Abstract
This paper considers the construction of a generalized computational experiment for solving verification problems. The problem of comparative accuracy assessment of numerical methods is currently acquiring special relevance due to the introduction of published standards and widespread use of software packages that include a large number of different solvers. A generalized computational experiment makes it possible to obtain a numerical solution for a class of problems determined by variation ranges of their governing parameters. Analysis of results represented as multidimensional arrays, where the number of measurements depends on the dimension of the space of governing parameters, requires the use of scientific visualization and visual analytics tools. Some approaches to the application of generalized computational experiments with and without a reference solution are discussed. An example of constructing error surfaces when comparing some solvers from the OpenFOAM software package is considered. The classical problem of an inviscid oblique shock wave is used as a basic problem. Certain variations of its main parameters—Mach number and angle of attack—are analyzed. In addition, we consider an example of the cone flow problem with variable Mach number, cone angle, and angle of attack. The concept of an error index is introduced as an integral characteristic of deviations from the exact solution for each solver in the class of problems under consideration.
Similar content being viewed by others
REFERENCES
Skeel, R., Thirteen ways to estimate global error, Numerische Mathematik, 1986, vol. 48, pp. 1–20.
Roy, C.J. and Oberkampf, W.L., A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing, Comput. Methods Appl. Mech. Eng., 2011, vol. 200, nos. 25–28, pp. 2131–2144.
Repin, S., A Posteriori Estimates for Partial Differential Equations, De Gruyter, 2008, vol. 4.
Repin, S., A unified approach to a posteriori error estimation based on duality error majorants, Math. Comput. Simul., 1999, vol. 50, nos. 1–4, pp. 305–321.
Repin, S. and Frolov, M., A posteriori error estimates for approximate solutions of elliptic boundary value problems, Comput. Math. Math. Phys., 2002, vol. 42, no. 12, pp. 1704–1716.
Synge, J.L., The Hypercircle in Mathematical Physics, Cambridge University Press, 1957.
Oden, J. and Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 2001, vol. 41, pp. 735–756.
Prudhomme, S. and Oden, J., On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors, Comput. Methods Appl. Mech. Eng., 1999, vol. 176, pp. 313–331.
Ainsworth, M. and Oden, J., A Posteriori Error Estimation in Finite Element Analysis, New York: Wiley Interscience, 2000.
Carpenter, M. and Casper, J., Accuracy of shock capturing in two spatial dimensions, AIAA J., 1999, vol. 37, no. 9, pp. 1072–1079. https://doi.org/10.2514/2.835
Babuska, I. and Osborn, J., Can a finite element method perform arbitrarily badly?, Math. Comput. Am. Math. Soc., 2000, vol. 69, no. 230, pp. 443–462.
Godunov, S.K., Manuzina, Yu.D., and Nazareva, M.A., Experimental analysis of convergence of the numerical solution to a generalized solution in fluid dynamics, Comput. Math. Math. Phys., 2011, vol. 51, pp. 88–95.
Linss, T. and Kopteva, N., A posteriori error estimation for a defect-correction method applied to convection-diffusion problems, Int. J. Numer. Anal. Model., 2009, vol. 1, no. 1, pp. 1–16.
Shokin, Yu.I., Method of Differential Approximation, Springer, 1983.
Banks, J., Hittinger, J., and Woodward, C., Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport, Comput. Methods Appl. Mech. Eng., 2012, vol. 213, pp. 1–15. https://doi.org/10.1016/j.cma.2011.11.021
Rauser, F., Marotzke, J., and Korn, P., Ensemble-type numerical uncertainty quantification from single model integrations, J. Comput. Phys., 2015, vol. 292, pp. 30–42. https://doi.org/10.1016/j.jcp.2015.02.043
Johnson, C., On computability and error control in CFD, Int. J. Numer. Methods Fluids, 1995, vol. 20, pp. 777–788. https://doi.org/10.1002/fld.1650200806
Babuska, I. and Rheinboldt, W., A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng., 1978, vol. 12, pp. 1597–1615. https://doi.org/10.1002/nme.1620121010
Roy, Ch. and Raju, A., Estimation of discretization errors using the method of nearby problems, AIAA J., 2007, vol. 45, no. 6, pp. 1232–1243. https://doi.org/10.2514/1.24282
Guide for the verification and validation of computational fluid dynamics simulations, American Institute of Aeronautics and Astronautics, AIAA-G-077-1998, Reston, VA, 1998.
