# zbMATH — the first resource for mathematics

Review of code and solution verification procedures for computational simulation. (English) Zbl 1072.65118
Summary: Computational simulation can be defined as any computer application which involves the numerical solution to a system of partial differential equations. A broad overview is given of verification procedures for computational simulation. The two aspects of verification examined are code verification and solution verification. Code verification is a set of procedures developed to find coding mistakes that affect the numerical discretization. The method of manufactured solutions combined with order of accuracy verification is recommended for code verification, and this procedure is described in detail. Solution verification is used to estimate the numerical errors that occur in every computational simulation.
Both round-off and iterative convergence errors are discussed, and a posteriori methods for estimating the discretization error are examined. Emphasis is placed on discretization error estimation methods based on Richardson extrapolation as they are equally applicable to any numerical method. Additional topics covered include calculating the observed order of accuracy, error bands, and practical aspects of mesh refinement.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65G20 Algorithms with automatic result verification 65Y20 Complexity and performance of numerical algorithms 35Q53 KdV equations (Korteweg-de Vries equations) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
CGM
Full Text:
##### References:
 [1] Guide for the verification and validation of computational fluid dynamics simulations, American Institute of Aeronautics and Astronautics, AIAA G-077-1998, Reston, VA, 1998 [2] Roache, P.J., Verification and validation in computational science and engineering, (1998), Hermosa Publishers New Mexico [3] Knupp, P.; Salari, K., () [4] Christensen, M.J.; Thayer, R.H., The project manager’s guide to software engineering’s best practices, (2001), IEEE Computer Society Los Alamitos, CA [5] Purdy, G.N., CVS pocket reference, (2003), O’Reilly and Associates [6] Anderson, D.A.; Tannehill, J.C.; Pletcher, R.H., Computational fluid mechanics and heat transfer, (1984), Hemisphere Publishing Corp New York, pp. 70-77 [7] Roache, P.J.; Steinberg, S., Symbolic manipulation and computational fluid dynamics, Aiaa j., 22, 10, 1390-1394, (1984) · Zbl 0547.76007 [8] P.J. Roache, P.M. Knupp, S. Steinberg, R.L. Blaine, Experience with benchmark test cases for groundwater flow, in: I. Celik, C.J. Freitas (Eds.), Benchmark Test Cases for Computational Fluid Dynamics, ASME FED 93, Book No. H00598, 1990, pp. 49-56 [9] Oberkampf, W.L.; Blottner, F.G., Issues in computational fluid dynamics code verification and validation, Aiaa j., 36, 5, 687-695, (1998), (see also W.L. Oberkampf, F.G. Blottner, D.P. Aeschliman, Methodology for computational fluid dynamics code verification/validation, AIAA Paper 95-2226, 1995) [10] K. Salari, P. Knupp, Code Verification by the Method of Manufactured Solutions, SAND 2000-1444, Sandia National Laboratories, Albuquerque, NM, 2000 [11] Roache, P.J., Code verification by the method of manufactured solutions, J. fluids eng., 124, 1, 4-10, (2002) [12] Roy, C.J.; Nelson, C.C.; Smith, T.M.; Ober, C.C., Verification of Euler/Navier-Stokes codes using the method of manufactured solutions, Int. J. numer. meth. fluids, 44, 6, 599-620, (2004) · Zbl 1067.76580 [13] T.G. Trucano, M.M. Pilch, W.L. Oberkampf, “On the Role of Code Comparisons in Verification and Validation,” SAND 2003-2752, Sandia National Laboratories, Albuquerque, NM, 2003 [14] Roy, C.J.; Blottner, F.G., Assessment of one- and two-equation turbulence models for hypersonic transitional flows, J. spacecraft rockets, 38, 5, 699-710, (2001) [15] Roy, C.J.; Blottner, F.G., Methodology for turbulence model validation: application to hypersonic flows, J. spacecraft rockets, 40, 3, 313-325, (2003) [16] Ferziger, J.H.; Peric, M., Further discussion of numerical errors in CFD, Int. J. numer. meth. fluids, 23, 12, 1263-1274, (1996) · Zbl 0892.76073 [17] J.R. Stewart, A posteriori error estimation for predictive models, Presentation at the Workshop on the Elements of Predictability, Johns Hopkins University, Baltimore, Maryland, November 13-14, 2003 [18] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates, part 2: error estimates and adaptivity, Int. J. numer. meth. eng., 33, 1365-1382, (1992) · Zbl 0769.73085 [19] Z. Zhang, A. Naga, A Meshless Gradient Recovery Method, Part I: Superconvergence Property, Research Report #2, Department of Mathematics, Wayne State University, 2002 [20] Eriksson, K.; Johnson, C., Error estimates and automatic time step control for nonlinear parabolic problems, SIAM J. numer. anal., 24, 1, 12-23, (1987) · Zbl 0618.65104 [21] Babuska, I.; Miller, A., Post-processing approach in the finite element method. part 3: a posteriori error estimates and adaptive mesh selection, Int. J. numer. meth. eng., 20, 12, 2311-2324, (1984) · Zbl 0571.73074 [22] Barth, T.J.; Larson, M.G., A-posteriori error estimation for higher order Godunov finite volume methods on unstructured meshes, (), 41-63 · Zbl 1062.65092 [23] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), Wiley New York · Zbl 1008.65076 [24] Babuska, I., Accuracy estimates and adaptive refinements in finite element computations, (1986), Wiley New York · Zbl 0663.65001 [25] Richardson, L.F., The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam, Trans. royal society London, ser. A, 210, 307-357, (1910) · JFM 42.0873.02 [26] Richardson, L.F., The deferred approach to the limit, Trans. royal society London, ser. A, 226, 229-361, (1927) [27] Roache, P.J., Perspective: a method for uniform reporting of grid refinement studies, J. fluids eng., 116, 405-413, (1994) [28] C.J. Roy, Grid convergence error analysis for mixed-order numerical schemes, AIAA Paper 2001-2606, 2001 [29] L. Eca, M. Hoekstra, An evaluation of verification procedures for CFD applications, in: 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 8-13, 2002 · Zbl 1349.76604 [30] Cadafalch, J.; Perez-Segarra, C.D.; Consul, R.; Oliva, A., Verification of finite volume computations on steady-state fluid flow and heat transfer, J. fluids eng., 24, 11-21, (2002) [31] D. Pelletier, P.J. Roache, Verification and validation of computational heat transfer (Chapter 13), in: W.J. Minkowycz, E.M. Sparrow, J.Y. Murthy (Eds.), Handbook of Numerical Heat Transfer, 2nd Ed., John Wiley and Sons, Inc., Hoboken, NJ, 2005 [32] Tautges, T.J., CGM: a geometry interface for mesh generation, analysis and other application, Eng. computers, 17, 3, 299-314, (2001) · Zbl 0983.68561 [33] Carpenter, M.H.; Casper, J.H., Accuracy of shock capturing in two spatial dimensions, Aiaa j., 37, 9, 1072-1079, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.