Errors and Uncertainties: Their Sources and Treatment

  • Christopher J. RoyEmail author
Part of the Simulation Foundations, Methods and Applications book series (SFMA)


There are numerous sources of error and uncertainty in modeling and simulation. Some of these sources arise because of inherent randomness existing in the system of interest, while others arise due to incomplete knowledge on the part of the person conducting the modeling and simulation activity. Other sources arise due to the fact that all models are imperfect reflections of reality. Finally, when models are sufficiently complex to require approximate numerical solutions (for example, when they take the form of partial differential equations), then the numerical approximations provide an additional source of error and uncertainty. This chapter discusses these different sources of error and uncertainty as well as methods to characterize and treat them. Techniques for rolling up these different uncertainty sources into a total prediction uncertainty are briefly discussed.


Model Simulation Error Uncertainty Validation Calibration Prediction 


  1. Ainsworth, M., & Oden, J. T. (2000). A posteriori error estimation in finite element analysis. New York: Wiley Interscience.CrossRefGoogle Scholar
  2. 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.Google Scholar
  3. ASME PTC 19.1-2005. (2005). Test Uncertainty.Google Scholar
  4. ASME. (2006). Guide for verification and validation in computational solid mechanics, American Society of Mechanical Engineers, ASME Standard V&V 10-2006, New York, NY.Google Scholar
  5. ASME. (2009). Standard for verification and validation in computational fluid dynamics and heat transfer, American Society of Mechanical Engineers, ASME Standard V&V 20-2009, New York, NY.Google Scholar
  6. Box, G. E. P. (1979). Robustness in the strategy of scientific model building. In R. L. Launer & G. N. Wilkinson (Eds.), Robustness in statistics (pp. 201–236). Academic Press.Google Scholar
  7. Beer, M., Ferson, S., & Kreinovich, V. (2013). Imprecise probabilities in engineering analysis. Mechanical Systems and Signal Processing, 37(1–2), 4–29.CrossRefGoogle Scholar
  8. Cao, J. (2005). Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow. Computers & Fluids, 34(8), 972–990.MathSciNetCrossRefGoogle Scholar
  9. Choudhary, A., & Roy, C. J. (2018). Verification and validation for multiphase flows. In G. H. Yeoh (Ed.), Handbook of multiphase flow science and technology. Springer. (to appear).Google Scholar
  10. Coleman, H. W., & Steele, W. G. (2009). Experimentation, validation, and uncertainty analysis for engineers (3rd ed.). New York: Wiley.CrossRefGoogle Scholar
  11. Despres, B. (2004). Lax theorem and finite-volume schemes. Mathematics of Computation, 73(247), 1203–1234.MathSciNetCrossRefGoogle Scholar
  12. Ferson, S., & Ginzburg, L. R. (1996). Different methods are needed to propagate ignorance and variability. Reliability Engineering and System Safety, 54, 133–144.CrossRefGoogle Scholar
  13. Ferson, S., & Hajagos, J. G. (2004). Arithmetic with uncertain numbers: Rigorous and (often) best possible answers. Reliability Engineering and System Safety, 85, 135–152.CrossRefGoogle Scholar
  14. Ferson, S., Oberkampf, W. L., & Ginzburg, L. (2008). Model validation and predictive capability for the thermal challenge problem. Computer Methods in Applied Mechanics and Engineering, 197, 2408–2430.CrossRefGoogle Scholar
  15. Geisser, S. (1993). Predictive inference. New York: Chapman and Hall. ISBN 0-412-03471-9.Google Scholar
  16. ISO Guide to the Expression of Uncertainty in Measurement. (1995). ISO, Geneva, Switzerland.Google Scholar
  17. Kennedy, M. C., & O’Hagan, A. (2001). Bayesian calibration of computer models. Journal of the Royal Statistical Society Series B—Statistical Methodology, 63(3), 425–450.MathSciNetCrossRefGoogle Scholar
  18. Kennedy, M. C., & O’Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1), 1–13.MathSciNetCrossRefGoogle Scholar
  19. Knupp, P. M., & Salari, K. (2003). Verification of computer codes in computational science and engineering. Boca Raton: Chapman and Hall/CRC. K. H. Rosen (ed.).Google Scholar
  20. Liu, Y., Chen, W., Arendt, P., & Huang, H.-Z. (2011). Toward a better understanding of model validation metrics. Journal of Mechanical Design, 133, 1–13.CrossRefGoogle Scholar
  21. Montgomery, D. C. (2017). Design and analysis of experiments (9th ed.). New Jersey: Wiley.Google Scholar
  22. Oberkampf, W. L., & Roy, C. J. (2010). Verification and validation in scientific computing. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  23. Oberkampf, W. L., & Smith, B. L. (2017) Assessment criteria for computational fluid dynamics model validation experiments. Journal of Verification, Validation, and Uncertainty Quantification, 2(3).Google Scholar
