Verification of Calculations involves error estimation, whereas Verification of Codes involves error evaluation, from known benchmark solutions. The best benchmarks are exact analytical solutions with sufficiently complex solution structure; they need not be realistic since Verification is a purely mathematical exercise. The Method of Manufactured Solutions (MMS) provides a straightforward and quite general procedure for generating such solutions. For complex codes, the method utilizes Symbolic Manipulation, but here it is illustrated with simple examples. When used with systematic grid refinement studies, which are remarkably sensitive, MMS produces strong Code Verifications with a theorem-like quality and a clearly defined completion point.

Issue Section:
Technical Papers
Keywords:
computational fluid dynamics, codes, error analysis, numerical analysis, boundary-value problems, convergence
Topics:
Boundary-value problems, Errors, Theorems (Mathematics), Computational fluid dynamics, Shock (Mechanics)
1.
Roache, P. J., 1998, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM.
2.
Roache, P. J., 1999, Fundamentals of Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, NM, Chapter 18.
3.
Roache
,
P. J.
,
Ghia
,
K.
, and
White
,
F.
,
1986
, “
Editorial Policy Statement on the Control of Numerical Accuracy
,”
ASME J. Fluids Eng.
,
108
, No. 1, Mar., p.
2
2
.
4.
Steinberg
,
S.
, and
Roache
,
P. J.
,
1985
, “
Symbolic Manipulation and Computational Fluid Dynamics
,”
J. Comput. Phys.
,
57
, No. 2, Jan., pp.
251
–
284
.
5.
Oberkampf, W. L., Blottner, F. G., and Aeschliman, D. P., 1995, “Methodology for Computational Fluid Dynamics Code Verification/Validation,” AIAA Paper 95-2226, 26th AIAA Fluid Dynamics Conference, 19–22 June, San Diego, California.
6.
Oberkampf, W. L., and Trucano, T. G., 2000, “Validation Methodology in Computational Fluid Dynamics,” AIAA Paper 2000–2549, 19–22 June, Denver, CO.
7.
Oberkampf, W. L., and Trucano, T. G., 2002, “Verification and Validation in Computational Fluid Dynamics,” AIAA Progress in Aerospace Sciences (to appear).
8.
Pelletier, D., and Ignat, L., 1995, “On the Accuracy of the Grid Convergence Index and the Zhu-Zienkiewicz Error Estimator,” pp. 31–36 in ASME FED Vol. 213, Quantification of Uncertainty in Computational Fluid Dynamics, Aug., R. W. Johnson and E. D. Hughes, eds.
9.
Reed
,
H. L.
,
Haynes
,
T. S.
, and
Saric
,
W. S.
,
1998
, “
CFD Validation Issues in Transition Modeling
,”
AIAA J.
,
36
, No. 5, May, pp.
742
–
749
. See also AIAA Paper 96-2051, 27th AIAA Fluid Dynamics Conference, 17–20 June 1996, New Orleans, Louisiana.
10.
Polya, G., 1957, How to Solve It; A New Aspect of Mathematical Method, Princeton University Press, Princeton, NJ.
11.
Dee, D. P., 1991, “Prescribed Solution Forcing Method for Model Verification in Hydraulic Engineering,” Proc. 1991 National Conference on Hydraulic Engineering, ASCE, Nashville, July 29-August 2.
12.
Ethier
,
C. R.
, and
Steinman
,
D. A.
,
1994
, “
Exact Fully 3D Navier-Stokes Solutions for Benchmarking
,”
Int. J. Fract.
,
19
, pp.
369
–
375
.
13.
Powers
,
J. M.
, and
Gonthier
,
K. A.
,
1992
, “
Reaction Zone Structure for Strong, Weak Overdriven and Weak Underdriven Oblique Detonations
,”
Phys. Fluids A
,
4
, No. 9, Sept., pp.
2082
–
2089
.
14.
Powers
,
J. M.
, and
Stewart
,
D. S.
,
1992
, “
Approximate Solutions for Oblique Detonations in the Hypersonic Limit
,”
AIAA J.
,
30
, No. 3, Mar., pp.
726
–
736
.
15.
Grismer
,
M. J.
, and
Powers
,
J. M.
,
1992
, “
Comparison of Numerical Oblique Detonation Solutions with an Asymptotic Benchmark
,”
AIAA J.
,
30
, No. 12, Dec., pp.
2985
–
2987
.
16.
Grismer
,
M. J.
, and
Powers
,
J. M.
,
1996
, “
Numerical Predictions of Oblique Detonation Stability Boundaries
,”
Shock Waves
,
6
, pp.
147
–
156
.
17.
Salari, K., and Knupp, P., 2000, “Code Verification by the Method of Manufactured Solutions,” SAND2000-1444, Sandia National Laboratories, Albuquerque, NM 87185, June.
18.
Salari, K., and Roache, P. J., 1990, “The Influence of Sweep on Dynamic Stall Produced by a Rapidly Pitching Wing,” AIAA Paper 90-0581.
19.
Roache, P. J., and Steinberg, S., 1985, “A New Approach to Grid Generation Using a Variational Formulation,” AIAA 85-1527-CP, Proceedings AIAA 7th Computational Fluid Dynamics Conference, 15–17 July, Cincinnati, Ohio.
20.
Steinberg
,
S.
, and
Roache
,
P. J.
,
1986
, “
Using MACSYMA to Write Fortran Subroutines
,”
Journal of Symbolic Computation
,
2
, pp.
213
–
216
. See also MACSYMA Newsletter, Vol. II-2, 1985, pp. 10–12.
21.
Steinberg
,
S.
, and
Roache
,
P. J.
,
1986
, “
Variational Grid Generation
,”
Numerical Methods for Partial Differential Equations
,
2
, pp.
71
–
96
.
22.
Steinberg, S., and Roache, P. J., 1986, “A Tool Kit of Symbolic Manipulation Programs for Variational Grid Generation,” AIAA Paper No. 86-0241, AIAA 24th Aerospace Sciences Meeting, 6–9 January 1986, Reno, Nevada.
23.
Steinberg, S., and Roache, P. J., 1986, “Grid Generation: A Variational and Symbolic-Computation Approach,” Proceedings Numerical Grid Generation in Fluid Dynamics Conference, July, Landshut, W. Germany.
24.
Steinberg
,
S.
, and
Roache
,
P. J.
,
1992
, “
Variational Curve and Surface Grid Generation
,”
J. Comput. Phys.
,
100
, No.
1
, pp.
163
–
178
.
25.
Roy, C. J., McWherter-Payne, M. A., and Oberkampf, W. L., 2000, “Verification and Validation for Laminar Hypersonic Flowfields,” AIAA 2000-2550, June, Denver.
26.
Roy, C. J., 2001, “Grid Convergence Error Analysis for Mixed-Order Numerical Schemes,” AIAA Paper 2001–2606, June, Anaheim. See also this issue of ASME JFE.
27.
Pautz, S. D., 2001, “Verification of Transport Codes by the Method of Manufactured Solutions: the ATTILA Experience,” Proc. ANS International Meeting on Mathematical Methods for Nuclear Applications, M&C 2001, Salt Lake City, Utah, Sept.
28.
AIAA, 1998, “Guide for the Verification and Validation of Computational Fluid Dynamics Simulations,” American Institute of Aeronautics and Astronautics, AIAA-G-077-1998, Reston, VA.
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