Standard for verification and validation in computational fluid dynamics and heat transfer, ASME V&V 20-2009, 2009.
GOST R 57700.12-2018: Numerical modeling of physical processes. Numerical modeling of supersonic inviscid gas flows. Software verification. National standard of the Russian Federation. 2018.
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. R. Soc. London: Ser. A, 2010, vol. 1908, pp. 307–357.
Roy, Ch.J., Grid convergence error analysis for mixed-order numerical schemes, AIAA J., 2003, vol. 41, no. 4, pp. 595–604.
Phillips, T.S. and Roy Ch.J., Richardson extrapolation-based discretization uncertainty estimation for computational fluid dynamics, ASME J. Fluids Eng., 2014, vol. 136, no. 12, p. 121401.
Ortner, C., A posteriori existence in numerical computations, SIAM J. Numer. Anal., 2009, vol. 47, no. 4, pp. 2550–2577.
Chernyshenko, S.I., Constantin, P., Robinson, J.C., and Titi, E.S., A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations, J. Math. Phys., 2007, vol. 48, p. 065204.
Bondarev, A., Analysis of space-time flow structures by optimization and visualization methods, Lect. Notes Comput. Sci., 2013, vol. 7870, pp. 158–168.
Bondarev, A. and Galaktionov, V., Parametric optimizing analysis of unsteady structures and visualization of multidimensional data, Int. J. Model., Simul., Sci. Comput., 2013.
Bondarev, A., On the construction of the generalized numerical experiment in fluid dynamics, Mathematica Montisnigri XLII , 2018, pp. 52–64.
Bondarev, A., On visualization problems in a generalized computational experiment, Sci. Visualization, 2019, vol. 11, no. 2, pp. 156–162. https://doi.org/10.26583/sv.11.2.12
Bondarev, A. and Kuvshinnikov, A., Analysis of the accuracy of OpenFOAM solvers for the problem of supersonic flow around a cone, Lect. Notes Comput. Sci., 2018, vol. 10862, pp. 221–230. https://doi.org/10.1007/978-3-319-93713-7_18
Bondarev, A., On the estimation of the accuracy of numerical solutions in CFD problems, Lect. Notes Comput. Sci., 2019, vol. 11540, pp. 325–333. https://doi.org/10.1007/978-3-030-22750-0_26
Bondarev, A. and Galaktionov, V., Generalized computational experiment and visual analysis of multidimensional data, Sci. Visualization, 2019, vol. 11, no. 4, pp. 102–114. https://doi.org/10.26583/sv.11.4.09
Alekseev, A., Bondarev, A., Galaktionov, V., and Kuvshinnikov, A., On the construction of a generalized computational experiment in verification problems, Matematica Montisnigri XLVIII , 2020, pp. 19–31. https://doi.org/10.20948/mathmontis-2020-48-3
OpenFOAM Foundation. http://www.openfoam.org. Accessed January 8, 2021.
Alekseev, A. and Bondarev, A., On exact solution enclosure on ensemble of numerical simulations, Mathematica Montisnigri XXXVIII , 2017, pp. 63–77.
