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The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic–quintic complex Ginzburg–Landau equations on complicated geometries

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Abstract

Our motivation in this contribution is to propose a new numerical algorithm for solving cubic–quintic complex Ginzburg-Landau (CQCGL) equations. The developed technique is based on the following stages. At the first step, the nonlinear CQCGL equation is splitted in the three problems that two of them don’t have the space derivative e.g problems (I) and (III) and one of them has the space derivative e.g Problem (II). At the second stage, the Problems (I) and (III) can be considered as two ODEs and they are solved by using a fourth-order exponential time differencing Runge-Kutta (ETDRK4) method to get a high-order numerical approximation. Furthermore, the Problem (II) is solved by using direct meshless finite volume method. The proposed method is a new high-order numerical procedure based on a truly meshless method for solving the complex PDEs on non-rectangular computational domains. Moreover, various samples are investigated that verify the efficiency of the new numerical scheme.

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References

  1. Abbaszadeh M, Dehghan M (2019) Meshless upwind local radial basis function-finite difference technique to simulate the time fractional distributed-order advection–diffusion equation. Eng Comput. https://doi.org/10.1007/s00366-019-00861-7

  2. Abbaszadeh M, Dehghan M (2019) The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations. Eng Anal Boundary Elem 106:300–308

    MathSciNet  MATH  Google Scholar 

  3. Abbaszadeh M, Dehghan M (2020) Direct meshless local Petrov–Galerkin (DMPLG) method for time-fractional fourth-order reaction–diffusion problem on complex domains. Comput Math Appl 79(3):876–888

    MathSciNet  MATH  Google Scholar 

  4. Abbaszadeh M, Dehghan M (2020) Investigation of the oldroyd model as a generalized incompressible Navier–Stokes equation via the interpolating stabilized element free Galerkin technique. App Numer Math 150:274–294

    MathSciNet  MATH  Google Scholar 

  5. Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C (2019) A direct meshless local collocation method for solving stochastic Cahn-Hilliard-Cook and stochastic Swift-Hohenberg equations. Eng Anal Boundary Elem 98:253–264

    MathSciNet  MATH  Google Scholar 

  6. Abbaszadeh M, Dehghan M (2020) An upwind local radial basis functions-differential quadrature (RBFs-DQ) technique to simulate some models arising in water sciences. Ocean Eng 197:106844

    Google Scholar 

  7. Agrawal GP (2000) Nonlinear fiber optics. Nonlinear Science at the Dawn of the 21st Century. Springer, New York, pp 195–211

    MATH  Google Scholar 

  8. Akhmediev N, Afanasjev V (1995) Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex Ginzburg–Landau equation. Phys Rev Lett 75(12):2320

    Google Scholar 

  9. Akhmediev N, Afanasjev V, Soto-Crespo J (1996) Singularities and special soliton solutions of the cubic–quintic complex Ginzburg–Landau equation. Phys Rev E 53(1):1190

    Google Scholar 

  10. Akhmediev N, Ankiewicz A (2008) Dissipative solitons: from optics to biology and medicine, vol 751. Springer, New York

  11. Akram G, Mahak N (2018) Application of the first integral method for solving (1+ 1) dimensional cubic–quintic complex Ginzburg–Landau equation. Optik 164:210–217

    Google Scholar 

  12. Ang W-T (2019) A boundary element approach for solving plane elastostatic equations of anisotropic functionally graded materials. Numer Methods Partial Differ Equ 35(4):1396–1411

    MathSciNet  MATH  Google Scholar 

  13. Aranson IS, Kramer L (2002) The world of the complex Ginzburg–Landau equation. Rev Mod Phys 74(1):99

    MathSciNet  MATH  Google Scholar 

  14. Atluri SN (2004) The Meshless Method (MLPG) for Domain and BIE Discretizations, Tech. Science. Pres 680

  15. Atluri SN, Shen S (2002) The meshless local Petrov–Galerkin (MLPG) method: a simple and less-costly alternative to the finite element and boundary element methods. Comput Model Eng Sci 3(1):11–51

    MathSciNet  MATH  Google Scholar 

  16. Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117–127

    MathSciNet  MATH  Google Scholar 

  17. Bao W, Jaksch D (2003) An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J Numer Anal 41(4):1406–1426

    MathSciNet  MATH  Google Scholar 

  18. Bao W, Jin S, Markowich PA (2002) On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J Comput Phys 175(2):487–524