  24. Pierce, N. A., & Giles, M. B. (2000). Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Review, 42(2), 247–264.MathSciNetCrossRefGoogle Scholar
  25. Phillips, T. S., & Roy, C. J. (2011). Residual methods for discretization error estimation. AIAA Paper 2011–3870.Google Scholar
  26. Phillips, T. S., & Roy, C. J. (2013). A new extrapolation-based uncertainty estimator for computational fluid dynamics. AIAA Paper 2013–0260.Google Scholar
  27. Queipo, N. V., Haftka, R. T., Shyy, W., Goel, T., Vaidyanathan, R., & Tucker, P. K. (2005). Surrogate-based analysis and optimization. Progress in Aerospace Sciences, 41, 1–28.CrossRefGoogle Scholar
  28. Roache, P. J., & Steinberg, S. (1984). Symbolic manipulation and computational fluid dynamics. AIAA Journal, 22(10), 1390–1394.MathSciNetCrossRefGoogle Scholar
  29. Roache, P. J. (1994). Perspective: A method for uniform reporting of grid refinement studies. Journal of Fluids Engineering, 116, 405–413.CrossRefGoogle Scholar
  30. Roache, P. J. (2009). Fundamentals of verification and validation. Socorro, New Mexico: Hermosa Publishers.Google Scholar
  31. Roy, C. J., Nelson, C. C., Smith, T. M., & Ober, C. C. (2004). Verification of Euler/Navier–stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44(6), 599–620.CrossRefGoogle Scholar
  32. Roy, C. J. (2005). Review of code and solution verification procedures for computational simulation. Journal of Computational Physics, 205, 131–156.CrossRefGoogle Scholar
  33. Roy, C. J. (2009). Strategies for driving mesh adaptation in CFD. AIAA Paper 2009–1302.Google Scholar
  34. Roy, C. J., & Oberkampf, W. L. (2011). A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Computer Methods in Applied Mechanics and Engineering, 200(25–28), 2131–2144.MathSciNetCrossRefGoogle Scholar
  35. Roy, C. J., & Balch, M. S. (2012). A holistic approach to uncertainty quantification with application to supersonic nozzle thrust. International Journal for Uncertainty Quantification, 2(4), 363–381.MathSciNetCrossRefGoogle Scholar
  36. Roy, C. J. (2015). Verification. In B. Engquist (Ed.), Encyclopedia of applied and computational mathematics (pp. 1530–1537). Heidelberg: Springer.CrossRefGoogle Scholar
  37. Roy, C. J., & Oberkampf, W. L. (2016). Verification and Validation in computational fluid dynamics. In R. W. Johnson (Ed.), Handbook of fluid dynamics (2nd Ed). Boca Raton: CRC Press.Google Scholar
  38. Shih, T. I.-P., & Williams, B. R. (2009). Development and evaluation of an a posteriori method for estimating and correcting grid-induced errors in solutions of the navier-stokes equations. AIAA Paper 2009–1499.Google Scholar
  39. Smith, R. C. (2013). Uncertainty quantification: Theory, implementation, and applications. SIAM.Google Scholar
  40. Skeel, R. D. (1986). Thirteen ways to estimate global error. Numerische Mathematik, 48, 1–20.MathSciNetCrossRefGoogle Scholar
  41. Stetter, H. J. (1978). The defect correction principle and discretization methods. Numerische Mathematik, 29, 425–443.MathSciNetCrossRefGoogle Scholar
  42. Stern, F., Wilson, R. V., Coleman, H. W., & Paterson, E. G. (2001). Comprehensive approach to verification and validation of cfd simulations—part 1: Methodology and procedures. Journal of Fluids Engineering, 123, 793–802.CrossRefGoogle Scholar
  43. Stewart, J. R., & Hughes, T. J. R. (1998). A tutorial in elementary finite element error analysis: A systematic presentation of a priori and a posteriori error estimates. Computer Methods in Applied Mechanics and Engineering, 158(1–2), 1–22.MathSciNetCrossRefGoogle Scholar
  44. Strang, G. (1986). Introduction to applied mathematics. Wellesley-Cambridge Press.Google Scholar
  45. Venditti, D. A., & Darmofal, D. L. (2000). Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one dimensional flow. Journal of Computational Physics, 164, 204–227.MathSciNetCrossRefGoogle Scholar
  46. Venditti, D. A., & Darmofal, D. L. (2003). Anisotropic grid adaptation for functional outputs: Application to two-dimensional viscous flows. Journal of Computational Physics, 187, 22–46.CrossRefGoogle Scholar
  47. Zhang, X. D., Trepanier, J.-Y., & Camarero, R. (2000). A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 185(1), 1–19.MathSciNetCrossRefGoogle Scholar
  48. Zienkiewicz, O. C., & Zhu, J. Z. (1992). The Superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33, 1365–1382.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Crofton Department of Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA

Personalised recommendations