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A POD reduced-order model based on spectral Galerkin method for solving the space-fractional Gray–Scott model with error estimate

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Abstract

This paper deals with developing a fast and robust numerical formulation to simulate a system of fractional PDEs. At the first stage, the time variable is approximated by a finite difference method with first-order accuracy. At the second stage, the spectral Galerkin method based upon the fractional Jacobi polynomials is employed to discretize the spatial variables. We apply a reduced-order method based upon the proper orthogonal decomposition technique to decrease the utilized computational time. The unconditional stability property and the order of convergence of the new technique are analyzed in detail. The proposed numerical technique is well known as the reduced-order spectral Galerkin scheme. Furthermore, by employing the Newton–Raphson method and semi-implicit schemes, the proposed method can be used for solving linear and nonlinear ODEs and PDEs. Finally, some examples are provided to confirm the theoretical results.

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References

  1. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    MATH  Google Scholar 

  2. Alber M, Glimm T, Hentschel H, Kazmierczak B, Zhang Y-T, Zhu J, Newman SA (2008) The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bull Math Biol 70(2):460–483

    MathSciNet  MATH  Google Scholar 

  3. Pavelchak I (2011) A numerical method for determining the localized initial condition in the Fitzhugh–Nagumo and Aliev–Panfilov models. Mosc Univ Comput Math Cybern 35(3):105

    MathSciNet  Google Scholar 

  4. Madzvamuse A, Maini PK, Wathen AJ (2005) A moving grid finite element method for the simulation of pattern generation by turing models on growing domains. J Sci Comput 24(2):247–262

    MathSciNet  MATH  Google Scholar 

  5. Coronel-Escamilla A, Gómez-Aguilar J, Torres L, Escobar-Jiménez R (2018) A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A Stat Mech Appl 491:406–424

    MathSciNet  Google Scholar 

  6. Bueno-Orovio A, Kay D, Burrage K (2014) Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer Math 54(4):937–954

    MathSciNet  MATH  Google Scholar 

  7. Owolabi KM, Patidar KC (2014) Numerical solution of singular patterns in one-dimensional Gray–Scott-like models. Int J Nonlinear Sci Numer Simul 15(7–8):437–462

    MathSciNet  MATH  Google Scholar 

  8. Wang T, Song F, Wang H, Karniadakis GE (2019) Fractional Gray–Scott model: well-posedness, discretization, and simulations. Comput Methods Appl Mech Eng 347:1030–1049

    MathSciNet  MATH  Google Scholar 

  9. Saxena RK, Mathai AM, Haubold HJ (2014) Space-time fractional reaction-diffusion equations associated with a generalized Riemann–Liouville fractional derivative. Axioms 3(3):320–334

    MATH  Google Scholar 

  10. Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222

    MathSciNet  MATH  Google Scholar 

  11. Hendy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-020-01095-8

    Article  Google Scholar 

  12. Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Equ 22:558–576

    MathSciNet  MATH  Google Scholar 

  13. Roop JP (2004) Variational solution of the fractional advection dispersion equation, Ph.D thesis, Clemson University

  14. Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations. Springer, New York

    MATH  Google Scholar 

  15. Mao Z, Karniadakis GE (2018) A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J Numer Anal 56(1):24–49

    MathSciNet  MATH  Google Scholar 

  16. Luo Z, Li H, Zhou Y, Xie Z (2012) A reduced finite element formulation based on POD method for two-dimensional solute transport problems. J Math Anal Appl 385(1):371–383

    MathSciNet  MATH  Google Scholar 

  17. Hamid M, Usman M, Zubair T, Haq RU, Wang W (2019) Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative. Eur Phys J Plus 134:484. https://doi.org/10.1140/epjp/i2019-12871-y

    Article  Google Scholar 

  18. Hamid M, Usman M, Zubair T, Mohyud-Din ST (2018) Comparison of Lagrange multipliers for telegraph equations. Ain Shams Eng J 9:2323–2328

    Google Scholar 

  19. Hamid M, Zubair T, Usma M, Haq RU (2019) Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel. AIMS Math 4(5):1416–1429

    MathSciNet  MATH  Google Scholar 

  20. Usman M, Hamid M, Zubair T, Haq RU, Wang W, Liu MB (2020) Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials. Appl Math Comput 372:124985

    MathSciNet  MATH  Google Scholar 

  21. Usman M, Hamid M, Khalid MSU, Ul Haq R, Liu M (2020) A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics. Numer Methods Partial Differ Equ. https://doi.org/10.1002/num.22492 (In press)

    Article  MathSciNet  Google Scholar 

  22. Hamid M, Usman M, Haq RU, Wang W (2020) A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Phys A Stat Mech Appl 551:124227

