to approximate theexact solution.

In order to evaluate how well the above methods areable to approximate the exact solution in a global sense,the discrete error function (or discrete L2 norm) fromEq. (11) is employed with the approximated exact solu-tion substituted in place of the discrete solution φk,n.This discrete L2 norm is evaluated over interior pointson the coarsest mesh level (9×9 nodes). The boundarypoints are omitted since they employ Dirichlet boundaryconditions from the exact solution. As the estimated ex-act solution approaches the true analytical exact solu-tion, the discrete error norm will approach zero. Thenorms are presented for the conservative variables ρ, ρu,and ρv which are solved for in the Premo code (ρet isomitted for brevity).

The discrete L2 error norms for the estimated exactsolutions relative to the analytical exact solution are giv-en in Appendix C, Table A.1 for the three finest meshes(Meshes 1-3). Note that the Richardson extrapolation re-sults require on two meshes (Meshes 1 and 2), while themixed-order approaches and the observed order ap-proach require all three mesh levels. As expected, sec-ond-order Richardson extrapolation provides the bestestimates of the exact solution for the three conservativevariables. Results using the local Observed Order andthe modified mixed-order method (Mixed Order2) alsoprovide fairly good error estimates. The original mixed-order method and the truth mesh approach using Mesh 1provide the poorest error estimates.

The discrete error norms for Meshes 2, 3, and 4 arepresented in Table A.2. Again, second-order Richardsonextrapolation provides the best approximation to the ex-act solution, while the Observed Order approach is alsofairly good. Both variants of the mixed-order methodsgive poor results relative to Richardson extrapolation.The truth mesh approach based on Mesh 2 gives resultsfor ρ and ρv that are comparable to Richardson extrapo-lation, but the results for ρu are poor. The results shownin Table A.1 using Mesh 1 for the truth mesh are argu-able better than the Richardson extrapolation results us-ing Meshes 2 and 3, but at the cost of computing an

additional mesh level. Based on the above results, the best approach for

problems where the formal order of accuracy is verifiedglobally (as in Fig. 6) is to compute the finest mesh lev-el possible and then to employ Richardson extrapolationusing the formal order of accuracy to approximate theexact solution. This approximation to the exact solutioncan then be used to provide discretization error esti-mates in the numerical solutions.

Results: Driven Cavity FlowThe second case to be examined is the lid-driven cav-

ity. The lid velocity UREF is chosen as 69.44 m/s whichgives a lid Mach number of 0.2. By choosing the viscos-ity to be constant at 8.1716×10-3 N·s/m2, the Reynoldsnumber based on the cavity length L of 0.01 m isReL = 100. The initial temperature and pressure are300.0 K and atmospheric pressure (101,325.0 N/m2), re-spectively. The walls all employ no-slip boundary con-ditions and the wall temperatures are fixed at thestagnation temperature (302.4 K).

1. Accurate Numerical SolutionThe accurate numerical solution is computed using

the SACCARA code on a 128×128 cell mesh. Verticalvelocity contours and streamlines are given in Fig. 7.The primary vortex is clockwise and is situated slightlyabove and to the right of the center of the cavity. Oppo-site of the driven top wall are two counter-clockwisevortices which form in the corners.

x/L

y/L

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1v/UREF0.30.250.20.150.10.050-0.05-0.1-0.15-0.2-0.25-0.3-0.35-0.4

Driven Cavity: ReL = 100128x128 Cells

Fig. 7 Vertical velocity contours and streamlines for the lid-driven cavity (initial SACCARA solution).

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2. Analytic Curve Fit

The SACCARA solution on the 128×128 mesh isused as data for generating polynomial curve fits fromorder zero to four. The L2 error norms of the polynomialfits relative to the baseline SACCARA solution areshown in Fig. 8 (solid lines). Contrary to the channelflow case, the error norms have a larger magnitude forthe cavity and show little reduction with increasingpolynomial order. Also shown in Fig. 8 are results forthe subdomain consisting of the lower 7/8 of the cavity(dashed lines). The upper 1/8 of the cavity was removedto minimize the effects of the corner singularities at thelid-wall junctures. While the L2 error norms are signifi-cantly reduced, there is still only a small reduction as theorder of the polynomial is increased from zero to four.

The inability of the polynomial functions to give re-ductions in the error norms suggests that using polyno-mials to approximate the primitive flow variables is apoor choice. As a result, we would not expect our proce-dure to be as informative here as in the previous case ofchannel flow. If the global nature of the curve fit is to bemaintained, a different choice of basis functions is need-ed. Going to significantly higher-order polynomials islikely to result in an unstable process due to the well-known misbehavior of polynomial coefficients at higherorder. Some likely candidate basis functions include theChebychev polynomials (also used in Ref. 14) and Fou-rier expansions.