    MathSciNet  MATH  Google Scholar 

  19. Chen L, Li X (2020) A complex variable boundary element-free method for the Helmholtz equation using regularized combined field integral equations. Appl Math Lett 101:106067

    MathSciNet  MATH  Google Scholar 

  20. Cheng H, Peng M, Cheng Y (2018) The dimension splitting and improved complex variable element-free Galerkin method for 3-dimensional transient heat conduction problems. Int J Numer Meth Eng 114(3):321–345

    MathSciNet  Google Scholar 

  21. Cheng H, Peng M, Cheng Y (2018) A hybrid improved complex variable element-free Galerkin method for three-dimensional advection-diffusion problems. Eng Anal Boundary Elem 97:39–54

    MathSciNet  Google Scholar 

  22. Cheng H, Peng M, Cheng Y (2019) Analyzing wave propagation problems with the improved complex variable element-free Galerkin method. Eng Anal Boundary Elem 100:80–87

    MathSciNet  MATH  Google Scholar 

  23. Crasovan L-C, Malomed B, Mihalache D (2000) Stable vortex solitons in the two-dimensional Ginzburg–Landau equation. Phys Rev E 63(1):016605

    Google Scholar 

  24. Degond P, Jin S, Tang M (2008) On the time splitting spectral method for the complex Ginzburg–Landau equation in the large time and space scale limit. SIAM J Sci Comput 30(5):2466–2487

    MathSciNet  MATH  Google Scholar 

  25. Dehghan M, Abbaszadeh M (2019) The simulation of some chemotactic bacteria patterns in liquid medium which arises in tumor growth with blow-up phenomena via a generalized smoothed particle hydrodynamics (GSPH) method. Eng Comput 35(3):875–892

    Google Scholar 

  26. Dehghan M, Abbaszadeh M (2017) A local meshless method for solving multi-dimensional Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems arising in plasma physics. Eng Comput 33:961–981

    Google Scholar 

  27. Dehghan M, Abbaszadeh M (2017) Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross-Pitaevskii equation. Eng Comput 33:983–996

    Google Scholar 

  28. Dehghan M, Abbaszadeh M (2017) The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations. Eng Anal Boundary Elem 78:49–64

    MathSciNet  MATH  Google Scholar 

  29. Dehghan M, Taleei A (2010) A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput Phys Commun 181(1):43–51

    MATH  Google Scholar 

  30. Goldman D, Sirovich L (1995) A novel method for simulating the complex Ginzburg-Landau equation. Q Appl Math 53(2):315–333

    MathSciNet  MATH  Google Scholar 

  31. Kadalbajoo MK, Kumar A, PatiTripathi L (2016) A radial basis function based implicit–explicit method for option pricing under jump-diffusion models. Appl Numer Math 110:159–173

    MathSciNet  MATH  Google Scholar 

  32. Kadalbajoo MK, Kumar A, Tripathi LP (2018) Radial-basis-function-based finite difference operator splitting method for pricing American options. Int J Comput Math 95:2343–2359

    MATH  Google Scholar 

  33. Kumar A, Bhardwaj A, Dubey S (2020) A local meshless method to approximate the time-fractional telegraph equation. Eng Comput. https://doi.org/10.1007/s00366-020-01006-x

  34. Kumar A, Bhardwaj A, Kumar BVR (2019) A meshless local collocation method for time fractional diffusion wave equation. Comput Math Appl 78:1851–1861

    MathSciNet  MATH  Google Scholar 

  35. Kumar A, Bhardwaj A (2020) A local meshless method for time fractional nonlinear diffusion wave equation. Numer. Algor. https://doi.org/10.1007/s11075-019-00866-9

  36. Lei Z, Yin B, Liew K (2018) Bending and vibration behaviors of matrix cracked hybrid laminated plates containing CNTR-FG layers and FRC layers. Compos Struct 184:314–326

    Google Scholar 

  37. Li X, Li S (2020) A complex variable boundary point interpolation method for the nonlinear Signorini problem. Comput Math Appl 79:3297–3309