    MathSciNet  MATH  Google Scholar 

  23. Zaky MA, Hendy AS, Macias-Diaz JE (2020) Semi-implicit GalerkinLegendre spectral schemes for nonlinear time-space fractional diffusionreaction equations with smooth and nonsmooth solutions. J Sci Comput 82:1–27

    MathSciNet  Google Scholar 

  24. Zaky MA, Ameen IG (2020) A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra–Fredholm integral equations. Numer Algorithms 84:63–89

    MathSciNet  MATH  Google Scholar 

  25. Everson R, Sirovich L (1995) Karhunen–Loeve procedure for gappy data. J Opt Soc Am A 12(8):1657–1664

    Google Scholar 

  26. Dimitriu G, Stefanescu R, Navon IM (2015) POD-DEIM approach on dimension reduction of a multi-species host-parasitoid system. Acad Rom Sci 7(1):173–188

    MathSciNet  MATH  Google Scholar 

  27. Ravindran S (2000) Reduced-order adaptive controllers for fluid flows using POD. J Sci Comput 15(4):457–478

    MathSciNet  MATH  Google Scholar 

  28. Cao Y, Zhu J, Navon IM, Luo Z (2007) A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int J Numer Methods Fluids 53(10):1571–1583

    MATH  Google Scholar 

  29. Wang Y, Navon IM, Wang X, Cheng Y (2016) 2D Burgers equation with large Reynolds number using POD/DEIM and calibration. Int J Numer Methods Fluids 82(12):909–931

    MathSciNet  Google Scholar 

  30. Xiao D, Fang F, Pain CC, Navon IM, Salinas P, Muggeridge A (2015) Non-intrusive reduced order modeling of multi-phase flow in porous media using the POD-RBF method. J Comput Phys 1:1–25

    Google Scholar 

  31. Xiao D, Fang F, Buchan AG, Pain CC, Navon IM, Du J, Datum GH (2014) Non-linear model reduction for the Navier–Stokes equations using residual DEIM method. J Comput Phys 263:1–18

    MathSciNet  MATH  Google Scholar 

  32. Xiao D, Fang F, Du J, Pain C, Navon I, Buchan A, ElSheikh A, Hu G (2013) Non-linear Petrov–Galerkin methods for reduced order modelling of the Navier–Stokes equations using a mixed finite element pair. Comput Methods Appl Mech Eng 255:147–157

    MathSciNet  MATH  Google Scholar 

  33. Xiao D, Fang F, Pain CC, Datum GH (2015) Non-intrusive reduced-order modelling of the Navier–Stokes equations based on RBF interpolation. Int J Numer Methods Fluids 79:580–595

    MathSciNet  MATH  Google Scholar 

  34. Zhang P, Zhang XH, Xiang H, Song L (2016) A fast and stabilized meshless method for the convection-dominated convection–diffusion problems. Numer Heat Transf Part A Appl 70(4):420–431

    Google Scholar 

  35. Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16–30

    MathSciNet  MATH  Google Scholar 

  36. Xiao D (2019) Error estimation of the parametric non-intrusive reduced order model using machine learning. Comput Methods Appl Mech Eng 355:513–534

    MathSciNet  MATH  Google Scholar 

  37. Abbaszadeh M (2019) Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl Math Lett 88:179–185

    MathSciNet  MATH  Google Scholar 

  38. Abbaszadeh M, Dehghan M (2020) A POD-based reduced-order Crank–Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation. Appl Numer Math 158:271–291

    MathSciNet  MATH  Google Scholar 

  39. Abbaszadeh M, Dehghan M, Navon IM (2020) A proper orthogonal decomposition variational multiscale meshless interpolating element free Galerkin method for incompressible magnetohydrodynamics flow. Int J Numer Methods Fluids. https://doi.org/10.1002/fld.4834 (In press)

    Article  MathSciNet  Google Scholar 

  40. Dehghan M, Abbaszadeh M, Khodadadian A, Heitzinger C (2019) Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift–Hohenberg equation. Int J Numer Methods Heat Fluid Flow 29:2642–2665

    Google Scholar 

  41. Dehghan M, Abbaszadeh M (2018) An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation. Eng Anal Bound Elem 92:244–256

    MathSciNet  MATH  Google Scholar 

  42. Dehghan M, Abbaszadeh M (2018) A reduced proper orthogonal decomposition (POD) element free Galerkin (POD-EFG) method to simulate two-dimensional solute transport problems and error estimate. Appl Numer Math 126:92–112

    MathSciNet  MATH  Google Scholar 

  43. Dehghan M, Abbaszadeh M (2018) A combination of proper orthogonal decomposition discrete empirical interpolation method (PODDEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination. Comput Math Appl 75(4):1390–1412

    MathSciNet  MATH  Google Scholar 

  44. Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann Rev Fluid Mech 25(1):539–575

    MathSciNet  Google Scholar 

  45. Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: meshless interpolating element free Galerkin (IEFG) and finite element methods. Eng Anal Bound Elem 64:205–221

