Skip to main content
SpringerLink
Log in

Differential quadrature parallel algorithms for solving systems of convection-diffusion and reaction models

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A procedure to incorporate the domain decomposition concept over the finite difference and differential quadrature methods is presented. In this procedure, three numerical schemes are formulated for solving reaction-diffusion and convection-diffusion-reaction systems. The physical domain is subdivided, and the mesh generation is done locally in each subdomain. Numerical schemes are formulated by employing suitable combinations of differential quadrature and finite difference methods for spatial and temporal derivative approximations in each subdomain. Its essence is computing the solution at the subdomain interfaces (pseudo-boundaries) first and then the solution in the subdomains by using the estimated pseudo-boundary conditions. On a two-component reaction-diffusion system, convergence, stability, the effect of grid refinement on computational time and errors are investigated. These schemes are also extended to solve the system of Black-Scholes equations for pricing European options in a regime-switching economy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Bellman, R., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34(2), 235238 (1971)

    MathSciNet  Google Scholar 

  2. Bellman, R., Kashef, B., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Computat. Phys. 10(1), 40–52 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible navier-Stokes equations. Int. J. Numer. Methods Fluids 15(7), 791–798 (1992)

    MATH  Google Scholar 

  4. Bert, C.W., Xinwei, W., Striz, A.G.: Differential quadrature for static and free vibration analyses of anisotropic plates. Int. J. Solids Struct. 30 (13), 1737–1744 (1993)

    MATH  Google Scholar 

  5. Striz, A., Wang, X., Bert, C.: Harmonic differential quadrature method and applications to analysis of structural components. Acta Mech. 111 (1-2), 85–94 (1995)

    MATH  Google Scholar 

  6. Shu, C., Xue, H.: Explicit computation of weighting coefficients in the harmonic differential quadrature. Elsevier (1997)

  7. Shu, C., Chew, Y.: Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems. Commun. Numer. Methods Eng. 13(8), 643–653 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Korkmaz, A., Dağ, İ.: Shock wave simulations using sinc differential quadrature method. Eng. Comput. 28(6), 654–674 (2011)

    MATH  Google Scholar 

  9. Korkmaz, A., Daug, İ.: A differential quadrature algorithm for simulations of nonlinear Schrödinger equation. Comput. Math. Appl. 56(9), 2222–2234 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Liew, K., Huang, Y.: Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method. Int. J. Mech. Sci. 45(1), 95–114 (2003)

    MATH  Google Scholar 

  11. Ragb, O., Matbuly, M., Nassar, M.: Analysis of composite plates using moving least squares differential quadrature method. Appl. Math. Comput. 238, 225–236 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Başhan, A., Yağmurlu, N.M., Uçar, Y., Esen, A.: A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method. Int. J. Modern Phys. C 29(06), 1850043 (2018)

    MathSciNet  Google Scholar 

  13. Başhan, A.: A mixed algorithm for numerical computation of soliton solutions of the coupled KdV equation: Finite difference method and differential quadrature method. Appl. Math. Comput. 360, 42–57 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Başhan, A.: A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number. Physica A Stat. Mechan. Appl. 545, 123755 (2020)

    MathSciNet  Google Scholar 

  15. Başhan, A., Yağmurlu, N. M., Uçar, Y., Esen, A.: Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation. Numer. Methods Partial Diff. Equ. 37(1), 690–706 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Başhan, A., Esen, A.: Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times. Numer. Methods Partial Diff. Equ. 37(2), 1561–1582 (2021)

    MathSciNet  Google Scholar 

  17. Başhan, A.: A novel approach via mixed crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mkdv equation. Pramana 92(6), 1–17 (2019)

    Google Scholar 

  18. Başhan, A.: Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods. Math. Comput. Simul. 179, 111–125 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Aswin, V.S., Awasthi, A.: Polynomial based differential quadrature methods for the numerical solution of Fisher and Extended Fisher–Kolmogorov equations. Int. J. Appl. Comput. Math. 1–13 (2017)

  20. Shen, L., Young, D., Lo, D., Sun, C.: Local differential quadrature method for 2-d flow and forced-convection problems in irregular domains. Numerical Heat Transfer Part B: Fundamentals 55(2), 116–134 (2009)

