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Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this article, we consider the computational cost reduction of approximating a coupled system of time variant multiscale parameterized problems with mixed type conditions, in addition to the mathematical convergence analysis of this approximation. A triangular splitting based additive approach on equidistributed partition points are considered to reduce the computational cost. The objective of the discretization includes optimal quadratic approximation at the interior points of the domain and preserves this rate of accuracy for the mixed type boundary conditions. Convergence analysis on an adaptive mesh, generated by a specially chosen monitor function, shows that the present approach provides uniform linear accuracy in time and uniform quadratic accuracy in space for parabolic systems. Numerical experiments based on triangular splitting (diagonal or triangular forms) of the reaction matrix in comparison to its coupled form, strongly validate the optimal accurate approximation with reduced computational cost.

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Code and data availability

For the sake of reproducibility, all computations are carried out using Matlab. Code can be obtained from the author Sumit Saini, upon request.

References

  1. Thomas, G.: Towards an improved turbulence model for wave-current interactions. In: 2nd Annual Report to EU MAST-III Project The Kinematics and Dynamics of Wave-Current Interactions, Contract No MAS3-CT95-0011 (1998)

  2. Madden, N., Stynes, M., Thomas, G.: On the application of robust numerical methods to a complete-flow wave-current model. In: Proceedings of BAIL 2004 (2004)

  3. Shishkin, G.I.: Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations. Comput. Math. Math. Phys. 4(35), 429–446 (1995)

    MATH  Google Scholar 

  4. Barenblatt, G.I., Zheltov, I.P., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)

    Article  MATH  Google Scholar 

  5. Joshi, N., Lustri, C.J.: Generalized solitary waves in a finite-difference Korteweg-de Vries equation. Stud. Appl. Math. 142, 359–384 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kolokolnikov, T., Ward, M., Wei, J.: Pulse-splitting for some reaction-diffusion systems in one-space dimension (English summary). Stud. Appl. Math. 114, 115–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Miller, P.D.: Applied Asymptotic Analysis. American Mathematical Society, Providence (2006)

    Book  MATH  Google Scholar 

  8. Sun, W., Tang, T., Ward, M., Wei, J.: Numerical challenges for resolving spike dynamics for two one-dimensional reaction-diffusion systems. Stud. Appl. Math. 111, 41–84 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuehn, C.: Multiple Time Scale Dynamics. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  10. Das, P., Mehrmann, V.: Numerical solution of singularly perturbed parabolic convection-diffusion-reaction problems with two small parameters. BIT 56, 51–76 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Debela, H.G., Duressa, G.F.: Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition. Int. J. Numer. Methods Fluids 92, 12 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Vigo-Aguiar, J., Ramos, H.: A family of A-stable Runge-Kutta collocation methods of higher order for initial-value problems. IMA J. Numer. Anal. 27(4), 798–817 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Simos, T.E., Vigo-Aguiar, J.: A symmetric high order method with minimal phase-lag for the numerical solution of the Schrodinger equation. Int. J. Mod. Phys. C. 12(7), 1035–1042 (2001)

    Article  MATH  Google Scholar 

  14. Das, P., Rana, S.: Theoretical prospects of the solutions of fractional order weakly singular Volterra integro differential equations and their approximations with convergence analysis. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7369

    Article  MATH  Google Scholar 

  15. Amiraliyev, I.G., Amiraliyev, G.M.: Uniform difference method for parameterized singularly perturbed delay differential equations. Numer. Algorithms 52, 509–521 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ansari, A.R., Hegarty, A.F.: Numerical solution of a convection diffusion problem with Robin boundary conditions. J. Comput. Appl. Math. 156(1), 221–238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ramos, H., Vigo-Aguiar, J.: A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems. Int. J. Comput. Math. 85(3–4), 603–611 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ramos, H., Vigo-Aguiar, J., Natesan, S., García-Rubio, R., Queiruga, M.A.: Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm. J. Math. Chem. 48(1), 38–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Christy Roja, J., Tamilselvan, A., Geetha, N.: An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations. Int. J. Numer. Methods Fluids 92, 6 (2019)