Alekseev, A., Bondarev, A., and Kuvshinnikov, A., Verification on the ensemble of independent numerical solutions, Lect. Notes Comput. Sci., 2019, vol. 11540, pp. 315–324. https://doi.org/10.1007/978-3-030-22750-0_26
Alekseev, A. and Bondarev, A., Estimation of the distance between true and numerical solutions, Comput. Math. Math. Phys., 2019, vol. 59, no. 6, pp. 857–863. https://doi.org/10.1134/S0965542519060034
Alekseev, A., Bondarev, A., and Kuvshinnikov, A., On uncertainty quantification via the ensemble of independent numerical solutions, J. Comput. Sci., 2020, vol. 42, p. 101114. https://doi.org/10.1016/j.jocs.2020.101114
Alekseev, A., Bondarev, A., and Kuvshinnikov, A., Comparative analysis of the accuracy of OpenFOAM solvers for the oblique shock wave problem, Mathematica Montisnigri XLV , 2019, pp. 95–105. https://doi.org/10.20948/mathmontis-2019-45-8
Bondarev, A. and Kuvshinnikov, A., Parametric study of the accuracy of OpenFOAM solvers for the oblique shock wave problem, IEEE Proc. Inst. Syst. Program. Russ. Acad. Sci. Open Conf., 2019, pp. 108–112. https://doi.org/10.1109/ISPRAS47671.2019.00023
Bondarev, A.E. and Galaktionov, V.A., Multidimensional data analysis and visualization for time-dependent CFD problems, Program. Comput. Software, 2015, vol. 41, no. 5, pp. 247–252. https://doi.org/10.1134/S0361768815050023
Andreev, S.V., Bondarev, A.E., Bondarenko, A.V., Vizilter, Yu.V., Galaktionov, V.A., Gudkov, A.V., Zheltov, S.Yu., Zhukov, V.T., Ilovaiskaya, E.B., Knyaz, V.A., Manukovskii, K.V., Novikova, N.D., Ososkov, M.V., Silaev, N.Zh., and Feodoritova, O.B., A computational technology for constructing the optimal shape of a power plant blade assembly taking into account structural constraints, Program. Comput. Software, 2017, vol. 43, no. 6, pp. 345–352. https://doi.org/10.1134/S0361768817060020
Bondarev, A.E., Galaktionov, V.A., and Kuvshinnikov, A.E., Parallel solutions of parametric problems in gas dynamics using DVM/DVMH technology, Program. Comput. Software, 2020, vol. 46, no. 3, pp. 176–182. https://doi.org/10.1134/S0361768820030032
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 2000, no. 1, pp. 241–282. https://doi.org/10.1006/jcph.2000.6459
Greenshields, C., Wellerr, H., Gasparini, L., and Reese, J., Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows, Int. J. Numer. Methods Fluids, 2010, vol. 63, no. 1, pp. 1–21. https://doi.org/10.1002/fld.2069
Issa, R., Solution of the implicit discretized fluid flow equations by operator splitting, J. Comput. Phys., 1986, vol. 66, no. 1, pp. 40–65. https://doi.org/10.1016/0021-9991(86)90099-9
Kraposhin, M., Bovtrikova, A., and Strijhak, S., Adaptation of Kurganov–Tadmor numerical scheme for applying in combination with the PISO method in numerical simulation of flows in a wide range of Mach numbers, Proc. Comput. Sci., 2015, vol. 66, pp. 43–52. https://doi.org/10.1016/j.procs.2015.11.007
Kraposhin, M., Smirnova, E., Elizarova, T., and Istomina, M., Development of a new OpenFOAM solver using regularized gas dynamic equations, Comput. Fluids, 2018, vol. 166, pp. 163–175. https://doi.org/10.1016/j.compfluid.2018.02.010
Babenko, K.I., Voskresenskii, G.P., Lyubimov, A.N., and Rusanov, V.V., Prostranstvennoe obtekanie gladkikh tel ideal’nym gazom (Spatial Flow of Ideal Gas around Smooth Bodies), Moscow: Nauka, 1964.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by Yu. Kornienko
Rights and permissions
About this article
Cite this article
Alekseev, A.K., Bondarev, A.E., Galaktionov, V.A. et al. Generalized Computational Experiment and Verification Problems. Program Comput Soft 47, 177–184 (2021). https://doi.org/10.1134/S0361768821030026
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768821030026