Research Article
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Year 2023, Volume: 3 Issue: 4, 297 - 317, 30.12.2023
https://doi.org/10.53391/mmnsa.1387125

Abstract

References

  • [1] Painter, K.J. Mathematical models for chemotaxis and their applications in self-organisation phenomena. Journal of Theoretical Biology, 481, 162-182, (2019).
  • [2] Bellomo, N., Outada, N., Soler, J., Tao, Y. and Winkler, M. Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences, 32(04), 713-792, (2022).
  • [3] Trelles, J.P. Pattern formation and self-organization in plasmas interacting with surfaces. Journal of Physics D: Applied Physics, 49(39), 393002, (2016).
  • [4] Van Gorder, R.A. Influence of temperature on Turing pattern formation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476(2240), 20200356, (2020).
  • [5] Garzón-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M. Turing pattern formation for reaction-convection-diffusion systems in fixed domains submitted to toroidal velocity fields. Applied Mathematical Modelling, 35(10), 4913-4925, (2011).
  • [6] Sarra, S.A. A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains. Applied Mathematics and Computation, 218(19), 9853-9865, (2012).
  • [7] Chapwanya, M., Lubuma, J.M.S. and Mickens, R.E. Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences. Computers & Mathematics with Applications, 68(9), 1071-1082, (2014).
  • [8] Yücel, H., Stoll, M. and Benner, P. A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms. Computers & Mathematics with Applications, 70(10), 2414-2431, (2015).
  • [9] Hidayat, M.I.P. Meshless finite difference method with B-splines for numerical solution of coupled advection-diffusion-reaction problems. International Journal of Thermal Sciences, 165, 106933, (2021).
  • [10] Wang, J., Tong, X. and Song, Y. Dynamics and pattern formation in a reaction-diffusionadvection mussel-algae model. Zeitschrift fur angewandte Mathematik und Physik, 73(3), 117, (2022).
  • [11] Clavero, C., Shiromani, R. and Shanthi, V. A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection-reaction-diffusion PDEs. Journal of Computational and Applied Mathematics, 436, 115422, (2024).
  • [12] Cengizci, S., U˘gur, Ö. and Natesan, S. SUPG-YZβ computation of chemically reactive convection-dominated nonlinear models. International Journal of Computer Mathematics, 100(2), 283-303, (2023).
  • [13] Uzunca, M., Karasözen, B. and Manguoğlu, M. Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations. Computers & Chemical Engineering, 68, 24-37, (2014).
  • [14] Yücel, H., Stoll, M. and Benner, P. Discontinuous Galerkin finite element methods with shockcapturing for nonlinear convection dominated models. Computers & Chemical Engineering, 58, 278-287, (2013).
  • [15] Joshi, H. and Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation With Applications, 1(2), 84-94, (2021).
  • [16] Joshi, H. and Jha, B.K. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects. International Journal of Nonlinear Sciences and Numerical Simulation, 24(6), 2383-2403, (2021).
  • [17] Joshi, H., Yavuz, M. and Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
  • [18] Brooks, A.N. and Hughes, T.J. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32(1-3), 199-259, (1982).
  • [19] Hughes, T.J.R. and Brooks A.N. A multi-dimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows, (pp. 19-35). AMD-Vol. 34, New York, USA: ASME, (1979).
  • [20] Tezduyar, T.E. and Hughes, T.J.R. Development of time-accurate finite element techniques for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. NASA Technical Report NASA-CR-204772, NASA, 510, (1982).
  • [21] Tezduyar, T. and Hughes, T. Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations. In: Proceedings of AIAA 21st Aerospace Sciences Meeting, AIAA Paper 83-0125, Reno, Nevada, 1983.
  • [22] Hughes, T.J.R. and Tezduyar, T.E. Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Computer Methods in Applied Mechanics and Engineering, 45(1-3), 217-284, (1984).
  • [23] Hughes, T.J.R., Franca, L.P. and Mallet, M. A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering, 63(1), 97-112, (1987).
  • [24] Le Beau, G.J. and Tezduyar, T.E. Finite element computation of compressible flows with the SUPG formulation. In: Advances in Finite Element Analysis in Fluid Dynamics, (pp. 21-27). FED-Vol. 123, New York, USA: ASME, (1991).
  • [25] Tezduyar, T.E. and Park, Y. Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Computer Methods in Applied Mechanics and Engineering, 59(3), 307-325, (1986).
  • [26] Tezduyar, T.E. Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces. Computers & Fluids, 36(2), 191-206, (2003).
  • [27] Tezduyar, T.E. Finite element methods for fluid dynamics with moving boundaries and interfaces. In: E Stein, RD Borst, TJR Hughes (Eds.), Encyclopedia of Computational Mechanics, Volume 3: Fluids, Wiley, (2004).
  • [28] Rispoli, F., Corsini, A. and Tezduyar, T.E. Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Computers & Fluids, 36(1), 121-126, (2007).
  • [29] Tezduyar, T.E. Computation of moving boundaries and interfaces and stabilization parameters. International Journal for Numerical Methods in Fluids, 43(5), 555-575, (2003).