    MathSciNet  MATH  Google Scholar 

  46. Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703–717

    MathSciNet  MATH  Google Scholar 

  47. Bu W, Tang Y, Yang J (2014) Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J Comput Phys 276:26–38

    MathSciNet  MATH  Google Scholar 

  48. Bu W, Tang Y, Wu Y, Yang J (2015) Crank–Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh–Nagumo monodomain model. Appl Math Comput 257:355–364

    MathSciNet  MATH  Google Scholar 

  49. Zhuang P, Liu F, Turner I, Gu Y (2014) Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Appl Math Model 38(15–16):3860–3870

    MathSciNet  MATH  Google Scholar 

  50. Zhao Z, Li C (2012) Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219(6):2975–2988

    MathSciNet  MATH  Google Scholar 

  51. Deng W (2008) Finite element method for the space and time fractional Fokker–Planck equation. SIAM J Numer Anal 47(1):204–226

    MathSciNet  MATH  Google Scholar 

  52. Dehghan M, Abbaszadeh M (2017) A finite element method for the numerical solution of Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng Comput 33:587–605

    Google Scholar 

  53. Caputo M (1995) Mean fractional-order-derivatives differential equations and filters. Annali dellUniversita di Ferrara 41:73–84

    MathSciNet  MATH  Google Scholar 

  54. Mainardi F, Mura A, Gorenflo R, Stojanovic M (2007) The two forms of fractional relaxation of distributed order. J Vib Control 13:1249–1268

    MathSciNet  MATH  Google Scholar 

  55. Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424–438

    MathSciNet  MATH  Google Scholar 

  56. Bhrawy A, Zaky M (2017) An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl Numer Math 111:197–218

    MathSciNet  MATH  Google Scholar 

  57. Li Y, Sheng H, Chen YQ (2010) On distributed order low-pass filter. In: Proceedings of 2010 IEEE/ASME international conference on mechatronic and embedded systems and applications IEEE, Qingdao, China. https://doi.org/10.1109/MESA.2010.5552095

    Book  Google Scholar 

  58. Pindza E, Owolabi KM (2016) Fourier spectral method for higher order space fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simul 40:112–128

    MathSciNet  MATH  Google Scholar 

  59. Yuan Z, Nie Y, Liu F, Turner I, Zhang G, Gu Y (2016) An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach. Appl Math Model 40(17–18):7816–7829

    MathSciNet  MATH  Google Scholar 

  60. Jia J, Wang H (2016) A fast finite volume method for conservative space-fractional diffusion equations in convex domains. J Comput Phys 310:63–84

    MathSciNet  MATH  Google Scholar 

  61. Fan W, Liu F, Jiang X, Turner I (2017) A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Fract Calculus Appl Anal 20(2):352–383

    MathSciNet  MATH  Google Scholar 

  62. Zhao X, Sun Z-Z, Hao Z-P (2014) A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J Sci Comput 36(6):A2865–A2886

    MathSciNet  MATH  Google Scholar 

  63. Bu W, Tang Y, Wu Y, Yang J (2015) Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J Comput Phys 293:264–279

    MathSciNet  MATH  Google Scholar 

  64. Atangana A, Shafiq A (2019) Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 127:226–243

    MathSciNet  MATH  Google Scholar 

  65. Ma Z-P (2013) Stability and hopf bifurcation for a three-component reaction-diffusion population model with delay effect. Appl Math Model 37(8):5984–6007

    MathSciNet  MATH  Google Scholar 

  66. Luo Z, Li H, Sun P, An J, Navon IM (2013) A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems. Math Comput Simul 89:50–68

    MathSciNet  MATH  Google Scholar 

  67. Luo Z, Chen J, Zhu J, Wang R, Navon I (2007) An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model. Int J Numer Methods Fluids 55(2):143–161

    MathSciNet  MATH  Google Scholar 

  68. Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transfer 84:729–739

    Google Scholar 

  69. Pearson JE (1993) Complex patterns in a simple system. Science 261:189–192

    Google Scholar 

  70. Kochubei AN (2008) Distributed order calculus and equations of ultraslow diffusion. J Math Anal Appl 340:252–281

    MathSciNet  MATH  Google Scholar 

  71. Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transf 84:729–739

    Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave, Tehran, 15914, Iran

    Mostafa Abbaszadeh & Mehdi Dehghan

  2. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306-4120, USA

    Ionel Michael Navon

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  1. Mostafa Abbaszadeh
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  2. Mehdi Dehghan
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  3. Ionel Michael Navon
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Correspondence to Mostafa Abbaszadeh.

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Abbaszadeh, M., Dehghan, M. & Navon, I.M. A POD reduced-order model based on spectral Galerkin method for solving the space-fractional Gray–Scott model with error estimate. Engineering with Computers 38, 2245–2268 (2022). https://doi.org/10.1007/s00366-020-01195-5

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