10.1007/s40096-022-00496-8

An algorithm for the Burgers’ equation using barycentric collocation method with a high-order exponential Lie-group scheme

  • Muaz Seydaoğlu*1,
  1. Department of Mathematics, Faculty of Art and Science, Muş Alparslan University, Muş, 49100, TR

Published in Issue 2022-11-08

How to Cite

Seydaoğlu, M. (2022). An algorithm for the Burgers’ equation using barycentric collocation method with a high-order exponential Lie-group scheme. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00496-8
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Abstract

Abstract In order to approximate the solution of the one-dimensional Burgers’ equation, an accurate algorithm is developed on the combination of the barycentric collocation technique and a high-order group preserving method for space and time discretization, respectively. We have performed this algorithm on two different test examples for various values of viscosity parameters studied in the literature. The comparisons of the numerical results manifest the improved accuracy of the new algorithm.

Keywords

  • Burgers’ equation,
  • Barycentric interpolation,
  • Lie-Group method,
  • Group preserving scheme

References

  1. Bateman (1915) Some recent researches on the motion of fluids (pp. 163-170)
  2. Burger (1948) A mathematical model illustrating the theory of turbulence (pp. 171-199)
  3. Cordero et al. (2015) Numerical solution of turbulence problems by solving Burgers’ equation (pp. 224-233)
  4. Bonkile et al. (2018) A systematic literature review of Burgers’ equation with recent advances (pp. 1-21)
  5. Seydaoğlu et al. (2016) Numerical solution of Burgers’ equation with high order splitting methods (pp. 410-421)
  6. Baltensperger et al. (1999) Exponential convergence of a linear rational interpolant between transformed Chebyshev points (pp. 1109-1120)
  7. Berrut et al. (2005) Recent developments in Barycentric rational interpolation (pp. 27-51)
  8. Berrut and Kliein (2014) Recent advances in linear Barycentric rational interpolation (pp. 95-107)
  9. Kliein and Berrut (2012) Linear rational finite differences from derivatives of Barycentric rational interpolants (pp. 643-656)
  10. Ma et al. (2016) A meshless collocation approach with Barycentric rational interpolation for two-dimensional hyperbolic telegraph equation (pp. 236-248)
  11. Liu et al. (2018) Barycentric interpolation collocation method for solving the coupled viscous Burgers’ equations (pp. 2162-2173)
  12. Oruç (2021) Two meshless methods based on pseudo spectral delta-shaped basis functions and Barycentric rational interpolation for numerical solution of modified Burgers equation (pp. 461-479)
  13. Elgindy and Dahy (2018) High-order numerical solution of viscous Burgers’ equation using a Cole-Hopf Barycentric Gegenbauer integral pseudospectral method (pp. 6226-6251)
  14. Dahy and Elgindy (2022) High-order numerical solution of viscous Burgers’ equation using an extended Cole-Hopf Barycentric Gegenbauer integral Pseudospectral method (pp. 446-464)
  15. Kapoor and Joshi (2022) Numerical solution of coupled 1D Burgers’ equation by employing Barycentric Lagrange interpolation basis function based differential quadrature method (pp. 263-283)
  16. Mittal and Rohila (2022) A numerical study of the Burgers’ and Fisher’s equations using Barycentric interpolation method https://doi.org/10.1108/HFF-03-2022-0166
  17. Zhao et al. (2022) Barycentric rational collocation method for Burgers’ equation https://doi.org/10.1155/2022/2177998
  18. Li (2022) Linear Barycentric rational collocation method for solving non-linear partial differential equations (pp. 1-19)
  19. Iserles, A., Quispel, G.R.W.: Why Geometric Numerical Integration? Discrete Mechanics, Geometric Integration and Lie-Butcher Series, Springer Proceedings in Mathematics and Statistics