  • [30] Tezduyar, T.E. and Senga, M. Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Computer Methods in Applied Mechanics and Engineering, 195(13-16), 1621-1632, (2006).
  • [31] Tezduyar, T.E. and Senga, M. SUPG finite element computation of inviscid supersonic flows with YZβ shock-capturing. Computers & Fluids, 36(1), 147-159, (2007).
  • [32] Tezduyar, T.E., Senga, M. and Vicker, D. Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZβ shock-capturing. Computational Mechanics, 38, 469-481, (2006).
  • [33] Bazilevs, Y., Calo, V.M., Tezduyar, T.E. and Hughes, T.J.R. YZβ discontinuity capturing for advection-dominated processes with application to arterial drug delivery. International Journal for Numerical Methods in Fluids, 54(6-8), 593-608, (2007).
  • [34] Cengizci, S. and Uğur, Ö. A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers. Applied Mathematics and Computation, 442, 127705, (2023).
  • [35] Cengizci, S. and Uğur, Ö. SUPG formulation augmented with YZβ shock-capturing for computing shallow-water equations. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 103(6), e202200232, (2023).
  • [36] Cengizci, S., Uğur, Ö. and Natesan, S. A SUPG formulation augmented with shock-capturing for solving convection-dominated reaction-convection-diffusion equations. Computational and Applied Mathematics, 42, 235, (2023).
  • [37] Tezduyar, T.E., Ramakrishnan, S. and Sathe, S. Stabilized formulations for incompressible flows with thermal coupling. International Journal for Numerical Methods in Fluids, 57(9), 1189- 1209, (2008).
  • [38] John, V. and Knobloch, P. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I-A review. Computer Methods in Applied Mechanics and Engineering, 196(17-20), 2197-2215, (2007).
  • [39] John, V. and Knobloch, P. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II-Analysis for P1 and Q1 finite elements. Computer Methods in Applied Mechanics and Engineering, 197(21-24), 1997-2014, (2008).
  • [40] John, V., Knobloch, P. and Novo, J. Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?. Computing and Visualization in Science, 19, 47-63, (2018).
  • [41] Ak, T., Karakoç, S.B.G. and Biswas, A. Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation. Scientia Iranica B, 24(3), 1148-1159, (2017).
  • [42] Bhowmik, S.K. and Karakoc, S.B.G. Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method. Numerical Methods for Partial Differential Equations, 35(6), 2236-2257, (2019).
  • [43] Karakoc, S.B.G., Saha, A. and Sucu, D. A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Korteweg-de Vries equation. Chinese Journal of Physics, 68, 605-617, (2020).
  • [44] Shakib, F. Finite element analysis of the compressible Euler and Navier-Stokes equations. Ph.D. Thesis, Department of Mechanical Engineering, Stanford University: California, (1988).
  • [45] Franca, L.P., Frey, S.L. and Hughes, T.J. Stabilized finite element methods: I. Application to the advective-diffusive model. Computer Methods in Applied Mechanics and Engineering, 95(2), 253-276, (1992).
  • [46] Tezduyar, T.E. Stabilized finite element formulations for incompressible flow computations. Advances in Applied Mechanics, 28, 1-44, (1991).
  • [47] Donea, J. and Huerta, A. Finite Element Methods for Flow Problems. John Wiley & Sons: England, (2003).
  • [48] Abali, B.E. Computational Reality: Solving Nonlinear and Coupled Problems in Continuum Mechanics. Advanced Structured Materials, Springer: Singapore, (2016).
  • [49] Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A. et al. The FEniCS project version 1.5. Archive of Numerical Software, 3(100), 9-23, (2015).
  • [50] Logg, A., Mardal, K.A. and Wells, G. Automated solution of differential equations by the finite element method: The FEniCS book (Vol. 84). Lecture Notes in Computational Science and Engineering, Springer-Verlag: Berlin, Heidelberg, (2012).
  • [51] Zhang, J. and Yan, G. Lattice Boltzmann simulation of pattern formation under cross-diffusion. Computers & Mathematics with Applications, 69(3), 157-169, (2015).
  • [52] Schnakenberg, J. Simple chemical reaction systems with limit cycle behaviour. Journal of Theoretical Biology, 81(3), 389-400, (1979).
  • [53] Garzón-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M. Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: First example. Applied Mathematical Modelling, 36(10), 5029-5045, (2013).
  • [54] Koppel, J.V.D., Rietkerk, M., Dankers, N. and Herman, P.M.J. Scale-dependent feedback and regular spatial patterns in young mussel beds. The American Naturalist, 165(3), E66-E77, (2005).
  • [55] Jones, D.A., Smith, H.L., Dung, L. and Ballyk, M. Effects of random motility on microbial growth and competition in a flow reactor. SIAM Journal on Applied Mathematics, 59(2), 573-596, (1998).
  • [56] Murray, J.D. Mathematical Biology II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, New York, NY, USA: Springer, (2013).
  • [57] Vanegas, J.C., Landinez, N.S. and Garzón-Alvarado, D.A. Modelo matemático de la coagulación en la interfase hueso implante dental. Revista Cubana de Investigaciones Biomédicas, 28(3), 167-191, (2009).

