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Improving the precision of discrete numerical solutions using the generalized integral transform technique

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Abstract

Most spatial discretizations applied by finite difference and volume methods to advective/convective terms introduce significant diffusive and dispersive numerical errors, where the former is often required to improve the numerical stability to the resulting scheme. This paper presents a new approach that employs integral transforms to improve the accuracy of available discrete numerical solutions. The method uses a known and inaccurate numerical solution to filter the original governing equations, solves the resulting problem using integral transforms, and then uses it to correct the original numerical solution. Before doing so, these numerical solutions are rewritten as a series representation, which is done using the same eigenfunction basis employed by the integral transformation. Different test cases, based on the unsteady and one-dimensional wave motion described by the nonlinear viscous Burgers’ equation in a finite domain, are selected to evaluate the proposed approach. These test cases involve numerical solutions with different types of errors, and are compared with a highly accurate numerical solution for verification. The integral transform technique, in combination with different numerical filters, showed that inaccurate numerical solutions can in fact be corrected by this methodology, as very satisfactory errors were generally obtained. This is true for smooth enough numerical solutions, i.e., the procedure is not advisable to problems that lead to discontinuous numerical solutions.

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References

  1. Almeida AR, Cotta RM (1999) On the integral transform solution of convection–diffusion problems within unbounded domain. J Franklin Inst 336(5):821–832

    Google Scholar 

  2. Altun AH, Bilir Ş, Ateş A (2016) Transient conjugated heat transfer in thermally developing laminar flow in thick walled pipes and minipipes with time periodically varying wall temperature boundary condition. Int J Heat Mass Transf 92:643–657

    Google Scholar 

  3. Carpenter MH, Gottlieb D, Abarbanel S (1993) The stability of numerical boundary treatments for compact high-order finite-difference schemes. J Comput Phys 108(2):272–295

    MATH  Google Scholar 

  4. Cotta RM (1993) Integral transforms in computational heat and fluid flow. CRC Press, Boca Raton

    MATH  Google Scholar 

  5. Cotta RM (1994) Benchmark results in computational heat and fluid flow: the integral transform method. Int J Heat Mass Transf 37:381–393

    MATH  Google Scholar 

  6. Cotta RM (1996) Integral transforms in transient convection: benchmarks and engineering simulations. In: ICHMT international symposium on transient convective heat transfer, pp 433–453

  7. Cotta RM (1998) The integral transform method in thermal and fluids sciences and engineering. Begell House Publishers, Danbury

    MATH  Google Scholar 

  8. Cotta RM, Mikhailov MD (1997) Heat conduction: lumped analysis, integral transforms, symbolic computation. Wiley, Chichester

    Google Scholar 

  9. Cotta RM, Naveira-Cotta CP, Knupp DC (2018) Convective eigenvalue problems for convergence enhancement of eigenfunction expansions in convection–diffusion problems. J Therm Sci Eng Appl 10(2):021009

    Google Scholar 

  10. Cotta RM, Naveira-Cotta CP, Knupp DC, Zotin JLZ, Pontes PC, Almeida AP (2017) Recent advances in computational–analytical integral transforms for convection–diffusion problems. Heat Mass Transf 54:2475–2496

    Google Scholar 

  11. Da Silva EF, Cotta RM (1996) Benchmark results for internal forced convection through integral transformation. Int Commun Heat Mass Transf 23(7):1019–1029

    Google Scholar 

  12. Gottlieb S, Shu C (1998) Total variation diminishing Runge–Kutta schemes. Math Comput 67:73–85

    MATH  Google Scholar 

  13. Guerrero JSP, Cotta RM (1996) Benchmark integral transform results for flow over a backward-facing step. Comput Fluids 25(5):527–540

    MATH  Google Scholar 

  14. Knupp DC, Mascouto FS, Abreu LAS, Naveira-Cotta CP, Cotta RM (2018) Conjugated heat transfer in circular microchannels with slip flow and axial diffusion effects. Int Commun Heat Mass Transf 91:225–233

    Google Scholar 

  15. Laney CB (1998) Computational gasdynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  16. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42

    MATH  Google Scholar 

  17. Liu XD, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115(1):200–212

    MATH  Google Scholar 

  18. Macedo EN, Cotta RM, Orlande HRB (1999) Local-instantaneous filtering in the integral transform solution of nonlinear diffusion problems. Comput Mech 23(5):524–532

    MATH  Google Scholar 

  19. Mikhailov MD, Ozisik MN (1984) Unified analysis and solutions of heat and mass diffusion. Wiley, New York, p 524. ISBN: 0-471-89830-9

    Google Scholar 

  20. Nyhoff LR, Leestma S (1997) Fortran 90 for engineers and scientists. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  21. Özişik MN (1993) Heat conduction, 2nd edn. Wiley Interscience, New York

    Google Scholar 

  22. Özkan F, Wörner M, Wenka A, Soyhan HS (2007) Critical evaluation of cfd codes for interfacial simulation of bubble-train flow in a narrow channel. Int J Numer Methods Fluids 55(6):537–564

    MATH  Google Scholar 

  23. Pinheiro IF, Serrano HL, Sphaier LA, Peixoto FC, Silva VNH (2019) Integral transform analysis of heat and mass diffusion in chemically reacting systems with Michaelis–Menten kinetics. Int Commun Heat Mass Transf 100:20–26

    Google Scholar 

  24. Pinheiro IF, Sphaier LA, Alves LSdB (2018) Integral transform solution of integro-differential equations in conduction–radiation problems. Numer Heat Transf A Appl 73(2):94–114

    Google Scholar 

  25. Pletcher RH, Tannehill JC, Anderson D (2012) Computational fluid mechanics and heat transfer. CRC Press, Boca Raton

    MATH  Google Scholar 

  26. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) 1992. numerical recipes in fortran 77: the art of scientific computing

  27. Santos CAC, Quaresma JNN, Lima JA (2001) Convective heat transfer in ducts: the integral transform approach. Editora E-papers

  28. Shukla RK, Zhong X (2005) Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation. J Comput Phys 204(2):404–429

    MATH  Google Scholar 

  29. Wesseling P (1995) Introduction to multigrid methods. Technical Report, ICASE

  30. Wolfram S (2003) The mathematica book, 5th edn. Wolfram Media/Cambridge University Press, New York/Champaign

    MATH  Google Scholar 

  31. Zhong X, Tatineni M (2003) High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition. J Comput Phys 190(2):419–458

    MATH  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support provided by the Brazilian Government Funding Agencies, CAPES, CNPq, and FAPERJ.

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Authors and Affiliations

  1. PGMEC/TEM/UFF, Rua Passo da Pátria 156, Bloco E, Niterói, RJ, 24210-240, Brazil

    Isabela F. Pinheiro, Ricardo D. Santos, Leandro A. Sphaier & Leonardo S. de B. Alves

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  1. Isabela F. Pinheiro
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  2. Ricardo D. Santos
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  3. Leandro A. Sphaier
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  4. Leonardo S. de B. Alves
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Correspondence to Leonardo S. de B. Alves.

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Pinheiro, I.F., Santos, R.D., Sphaier, L.A. et al. Improving the precision of discrete numerical solutions using the generalized integral transform technique. J Braz. Soc. Mech. Sci. Eng. 42, 329 (2020). https://doi.org/10.1007/s40430-020-02346-x

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  • DOI: https://doi.org/10.1007/s40430-020-02346-x

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