    Google Scholar 

  21. Civan, F., Sliepcevich, C.M.: Application of differential quadrature to solution of pool boiling cavities. In: Proceedings of the Oklahoma Academy of Science, vol. 65, pp 73–78 (1985)

  22. Shu, C., Richard, B.: Parallel simulation of incompressible viscous flows by generalized differential quadrature. Comput. Syst. Eng. 3(1-4), 271–281 (1992)

    Google Scholar 

  23. Zong, Z., Lam, K.Y., Zhang, Y.: A multidomain differential quadrature approach to plane elastic problems with material discontinuity. Math. Comput. Model. 41(4-5), 539–553 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Striz, A.G., Weilong, C., Bert, C.W.: Static analysis of structures by the quadrature element method (QEM). Int. J. Solids Struct. 31(20), 2807–2818 (1994)

    MATH  Google Scholar 

  25. Wang, X., Gu, H.: Static analysis of frame structures by the differential quadrature element method. Int. J. Numer. Methods Eng. 40(4), 759–772 (1997)

    MATH  Google Scholar 

  26. Liu, F.-L., Liew, K.: Static analysis of reissner-Mindlin plates by differential quadrature element method. J. Appl. Mechan. 65(3), 705–710 (1998)

    Google Scholar 

  27. Zhong, H., He, Y.: A note on incorporation of domain decomposition into the differential quadrature method. Int. J. Numer. Methods Biomed. Eng. 19(4), 297–306 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Chen, C.-N.: Discrete Element Analysis Methods of Generic Differential Quadratures, vol. 25. Springer, Chicago (2008)

    Google Scholar 

  29. Wang, X.: Novel differential quadrature element method for vibration analysis of hybrid nonlocal euler–Bernoulli beams. Appl. Math. Lett. 77, 94–100 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Torabi, K., Afshari, H., Aboutalebi, F.H.: A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions. Comput. Math. Appl. 67(3), 527–541 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Zhong, H., Yu, T.: A weak form quadrature element method for plane elasticity problems. Appl. Math. Model. 33(10), 3801–3814 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Zhong, H., Pan, C., Yu, H.: Buckling analysis of shear deformable plates using the quadrature element method. Appl. Math. Model. 35(10), 5059–5074 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Shen, Z., Zhong, H.: Static and vibrational analysis of partially composite beams using the weak-form quadrature element method. Math. Probl. Eng. 2012 (2012)

  34. Jin, C., Wang, X.: Weak form quadrature element method for accurate free vibration analysis of thin skew plates. Comput. Math. Appl. 70(8), 2074–2086 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Yuan, S., Du, J.: Upper bound limit analysis using the weak form quadrature element method. Appl. Math. Model. 56, 551–563 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Eftekhari, S., Khani, M.: A coupled finite element-differential quadrature element method and its accuracy for moving load problem. Appl. Math. Model. 34(1), 228–237 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Aswin, V.S., Awasthi, A., Rashidi, M.M.: A differential quadrature based numerical method for highly accurate solutions of Burgers’ equation. Numerical Methods for Partial Differential Equations (2017)

  38. Wu, X., Shen, Y.: Differential quadrature domain decomposition method for a class of parabolic equations. Comput. Math. Appl. 48(12), 1819–1832 (2004)

    MathSciNet  MATH  Google Scholar 

  39. Wu, X., Liu, S.: Differential quadrature domain decomposition method for problems on a triangular domain. Numer. Methods Partial Diff. Equ. 21(3), 574–585 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Aswin, V.S., Awasthi, A., Anu, C.: A comparative study of numerical schemes for convection-diffusion equation. Procedia Eng. 127, 621–627 (2015)

    Google Scholar 

  41. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    MathSciNet  MATH  Google Scholar 

  42. Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42(2), 281–300 (1987)

    MATH  Google Scholar 

  43. Constantinides, G.M.: Financial Derivatives: Futures, Forwards, Swaps, Options, Corporate Securities, And Credit Default Swaps. World Scientific Lecture Notes In Economics. World Scientific Publishing Company. https://books.google.co.in/books?id=sBi3CgAAQBAJ (2014)

  44. Duan, J.-C., Popova, I., Ritchken, P., et al.: Option pricing under regime switching. Quant. Finance 2(116–132), 209 (2002)