An enhanced SUPG-stabilized finite element formulation for simulating natural phenomena governed by coupled system of reaction-convection-diffusion equations

Year 2023, Volume: 3 Issue: 4, 297 - 317, 30.12.2023
https://doi.org/10.53391/mmnsa.1387125

Abstract

Many phenomena arising in nature, science, and industry can be modeled by a coupled system of reaction-convection-diffusion (RCD) equations. Unfortunately, obtaining analytical solutions to RCD systems is typically not possible and, therefore, usually requires the use of numerical methods. On the other hand, since solutions to RCD-type equations can exhibit rapid changes and may have boundary/inner layers, classical computational tools yield approximations polluted with physically meaningless oscillations when convection dominates the transport process. Towards that end, in order to eliminate such numerical instabilities without sacrificing accuracy, this work employs a stabilized finite element formulation, the so-called streamline-upwind/Petrov-Galerkin (SUPG) method. The SUPG-stabilized formulation is then also supplemented with the YZ$\beta$ shock-capturing mechanism to achieve higher-quality approximations around sharp gradients. A comprehensive set of numerical test experiments, including cross-diffusion systems, the Schnakenberg reaction model, and mussel-algae interactions, is considered to reveal the robustness of the proposed formulation, which we call the SUPG-YZ$\beta$ formulation. Comparisons with reported studies reveal that the proposed formulation performs quite well without introducing excessive numerical dissipation.

