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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

An Unconditionally Stable Numerical Algorithm for Two-Dimensional Convection-Diffusion-Reaction Equations

Journal of Applied Nonlinear Dynamics 13(1) (2024) 37--47 | DOI:10.5890/JAND.2024.03.004

Chinedu Nwaigwe

Department of Mathematics, Rivers State University, Nigeria

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Abstract

An unconditionally stable finite difference scheme is developed and fully analyzed for two-dimensional convection-diffusion-reaction equations with nonlinear coefficients and external sources. The well-known difficulty in obtaining a stable second-order discretization of the convection term is resolved by first adopting a central discretization in space. The semi-discrete convection term is split into the positive and negative parts. Then a non-standard time-integration is proposed - by treating one part implicitly and the other explicitly. This allowed to maintain numerical stability, retain compactness of the stencil and avoid division by the diffusion coefficient, hence can be applied to purely convection (zero diffusion) problems. The solvability, consistency and stability of the scheme are established. Numerical results are provided to verify the theoretical properties. [\hfill Convection-diffusion equations\par \hfill Nonlinear coefficients\par \hfill Convergence analysis\par \hfill Stability\par \hfill Consistency\par \hfill Implicit scheme\par \hfill Two-sided bounds\par][Antonio Lopes][6 May 2022][25 October 2022][1 January 2024][2024 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction}\label{sec-intro} \begin{figure