3. Generation of Analytic Source Terms

Analytical source terms were generated using thefourth-order polynomial solutions and the entire range

of initial SACCARA solutions for the driven cavity. TheL2 norms of the source terms for each of the governingequations are given in Fig. 9. For this case, the smallestsource terms are found when the 16×16 cell SACCARAsolution was used as the data for the curve fitting proce-dure. The fact that the size of the source term does notdrop as the initial mesh is refined further supports thehypothesis that the polynomial-based curve fit is notsufficient for this case. The source terms for the y-mo-mentum equation using the fourth-order polynomial fitsare presented in Fig. 10 for initial SACCARA solutionon a 16×16 cell mesh and a 256×256 cell mesh. The am-plitude of the source term from the finer initial mesh issignificantly larger near the boundaries.

Vertical velocity contours and streamlines from thefourth-order polynomial fit based on the 16×16 cellSACCARA solution are presented in Fig. 11. The verti-cal velocity contours are somewhat smoothed out by thepolynomial fit relative to the original solution (seeFig. 7), and while the overall vortex does appear to besimilar in structure, the secondary vortices are signifi-cantly larger and located farther from the floor of thecavity. Based on the preceding analysis, the generatedanalytical solution is not as “near” to the original drivencavity problem as we would like.

ConclusionsA methodology was presented for the generation of

exact solutions to slight perturbations of the Navier-Stokes equations. This methodology was successfullydemonstrated for fully developed compressible flow in a

Polynomial Order

L2Norm

0 1 2 3 410-2

10-1

100

101

u (original domain)v (original domain)p (original domain)u (truncated domain)v (truncated domain)p (truncated domain)

Driven CavityReL = 100Baseline Solution:

128x128 Cells

Fig. 8 Error in the polynomial fits relative to the 128×128 cell SACCARA solution.

h

L2Norm

100 101 102100

101

102

103

104

105

106

107

massxmtmymtmenergy2nd Order Slope

Driven CavityReL = 1004th Order LSQ Fit

Fig. 9 L2 norm of the source terms for the fourth-order polynomial fit versus element size of the initial mesh.

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channel. This case required only third-order polynomialfunctions to adequately represent the primitive solutionvariables. A preliminary investigation of the more com-plex lid-driven cavity flow showed that, as expected, theglobal polynomial fits did not capture the solution well.This failure illustrates the need for incorporating expertknowledge in our process. An important goal of con-tinuing work is to develop the process to the point wheresuch expert analysis will not be required. For example,the lid-driven cavity problem may simply require a

more appropriate set of interpolating functions.There are many avenues for future work. As previ-

ously stated, we will examine other classes of approxi-mation functions. In addition, special treatment ofsingularities may be needed. In the case of the drivencavity, there are two singularities at beginning and end-ing of the driven lid. As posed, these singularities giverise to discontinuities in both velocity and pressure. Lo-cal treatment of singularities is required for the applica-tion of this methodology to realistic problems.

Appendix A: Polynomial FunctionsThe general form of a nth degree polynomial is

(A.1)

where represents the set of coefficients . For agiven mesh M and primitive variable we want to find a set of coefficients such that thepolynomial approximates the discretesolution C. To do this we use least squares optimization(via Matlab) to minimize

(A.2)

where i goes over all mesh points (including bound-aries), to find an optimal set of coefficients . In thecourse of our analysis, we repeated this process for ev-ery combination of polynomial order (n), mesh (M), andprimitive unknown (C).

Appendix B: Source TermThe source term for the mass conservation equation is

given below. This source term assumes that the solutionfor the primitive variables is given by a degree fourpolynomial. Source terms for the momentum and energyequations are more complex and are omitted for the sakeof brevity.

Appendix C: Error Estimator ResultsThe discrete L2 norm error between the extrapolated

solution and the analytical exact solution are presentedin Tables A.1-A.2 for the channel flow case.

AcknowledgementsWe would like to thank Fred Blottner and Jim Stew-

art of Sandia National Laboratories for providing ex-

0

0.5

1

x/L 00.5y/L

Driven Cavity: ReL = 1004th Order LSQ Polynomial FitBased on Initial 16x16 Cell Soln.

0

0.5

1

x/L 00.5y/L

X

Y

Z

ymtmeq350000300000250000200000150000100000500000-50000-100000-150000

Driven Cavity: ReL = 1004th Order LSQ Polynomial FitBased on Initial 256x256 Cell Soln.