    MathSciNet  MATH  Google Scholar 

  38. Li X, Dong H (2020) Error analysis of the meshless finite point method. Appl Math Comput 382:125326

    MathSciNet  MATH  Google Scholar 

  39. Li X (2014) Symmetric coupling of the meshless Galerkin boundary node and finite element methods for elasticity. Comput Model Eng Sci 97(6):483–507

    MathSciNet  MATH  Google Scholar 

  40. Li X, Zhu J (2009) A Galerkin boundary node method for two-dimensional linear elasticity. Comput Model Eng Sci 45:1–29

    MathSciNet  MATH  Google Scholar 

  41. Liang X, Khaliq AQ, Xing Y (2015) Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrödinger equations. Commun Comput Phys 17(2):510–541

    MathSciNet  MATH  Google Scholar 

  42. Liew K, Pan Z, Zhang L (2012) An overview of layerwise theories for composite laminates and structures: development, numerical implementation and application. Compos Struct 216(3):240–259

    Google Scholar 

  43. Liu D, Cheng Y (2019) The interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems. Eng Anal Boundary Elem 108:115–123

    MathSciNet  MATH  Google Scholar 

  44. Liu F, Cheng Y (2018) The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels. Int J Appl Mech 10(04):1850047

    Google Scholar 

  45. Liu F, Wu Q, Cheng Y (2019) A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems. Int J Appl Mech 11(01):1950006

    Google Scholar 

  46. Ilati M, Dehghan M (2017) Application of direct meshless local Petrov-Galerkin (DMLPG) method for some Turing-type models. Eng Comput 33:107–124

    Google Scholar 

  47. Mazzia A, Pini G, Sartoretto F (2012) Numerical investigation on direct MLPG for 2D and 3D potential problems. Comput Model Eng Sci (CMES) 88(3):183–209

    MathSciNet  MATH  Google Scholar 

  48. Mihalache D, Mazilu D (2009) Three-dimensional Ginzburg–Landau solitons: collision scenarios. Rom Rep Phys 61:175–189

    Google Scholar 

  49. Mihalache D, Mazilu D, Lederer F, Leblond H, Malomed B (2007) Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation. Phys Rev A 75(3):033811

    Google Scholar 

  50. Mihalache D, Mazilu D, Lederer F, Leblond H, Malomed B (2008) Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation. Phys Rev A 77(3):033817

    Google Scholar 

  51. Mirzaei D (2016) A greedy meshless local Petrov–Galerkin methodbased on radial basis functions. Numer Methods Partial Differ Equ 32(3):847–861

    MathSciNet  MATH  Google Scholar 

  52. Mirzaei D, Schaback R (2013) Direct meshless local Petrov–Galerkin (DMLPG) method: a generalized MLS approximation. Appl Numer Math 68:73–82

    MathSciNet  MATH  Google Scholar 

  53. Mirzaei D, Schaback R (2014) Solving heat conduction problems by the direct meshless local Petrov–Galerkin (DMLPG) method. Numer Algorithms 65(2):275–291

    MathSciNet  MATH  Google Scholar 

  54. Mirzaei D, Schaback R, Dehghan M (2012) On generalized moving least squares and diffuse derivatives. IMA Journal of Numerical Analysis 32(3):983–1000

    MathSciNet  MATH  Google Scholar 

  55. Ooi EH, Ooi ET, Ang WT (2015) Numerical investigation of the meshless radial basis integral equation method for solving 2D anisotropic potential problems. Eng Anal Bound Elem 53:27–39

    MathSciNet  MATH  Google Scholar 

  56. Osman M, Lu D, Khater M, Attia R (2019) Complex wave structures for abundant solutions related to the complex Ginzburg-Landau model. Optik 192:162927

    Google Scholar 

  57. Petviashvili VI, Sergeev AM (1984) Spiral solitons in active media with an excitation threshold. DoSSR 276(6):1380–1384

    Google Scholar 

  58. Qu W, Fan CM, Li X (2020) Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions. Comput Math Appl 80:13–30

    MathSciNet  MATH  Google Scholar 

  59. Ren H, Cheng Y (2012) A new element-free Galerkin method based on improved complex variable moving least-squares approximation for elasticity. International Journal of Computational Materials Science and Engineering 1(01):1250011