    MathSciNet  MATH  Google Scholar 

  45. Jeanne, O., Masson, P.: Currency crises, sunspots and Markov-switching regimes. J. Int Econom. 50(2), 327–350 (2000)

    Google Scholar 

  46. Cerra, V., Saxena, S.C.: Did output recover from the Asian crisis? IMF Staff. Pap. 52(1), 1–23 (2005)

    Google Scholar 

  47. Hamilton, J.D.: Rational-expectations econometric analysis of changes in regime: an investigation of the term structure of interest rates. J. Econ. Dyn. Control. 12(2-3), 385–423 (1988)

    MathSciNet  MATH  Google Scholar 

  48. Zhu, S. -P., Badran, A., Lu, X.: A new exact solution for pricing European options in a two-state regime-switching economy. Comput. Math. Appl. 64(8), 2744–2755 (2012). https://doi.org/10.1016/j.camwa.2012.08.005

    MathSciNet  MATH  Google Scholar 

  49. Tauryawati, M.L., Imron, C., Putri, E.R.: Finite volume method for pricing European call option with regime-switching volatility. In: Journal of Physics: Conference Series, vol. 974, p 012024. IOP Publishing (2018)

  50. Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mechan. Rev. 49(1), 1–28 (1996)

    Google Scholar 

  51. Shu, C.: Differential Quadrature and Its Application in Engineering, 1st edn. Springer, Berlin (2000). https://books.google.co.in/books?id=wMFOpGzv1-IC

    MATH  Google Scholar 

  52. Polyanin, A.D.: EqWorld. http://eqworld.ipmnet.ru/en/solutions/syspde/spde0101.pdf (2004)

  53. Chen, W.: Differential quadrature method and its applications in engineering. Department of Mechanical Engineering Shanghai Jiao Tong University (1996)

  54. Ersoy, O., Dag, I.: Numerical solutions of the reaction diffusion system by using exponential cubic b-spline collocation algorithms. Open Physics 13(1) (2015)

  55. Sahin, A.: Numerical Solutions of the Reaction-Diffusion Equations with B-Spline Finite Element Method. PhD thesis, Department of Mathematics. Eskişehir Osmangazi University (2009)

  56. Yatim, S., Asnor, A., Ibrahim, Z.: Stability region of two-point variable step–block backward differentiation formulae. J. Algorithm Comput. Technol. 11(2), 192–198 (2017)

    MathSciNet  Google Scholar 

  57. Guo, X.: Information and option pricings. Quant. Finance 1(1), 38–44 (2001)

    MathSciNet  MATH  Google Scholar 

  58. Fuh, C.-D., Ho, K.W.R., Hu, I., Wang, R. -H.: Option pricing with markov switching. J. Data Sci. 10(3), 483–509 (2012)

    MathSciNet  Google Scholar 

  59. Zeng, X.-C., Guo, I., Zhu, S. -P.: Pricing European options on regime-switching assets: a comparative study of Monte Carlo and finite-difference approaches. The ANZIAM J. 59(2), 183–199 (2017)

    MathSciNet  MATH  Google Scholar 

  60. Yuen, F.L., Yang, H.: Option pricing with regime switching by trinomial tree method. J. Comput. Appl. Math. 233(8), 1821–1833 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the anonymous reviewers for their productive inputs, comments, and suggestions. In addition, authors are grateful to Mr. Raghu H Venkatesh of the Center for Development of Advanced Computing in Bangalore, India, for his guidance during this project.

Funding

The authors received financial support from the Kerala State Council for Science, Technology and Environment (KSCSTE) and The Council of Scientific and Industrial Research (CSIR), India.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Calicut, Calicut, 673601, Kerala, India

    Aswin V.S., Riyasudheen T.K. & Ashish Awasthi

Authors
  1. Aswin V.S.
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riyasudheen T.K.
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ashish Awasthi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashish Awasthi.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

V.S., A., T.K., R. & Awasthi, A. Differential quadrature parallel algorithms for solving systems of convection-diffusion and reaction models. Numer Algor 93, 321–346 (2023). https://doi.org/10.1007/s11075-022-01416-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01416-6

Keywords

Search

Navigation