  20. 267
  21. , pp. 805–882 (2018)
  22. Hairer et al. (2006) Springer-Verlag
  23. Blanes and Casas (2016) CRC Press
  24. Liu (2001) Cone of non-linear dynamical system and group preserving schemes (pp. 1047-1068)
  25. Liu (2013) A method of Lie-symmetry GL(n,R)documentclass[12pt]{minimal}
  26. usepackage{amsmath}
  27. usepackage{wasysym}
  28. usepackage{amsfonts}
  29. usepackage{amssymb}
  30. usepackage{amsbsy}
  31. usepackage{mathrsfs}
  32. usepackage{upgreek}
  33. setlength{oddsidemargin}{-69pt}
  34. begin{document}$$GL(n, R)$$end{document} for solving non-linear dynamical systems (pp. 85-95)
  35. Lee and Liu (2009) The fourth-order group preserving methods for the integrations of ordinary differential equations (pp. 1-26)
  36. Chen (2016) High order implicit and explicit Lie-group schemes for solving backward heat conduction problems (pp. 1016-1029)
  37. Abbasbandy and Hajiketabi (2021) A simple, efficient and accurate new Lie-group shooting method for solving nonlinear boundary value problems (pp. 761-781)
  38. Abbasbandy and Hajiketabi (2022) A fictitious time Lie-group integrator for the Brinkman–Forchheimer momentum equation modeling flow of fully developed forced convection (pp. 1563-1563)
  39. Polat and Oruç (2021) A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg-de Vries (KdV) equation
  40. Liu (2006) A group preserving scheme for Burgers’ equation with very large Reynolds number (pp. 197-211)
  41. Liu (2006) An efficient backward group preserving scheme for the backward in time Burgers’ equation (pp. 55-65)
  42. Seydaoğlu (2019) A meshless method for Burgers’ equation using multiquadric radial basis functions with a Lie-group integrator
  43. Hajiketabi and Abbasbandy (2018) The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: application to the heat equation (pp. 36-46)
  44. Hajiketabi et al. (2018) The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers’ equation in arbitrary domains (pp. 223-243)
  45. Berrut and Trefethen (2004) Barycentric Lagrange interpolation (pp. 501-517)
  46. Salzer (1972) Lagrangian interpolation at the Chebyshev points xn,v≡cos(vπ/n),v=0(1)ndocumentclass[12pt]{minimal}
  47. usepackage{amsmath}
  48. usepackage{wasysym}
  49. usepackage{amsfonts}
  50. usepackage{amssymb}
  51. usepackage{amsbsy}
  52. usepackage{mathrsfs}
  53. usepackage{upgreek}
  54. setlength{oddsidemargin}{-69pt}
  55. begin{document}$$x_{n, v} equiv cos(vpi /n), v = 0(1)n$$end{document}; some unnoted advantages (pp. 156-159)
  56. Trefethen (2000) Siam
  57. Cole (1951) On a quasi-linear parabolic equation in aerodynamics (pp. 225-236)
  58. Hopf (1950) The partial differential equationut+uux=μuxxdocumentclass[12pt]{minimal}
  59. usepackage{amsmath}
  60. usepackage{wasysym}
  61. usepackage{amsfonts}
  62. usepackage{amssymb}
  63. usepackage{amsbsy}
  64. usepackage{mathrsfs}
  65. usepackage{upgreek}
  66. setlength{oddsidemargin}{-69pt}
  67. begin{document}$$u_t+uu_x=mu u_{xx}$$end{document} (pp. 201-230)
  68. Öziş et al. (2003) A finite element approach for solution of Burgers’ equation (pp. 417-428)
  69. Kutluay et al. (1999) Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods (pp. 251-261)
  70. Verma et al. (2020) A high-order weakly L-stable time integration scheme with an application to Burgers’ equation
  71. Jiwari et al. (2019) Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems (pp. 1142-1168)
  72. Hussain (2021) Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation (pp. 1-49)
  73. Ganaie and Kukreja (2014) Numerical solution of Burgers’ equation by cubic Hermite collocation method (pp. 571-581)
  74. Mittal and Rohila (2017) A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method (pp. 673-695)
  75. Mukundan and Awasthi (2015) Efficient numerical techniques for Burgers’ equation (pp. 282-297)