References

  • [1] Painter, K.J. Mathematical models for chemotaxis and their applications in self-organisation phenomena. Journal of Theoretical Biology, 481, 162-182, (2019).
  • [2] Bellomo, N., Outada, N., Soler, J., Tao, Y. and Winkler, M. Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences, 32(04), 713-792, (2022).
  • [3] Trelles, J.P. Pattern formation and self-organization in plasmas interacting with surfaces. Journal of Physics D: Applied Physics, 49(39), 393002, (2016).
  • [4] Van Gorder, R.A. Influence of temperature on Turing pattern formation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476(2240), 20200356, (2020).
  • [5] Garzón-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M. Turing pattern formation for reaction-convection-diffusion systems in fixed domains submitted to toroidal velocity fields. Applied Mathematical Modelling, 35(10), 4913-4925, (2011).
  • [6] Sarra, S.A. A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains. Applied Mathematics and Computation, 218(19), 9853-9865, (2012).
  • [7] Chapwanya, M., Lubuma, J.M.S. and Mickens, R.E. Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences. Computers & Mathematics with Applications, 68(9), 1071-1082, (2014).
  • [8] Yücel, H., Stoll, M. and Benner, P. A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms. Computers & Mathematics with Applications, 70(10), 2414-2431, (2015).
  • [9] Hidayat, M.I.P. Meshless finite difference method with B-splines for numerical solution of coupled advection-diffusion-reaction problems. International Journal of Thermal Sciences, 165, 106933, (2021).
  • [10] Wang, J., Tong, X. and Song, Y. Dynamics and pattern formation in a reaction-diffusionadvection mussel-algae model. Zeitschrift fur angewandte Mathematik und Physik, 73(3), 117, (2022).
  • [11] Clavero, C., Shiromani, R. and Shanthi, V. A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection-reaction-diffusion PDEs. Journal of Computational and Applied Mathematics, 436, 115422, (2024).
  • [12] Cengizci, S., U˘gur, Ö. and Natesan, S. SUPG-YZβ computation of chemically reactive convection-dominated nonlinear models. International Journal of Computer Mathematics, 100(2), 283-303, (2023).
  • [13] Uzunca, M., Karasözen, B. and Manguoğlu, M. Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations. Computers & Chemical Engineering, 68, 24-37, (2014).
  • [14] Yücel, H., Stoll, M. and Benner, P. Discontinuous Galerkin finite element methods with shockcapturing for nonlinear convection dominated models. Computers & Chemical Engineering, 58, 278-287, (2013).
  • [15] Joshi, H. and Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation With Applications, 1(2), 84-94, (2021).
  • [16] Joshi, H. and Jha, B.K. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects. International Journal of Nonlinear Sciences and Numerical Simulation, 24(6), 2383-2403, (2021).
  • [17] Joshi, H., Yavuz, M. and Stamova, I. Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bulletin of Biomathematics, 1(1), 24-39, (2023).
  • [18] Brooks, A.N. and Hughes, T.J. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32(1-3), 199-259, (1982).
  • [19] Hughes, T.J.R. and Brooks A.N. A multi-dimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows, (pp. 19-35). AMD-Vol. 34, New York, USA: ASME, (1979).
  • [20] Tezduyar, T.E. and Hughes, T.J.R. Development of time-accurate finite element techniques for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. NASA Technical Report NASA-CR-204772, NASA, 510, (1982).
  • [21] Tezduyar, T. and Hughes, T. Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations. In: Proceedings of AIAA 21st Aerospace Sciences Meeting, AIAA Paper 83-0125, Reno, Nevada, 1983.
  • [22] Hughes, T.J.R. and Tezduyar, T.E. Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Computer Methods in Applied Mechanics and Engineering, 45(1-3), 217-284, (1984).
  • [23] Hughes, T.J.R., Franca, L.P. and Mallet, M. A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering, 63(1), 97-112, (1987).
  • [24] Le Beau, G.J. and Tezduyar, T.E. Finite element computation of compressible flows with the SUPG formulation. In: Advances in Finite Element Analysis in Fluid Dynamics, (pp. 21-27). FED-Vol. 123, New York, USA: ASME, (1991).
  • [25] Tezduyar, T.E. and Park, Y. Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Computer Methods in Applied Mechanics and Engineering, 59(3), 307-325, (1986).
  • [26] Tezduyar, T.E. Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces. Computers & Fluids, 36(2), 191-206, (2003).
  • [27] Tezduyar, T.E. Finite element methods for fluid dynamics with moving boundaries and interfaces. In: E Stein, RD Borst, TJR Hughes (Eds.), Encyclopedia of Computational Mechanics, Volume 3: Fluids, Wiley, (2004).
  • [28] Rispoli, F., Corsini, A. and Tezduyar, T.E. Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Computers & Fluids, 36(1), 121-126, (2007).
  • [29] Tezduyar, T.E. Computation of moving boundaries and interfaces and stabilization parameters. International Journal for Numerical Methods in Fluids, 43(5), 555-575, (2003).
  • [30] Tezduyar, T.E. and Senga, M. Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Computer Methods in Applied Mechanics and Engineering, 195(13-16), 1621-1632, (2006).
  • [31] Tezduyar, T.E. and Senga, M. SUPG finite element computation of inviscid supersonic flows with YZβ shock-capturing. Computers & Fluids, 36(1), 147-159, (2007).