    Google Scholar 

  60. Ren H, Cheng Y, Zhang W (2010) An interpolating boundary element-free method (IBEFM) for elasticity problems. Sci Chin Phys Mech Astron 53(4):758–766

    Google Scholar 

  61. Sartoretto F, Mazzia A, Pini G (2014) The DMLPG meshless technique for Poisson problems. Appl Math Sci 8(164):8233–8250

    Google Scholar 

  62. Shi Y, Dai Z, Li D (2009) Application of exp-function method for 2d cubic-quintic Ginzburg-Landau equation. Appl Math Comput 210(1):269–275

    MathSciNet  MATH  Google Scholar 

  63. Skarka V, Aleksić N (2006) Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations. Phys Rev Lett 96(1):013903

    Google Scholar 

  64. Sladek J, Sladek V, Hon Y (2006) Inverse heat conduction problems by meshless local Petrov–Galerkin method. Eng Anal Boundary Elem 30(8):650–661

    MATH  Google Scholar 

  65. Taleei A, Dehghan M (2014) Direct meshless local Petrov–Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic. Comput Methods Appl Mech Eng 278:479–498

    MathSciNet  MATH  Google Scholar 

  66. Taleei A, Dehghan M (2015) An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions. Numer Methods Partial Differ Equ 31(4):1031–1053

    MathSciNet  MATH  Google Scholar 

  67. Towers I, Buryak AV, Sammut RA, Malomed BA, Crasovan L-C, Mihalache D (2001) Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation. Phys Lett A 288(5–6):292–298

    MATH  Google Scholar 

  68. Wainblat G, Malomed BA (2009) Interactions between two-dimensional solitons in the diffractive-diffusive Ginzburg–Landau equation with the cubic-quintic nonlinearity. Physica D 238(14):1143–1151

    MathSciNet  MATH  Google Scholar 

  69. Wang H (2005) Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl Math Comput 170(1):17–35

    MathSciNet  MATH  Google Scholar 

  70. Wang H (2010) An efficient Chebyshev-tau spectral method for Ginzburg–Landau-Schrödinger equations. Comput Phys Commun 181(2):325–340

    MATH  Google Scholar 

  71. Wang S, Zhang L (2013) An efficient split-step compact finite difference method for cubic–quintic complex Ginzburg–Landau equations. Comput Phys Commun 184(6):1511–1521

    MathSciNet  MATH  Google Scholar 

  72. Xu Q, Chang Q (2011) Difference methods for computing the Ginzburg–Landau equation in two dimensions. Numer Methods Partial Differ Equ 27(3):507–528

    MathSciNet  MATH  Google Scholar 

  73. Yıldırım Y, Biswas A, Khan S, Alshomrani AS, Belic MR (2020) Optical solitons with differential group delay for complex Ginzburg-Landau equation having kerr and parabolic laws of refractive index. Optik 202:163737

    Google Scholar 

  74. Yu S, Peng M, Cheng H, Cheng Y (2019) The improved element-free Galerkin method for three-dimensional elastoplasticity problems. Eng Anal Boundary Elem 104:215–224

    MathSciNet  MATH  Google Scholar 

  75. Zhang Y, Bao W, Du Q (2007) Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation. Eur J Appl Math 18(5):607–630

    MATH  Google Scholar 

  76. Zhang T, Li X (2019) Meshless analysis of Darcy flow with a variational multiscale interpolating element-free Galerkin method. Eng Anal Bound Elem 100:237–245

    MathSciNet  MATH  Google Scholar 

  77. Zhang T, Li X (2020) Variational multiscale interpolating element-free Galerkin method for the nonlinear Darcy–Forchheimer model. Comput Math Appl 79:363–377

    MathSciNet  MATH  Google Scholar 

  78. Zhang T, Li X (2020) Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems. Appl Math Comput 380:125306

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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

    Mostafa Abbaszadeh & Mehdi Dehghan

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  1. Mostafa Abbaszadeh
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Correspondence to Mostafa Abbaszadeh.

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Abbaszadeh, M., Dehghan, M. The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic–quintic complex Ginzburg–Landau equations on complicated geometries. Engineering with Computers 38, 1543–1557 (2022). https://doi.org/10.1007/s00366-020-01089-6

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