References

  1. [1] Hernandez-Coronado, H., Coronado, M., and Castillo-Negrete, D. D. (2017), On the anisotropic advection-diffusion equation with time dependent coefficients, Revista mexicana de f{\isica}, 63(1), 40-48.
  2. [2] Zhang, Z., Johnson, D.L., and Schwartz, L.M. (2011), Simulating the time-dependent diffusion coefficient in mixed-pore-size materials, Physical Review E, 84(3), 031129.
  3. [3] Shvidler, M. and Karasaki, K. (2003), Exact averaging of stochastic equations for transport in random velocity field, Transport in porous media, 50(3), 223-241.
  4. [4] Numbere, D.T. and Erkal, A. (1998), A model for tracer flow in heterogeneous porous media, In SPE Asia Pacific Conference on Integrated Modelling for Asset Management, Society of Petroleum Engineers, 39705.
  5. [5] Gallouët, T., Hérard, J.M., and Seguin, N. (2004), Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Mathematical Models and Methods in Applied Sciences, 14(05), 663-700.
  6. [6] Kuzmin, D. (2010), A guide to numerical methods for transport equations, University Erlangen-Nuremberg.
  7. [7] Tirabassi, T., Vilhena, M.T., Buske, D., and Degrazia, G.A. (2013), An analytical air pollution model with time dependent eddy diffusivity, Journal of Environmental Protection, 4(8), 16.
  8. [8] Costa, C.P., Tirabassi, T., Vilhena, M., and Moreira, D.M. (2012), A general formulation for pollutant dispersion in the atmosphere, Journal of Engineering Mathematics, 74(1), 159-173.
  9. [9] Latour, L.L., Svoboda, K., Mitra, P.P., and Sotak, C.H. (1994), Time-dependent diffusion of water in a biological model system, Proceedings of the National Academy of Sciences, 91(4), 1229-1233.
  10. [10] Latour, L.L., Mitra, P.P., Kleinberg, R.L., and Sotak, C.H. (1993), Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio, Journal of Magnetic Resonance, Series A, 101(3), 342-346.
  11. [11] Dremkova, E. and Ehrhardt, M. (2011), A high-order compact method for nonlinear Black-Scholes option pricing equations of American options, International Journal of Computer Mathematics, 88(13), 2782-2797.
  12. [12] Le Minh, H., Truong, T.H.H., and Dang, N.H.T. (2019), Finite-difference scheme for initial boundary value problems in financial mathematics, VNU Journal of Science: Mathematics-Physics, 35(4).
  13. [13] Hyong-Chol, O., Jo, J., and Kim, J., (2016), General properties of solutions to inhomogeneous Black--Scholes equations with discontinuous maturity payoffs, Journal of Differential Equations, 260(4), 3151-3172.
  14. [14] Hanh, T.T.H. and Thanh, D.N.H. (2020), Monotone finite-difference schemes with second order approximation based on regularization approach for the Dirichlet boundary problem of the Gamma equation, IEEE Access, 8, 45119-45132.
  15. [15] Toro, E.F. (2001), Shock-capturing methods for free-surface shallow flows, Wiley-Blackwell.
  16. [16] Toro, E.F. (1999), {Riemann solvers and numerical methods for fluid dynamics: a practical introduction}, Springer Science $\&$ Business Media.
  17. [17] Leveque, R.J. (2002), {Finite volume methods for hyperbolic problems}, Cambridge University Press.
  18. [18] Chinyoka, T. and Makinde, D.O. (2015), Unsteady and porous media flow of reactive non-Newtonian fluids subjected to buoyancy and suction/injection, International Journal of Numerical Methods for Heat $\&$ Fluid Flow, 25(7), 1682-1704.
  19. [19] Weli, A. and Nwaigwe, C. (2020), Numerical analysis of channel flow with velocity-dependent suction and nonlinear heat source, Journal of Interdisciplinary Mathematics, 23(5), 987-1008.
  20. [20] Ferziger, J.H., Perić, M., and Street, R.L. (2002), Computational Methods for Fluid Dynamics (Vol. 3, pp. 196-200), Berlin: springer.
  21. [21] Anderson, J.D. and Wendt, J. (1995), Computational Fluid Dynamics (Vol. 206, p. 332), New York: McGraw-Hill.
  22. [22] Versteeg, H.K. and Malalasekera, W. (2007), An introduction to computational fluid dynamics: the finite volume method, Pearson education.
  23. [23] Nwaigwe, C. and Orji, C. (2019), Second-order non-oscillatory scheme for simulating a pressure-driven flow, Journal of the Nigerian Association of Mathematical Physics, 52(1), 53-58.
  24. [24] Nwaigwe, C. and Weli, A. (2019), Analysis of two finite difference schemes for a channel flow problem, Asian Research Journal of Mathematics, 15, 1-14.
  25. [25] Shukla, A., Singh, A.K., and Singh, P. (2011), A recent development of numerical methods for solving convection-diffusion problems, Applied Mathematics, 1(1), 1-12.
  26. [26] Shu, C.W. (2009), High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM review, 51(1), 82-126.
  27. [27] Shen, C., Qiu, J.M., and Christlieb, A. (2011), Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations, Journal of Computational Physics, 230(10), 3780-3802.
  28. [28] Zhong, X. and Shu, C. W. (2013), A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics, 232(1), 397-415.
  29. [29] Balsara, D.S., Garain, S., and Shu, C.W. (2016), An efficient class of WENO schemes with adaptive order, Journal of Computational Physics, 326, 780-804.
  30. [30] Shu, C.W. (2016), High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments, Journal of Computational Physics, 316, 598-613.
  31. [31] Lefèvre, V., Garnica, A., and Lopez-Pamies, O. (2019), A WENO finite-difference scheme for a new class of Hamilton-Jacobi equations in nonlinear solid mechanics, Computer Methods in Applied Mechanics and Engineering, 349, 17-44.
  32. [32] Zhu, J., Qiu, J., and Shu, C. W. (2020), High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters, Journal of Computational Physics, 404, 109105.
  33. [33] Matus, P. and Vulkov, L. G. (2017), Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations, Journal of Computational and Applied Mathematics, 310, 186-199.
  34. [34] Matus, P. (2014), On convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions, Computational Methods in Applied Mathematics, 14(3), 361-371.
  35. [35] Samarskii, A.A. (2001), The theory of difference schemes, Marcel Dekker, New York.
  36. [36] Chen, J. and Ge, Y. (2018), High order locally one-dimensional methods for solving two-dimensional parabolic equations, Advances in Difference Equations, 2018(1), 1-17.
  37. [37] Nwaigwe, C. (2016), Coupling methods for 2d/1d shallow water flow models for flood simulations, (Doctoral dissertation, University of Warwick).

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