Fig. 10 Magnitude of the y-momentum source term for the “nearby” cavity flow (fourth-order polynomial fits).

x/L

y/L

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v/UREF0.30.250.20.150.10.050-0.05-0.1-0.15-0.2-0.25-0.3-0.35-0.4

Driven Cavity: ReL = 1004th Order LSQ Polynomial (16x16 Cells)

Fig. 11 Vertical velocity contours and streamlines for the lid-driven cavity (fourth-order polynomial fit to the

16×16 cell initial solution).

qn x y λ;,( ) λα β, xαyα β–

β 0=

α

∑α 0=

n

∑=

λ λα β,C ρ u v p, , , ∈

λC M,

qn x y λC M,;,( )

qn xi yi λC M,;,( ) C xi yi,( )–[ ]2

i M∈∑

λα β,C M,

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tremely helpful reviews of this paper.

References1. Muzaferija, S., and Gosman, D., “Finite-Volume

CFD procedure and Adaptive Error Control Strategy forGrids of Arbitrary Topology,” Journal of Computation-al Physics, Vol. 138, No. 2., 1997, pp. 766-787.

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Source term for the mass conservation equation using the degree four polynomial.

In[21]:= masseq = D@rho, tD + D@rho uvel, xD + D@rho vvel, yD

Out[21]= Hbux + 2 cux2 x + 3 dux3 x2 + 4 eux4 x3 + cuxy y + 2 dux2y x y +3 eux3y x2 y + duxy2 y2 + 2 eux2y2 x y2 + euxy3 y3L Harho0 + brhox x + crhox2 x2 +drhox3 x3 + erhox4 x4 + brhoy y + crhoxy x y + drhox2y x2 y + erhox3y x3 y +crhoy2 y2 + drhoxy2 x y2 + erhox2y2 x2 y2 + drhoy3 y3 + erhoxy3 x y3 + erhoy4 y4L +

Hbvy + cvxy x + dvx2y x2 + evx3y x3 + 2 cvy2 y +2 dvxy2 x y + 2 evx2y2 x2 y + 3 dvy3 y2 + 3 evxy3 x y2 + 4 evy4 y3L Harho0 + brhox x +crhox2 x2 + drhox3 x3 + erhox4 x4 + brhoy y + crhoxy x y + drhox2y x2 y + erhox3y x3 y +crhoy2 y2 + drhoxy2 x y2 + erhox2y2 x2 y2 + drhoy3 y3 + erhoxy3 x y3 + erhoy4 y4L +

Hbrhox + 2 crhox2 x + 3 drhox3 x2 + 4 erhox4 x3 + crhoxy y +2 drhox2y x y + 3 erhox3y x2 y + drhoxy2 y2 + 2 erhox2y2 x y2 + erhoxy3 y3L

Hau0 + bux x + cux2 x2 + dux3 x3 + eux4 x4 + buy y + cuxy x y + dux2y x2 y +eux3y x3 y + cuy2 y2 + duxy2 x y2 + eux2y2 x2 y2 + duy3 y3 + euxy3 x y3 + euy4 y4L +

Hbrhoy + crhoxy x + drhox2y x2 + erhox3y x3 + 2 crhoy2 y +2 drhoxy2 x y + 2 erhox2y2 x2 y + 3 drhoy3 y2 + 3 erhoxy3 x y2 + 4 erhoy4 y3L

Hav0 + bvx x + cvx2 x2 + dvx3 x3 + evx4 x4 + bvy y + cvxy x y + dvx2y x2 y + evx3y x3 y +cvy2 y2 + dvxy2 x y2 + evx2y2 x2 y2 + dvy3 y3 + evxy3 x y3 + evy4 y4L

Table A.1 Discrete L2 Error Norms for the Channel Flow (Meshes 1, 2, and 3)

Error Estimator ρ ρu ρv

2nd Order RE 5.3×10−7 9.9×10−6 7.9×10−7

Mixed Order1 7.4×10−7 4.7×10−5 8.6×10−6

Mixed Order2 5.8×10−7 9.9×10−6 4.0×10−6

Observed Order 6.6×10−7 1.3×10−5 1.5×10−6

Truth Mesh 4.7×10−7 1.4×10−4 2.0×10−6

Table A.2 Discrete L2 Error Norms for the Channel Flow (Meshes 2, 3, and 4)

Error Estimator ρ ρu ρv

2nd Order RE 6.3×10−7 6.3×10−5 9.7×10−6

Mixed Order1 2.7×10−5 2.4×10−4 1.3×10−4

Mixed Order2 2.0×10−5 6.3×10−5 5.1×10−5

Observed Order 1.0×10−6 9.0×10−5 1.7×10−5

Truth Mesh 6.5×10−7 6.0×10−4 9.2×10−6

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