  • [32] Tezduyar, T.E., Senga, M. and Vicker, D. Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZβ shock-capturing. Computational Mechanics, 38, 469-481, (2006).
  • [33] Bazilevs, Y., Calo, V.M., Tezduyar, T.E. and Hughes, T.J.R. YZβ discontinuity capturing for advection-dominated processes with application to arterial drug delivery. International Journal for Numerical Methods in Fluids, 54(6-8), 593-608, (2007).
  • [34] Cengizci, S. and Uğur, Ö. A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers. Applied Mathematics and Computation, 442, 127705, (2023).
  • [35] Cengizci, S. and Uğur, Ö. SUPG formulation augmented with YZβ shock-capturing for computing shallow-water equations. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 103(6), e202200232, (2023).
  • [36] Cengizci, S., Uğur, Ö. and Natesan, S. A SUPG formulation augmented with shock-capturing for solving convection-dominated reaction-convection-diffusion equations. Computational and Applied Mathematics, 42, 235, (2023).
  • [37] Tezduyar, T.E., Ramakrishnan, S. and Sathe, S. Stabilized formulations for incompressible flows with thermal coupling. International Journal for Numerical Methods in Fluids, 57(9), 1189- 1209, (2008).
  • [38] John, V. and Knobloch, P. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I-A review. Computer Methods in Applied Mechanics and Engineering, 196(17-20), 2197-2215, (2007).
  • [39] John, V. and Knobloch, P. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II-Analysis for P1 and Q1 finite elements. Computer Methods in Applied Mechanics and Engineering, 197(21-24), 1997-2014, (2008).
  • [40] John, V., Knobloch, P. and Novo, J. Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?. Computing and Visualization in Science, 19, 47-63, (2018).
  • [41] Ak, T., Karakoç, S.B.G. and Biswas, A. Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation. Scientia Iranica B, 24(3), 1148-1159, (2017).
  • [42] Bhowmik, S.K. and Karakoc, S.B.G. Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method. Numerical Methods for Partial Differential Equations, 35(6), 2236-2257, (2019).
  • [43] Karakoc, S.B.G., Saha, A. and Sucu, D. A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Korteweg-de Vries equation. Chinese Journal of Physics, 68, 605-617, (2020).
  • [44] Shakib, F. Finite element analysis of the compressible Euler and Navier-Stokes equations. Ph.D. Thesis, Department of Mechanical Engineering, Stanford University: California, (1988).
  • [45] Franca, L.P., Frey, S.L. and Hughes, T.J. Stabilized finite element methods: I. Application to the advective-diffusive model. Computer Methods in Applied Mechanics and Engineering, 95(2), 253-276, (1992).
  • [46] Tezduyar, T.E. Stabilized finite element formulations for incompressible flow computations. Advances in Applied Mechanics, 28, 1-44, (1991).
  • [47] Donea, J. and Huerta, A. Finite Element Methods for Flow Problems. John Wiley & Sons: England, (2003).
  • [48] Abali, B.E. Computational Reality: Solving Nonlinear and Coupled Problems in Continuum Mechanics. Advanced Structured Materials, Springer: Singapore, (2016).
  • [49] Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A. et al. The FEniCS project version 1.5. Archive of Numerical Software, 3(100), 9-23, (2015).
  • [50] Logg, A., Mardal, K.A. and Wells, G. Automated solution of differential equations by the finite element method: The FEniCS book (Vol. 84). Lecture Notes in Computational Science and Engineering, Springer-Verlag: Berlin, Heidelberg, (2012).
  • [51] Zhang, J. and Yan, G. Lattice Boltzmann simulation of pattern formation under cross-diffusion. Computers & Mathematics with Applications, 69(3), 157-169, (2015).
  • [52] Schnakenberg, J. Simple chemical reaction systems with limit cycle behaviour. Journal of Theoretical Biology, 81(3), 389-400, (1979).
  • [53] Garzón-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M. Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: First example. Applied Mathematical Modelling, 36(10), 5029-5045, (2013).
  • [54] Koppel, J.V.D., Rietkerk, M., Dankers, N. and Herman, P.M.J. Scale-dependent feedback and regular spatial patterns in young mussel beds. The American Naturalist, 165(3), E66-E77, (2005).
  • [55] Jones, D.A., Smith, H.L., Dung, L. and Ballyk, M. Effects of random motility on microbial growth and competition in a flow reactor. SIAM Journal on Applied Mathematics, 59(2), 573-596, (1998).
  • [56] Murray, J.D. Mathematical Biology II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, New York, NY, USA: Springer, (2013).
  • [57] Vanegas, J.C., Landinez, N.S. and Garzón-Alvarado, D.A. Modelo matemático de la coagulación en la interfase hueso implante dental. Revista Cubana de Investigaciones Biomédicas, 28(3), 167-191, (2009).
There are 57 citations in total.

Details

Primary Language English
Subjects Finite Element Analysis , Numerical and Computational Mathematics (Other)
Journal Section Research Articles
Authors

Süleyman Cengizci 0000-0002-4345-1253

Publication Date December 30, 2023
Submission Date November 6, 2023
Acceptance Date December 9, 2023
Published in Issue Year 2023 Volume: 3 Issue: 4

Cite

APA Cengizci, S. (2023). An enhanced SUPG-stabilized finite element formulation for simulating natural phenomena governed by coupled system of reaction-convection-diffusion equations. Mathematical Modelling and Numerical Simulation With Applications, 3(4), 297-317. https://doi.org/10.53391/mmnsa.1387125


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