    MathSciNet  Google Scholar 

  20. Das, P., Rana, S., Vigo-Aguiar, J.: Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature. Appl. Numer. Math. 148, 79–97 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, W., Ren, Y., Russell, R.D.: Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31, 709–730 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods. Springer, New York (2011)

    Book  MATH  Google Scholar 

  23. Das, P., Vigo-Aguiar, J.: Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. J. Comput. Appl. Math. 354, 533–544 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Clavero, C., Gracia, J.L.: A high order HODIE finite difference scheme for 1d parabolic singularly perturbed reaction diffusion problems. Appl. Math. Comput. 218(9), 5067–5080 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Clavero, C., Gracia, J.L.: Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction diffusion systems. Comput. Math. Appl. 67(3), 655–670 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Shakti, D., Mohapatra, J., Das, P., Vigo-Aguiar, J.: A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction-diffusion problems with arbitrary small diffusion terms. J. Comput. Appl. Math. 404, 113167 (2022). https://doi.org/10.1016/j.cam.2020.113167

    Article  MathSciNet  MATH  Google Scholar 

  27. Kumar, S., Sumit, H.R.: Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions. Appl. Math. Comput. (2021). https://doi.org/10.1016/j.amc.2020.125677

    Article  MathSciNet  MATH  Google Scholar 

  28. Kumar, S., Vigo-Aguiar, J.: A parameter-uniform grid equidistribution method for singularly perturbed degenerate parabolic convection diffusion problems. J. Comput. Appl. Math. 404, 113273 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions. Comput. Methods Appl. Math. 2(1), 3–25 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ishwariya, R., Miller, J.J., Valarmathi, S.: Parameter uniform essentially first order convergence of a fitted mesh method for a class of parabolic singularly perturbed Robin problem for a system of reaction-diffusion equations. Int. J. Biomath. 12(01), 1950001 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Das, P.: An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer. Algorithms 81, 465–487 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Das, P.: A higher order difference method for singularly perturbed parabolic partial differential equations. J. Differ. Equ. Appl. 24(3), 452–477 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Das, P.: Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J. Comput. Appl. Math. 290, 16–25 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gracia, J.L., Lisbona, F.J., O’Riordan, E.: A coupled system of singularly perturbed parabolic reaction-diffusion equations. Adv. Comput. Math. 32(1), 43 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Springer, Berlin (2009)

    MATH  Google Scholar 

  36. Beckett, G., Mackenzie, J.: On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution. J. Comput. Appl. Math. 131(1–2), 381–405 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. de Boor, C.: Good approximation by splines with variable knots. In: Spline Functions and Approximation Theory, pp. 57–72. Springer, Berlin (1973)

  38. Xu, X., Huang, W., Russell, R., Williams, J.: Convergence of de Boor’s algorithm for the generation of equidistributing meshes. IMA J. Numer. Anal. 31(2), 580–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Huang, W., Zheng, L., Zhan, X.: Adaptive moving mesh methods for simulating one-dimensional groundwater problems with sharp moving fronts. Int. J. Numer. Methods Eng. 54, 11 (2002)

    Article  MATH  Google Scholar 

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Funding

This research is supported by the Science and Engineering Research Board (SERB) under the Project Grant No. MTR/2021/000797 for the author Pratibhamoy Das.

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Author notes

  1. Pratibhamoy Das and Sunil Kumar contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, M.P., 462003, India

    Sumit Saini

  2. Department of Mathematics, Indian Institute of Technology, Patna, Bihar, 801106, India

    Pratibhamoy Das

  3. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, 221005, India

    Sunil Kumar

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  1. Sumit Saini
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  3. Sunil Kumar
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Saini, S., Das, P. & Kumar, S. Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 66 (2023). https://doi.org/10.1007/s13398-023-01397-8

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  • DOI: https://doi.org/10.1007/s13398-023-01397-8

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