Engineering with Computers

, Volume 33, Issue 4, pp 897–918 | Cite as

The local radial point interpolation meshless method for solving Maxwell equations

Original Article
First Online:
Received:
Accepted:

Abstract

The Maxwell equations are basic equations of electromagnetic. In this paper we employed ADI–LRPIM (alternative direction implicit method is applied for approximating the time variable and the local radial point interpolation meshless method is used for space variable) to solve the two-dimensional time dependent Maxwell equations. This method consists of two stages for each time step implemented in alternative directions which are simple in computations. Local radial point interpolation method is a type of meshless method which uses a set of nodes scattered within the domain of the problem as well as a set of nodes scattered on the boundaries of the domain instead of using a predefined mesh to represent the problem domain and its boundaries, this feature makes, LRPIM to be flexible. Also it produces acceptable results for solving many partial differential equations. The proposed method is accurate and efficient, these features are illustrated by solving numerical examples in transverse magnetic and transverse electric fields. We used a kind of finite difference scheme for approximation of derivative terms in main relations to reduce errors and computational cost and eliminate integrals of weak form on internal boundaries by suitable selection of test function.

Keywords

Time-dependent partial differential equation Maxwell equations Meshless method and local weak form Local radial point interpolation method (LRPIM) Radial basis functions Alternative direction implicit (ADI) method 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors thank one of the reviewers for his useful comments. Also authors would like to thank Mostafa Abbaszadeh for his useful suggestions and nice comments that improved the paper.

References

  1. 1.
    Abbasbandy S, Naz R, Hayat T, Alsaedi A (2014) Numerical and analytical solutions for Falkner–Skan flow of MHD Maxwell fluid. Appl Math Comput 242(1):569–575MathSciNetMATHGoogle Scholar
  2. 2.
    Abbaszadeh M, Dehghan M (2015) A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method. Comput Math Appl 70:2493–2512MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ala G, Francomano E (2015) Numerical investigations of an implicit leapfrog time-domain meshless method. J Sci Comput 62:898–912MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ala G, Francomano E (2012) An improved smoothed particle electromagnetics method in 3D time domain simulations. Int J Numer Modell Electron Netw Dev Fields 25:325–337CrossRefGoogle Scholar
  5. 5.
    Ala G, Francomano E, Ganci S (2015) Unconditionally stable meshless integration of time-domain Maxwell’s curl equations. Appl Math Comput 255:157–164MathSciNetMATHGoogle Scholar
  6. 6.
    Binns KJ, Lawrenson PJ, Trowbridge CW (1993) The analytical and numerical solution of electric and magnetic fields. Wiley, USAGoogle Scholar
  7. 7.
    Buhmann MD (2003) Radial basis functions: theory and implementations. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  8. 8.
    Burden RL, Faires JD (2010) Numerical analysis. Cengage LearnGoogle Scholar
  9. 9.
    Chari MVK, Silvester PP (1980) Finite element in electrical and magnetic field problems. Wiley, USAGoogle Scholar
  10. 10.
    Chatterjee R (2003) Antenna theory and practice, vol 1996(19). New Age International, IndiaGoogle Scholar
  11. 11.
    Dehghan M (2002) A new ADI technique for two-dimensional parabolic equation with an integral condition. Comput Math Appl 43:1477–1488MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dehghan M, Abbaszadeh M, Mohebbi A (2015) A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model. Eng Anal Bound Elem 56:129–144MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput Phys Commun 181:772–786MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dehghan M, Ghesmati A (2010) Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation. Eng Anal Bound Elem 34:324–336MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dehghan M, Salehi R (2014) A meshless local Petrov–Galerkin method for the time-dependent Maxwell equations. J Comput Appl Math 268:93–110MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16–30MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fasshauer G (2007) Meshfree approximation methods with Matlab. World Scientific, DaverseCrossRefMATHGoogle Scholar
  18. 18.
    Fasshauer G, Zhang JG (2007) On choosing “optimal” shape parameters for RBF approximation. Numer Algorithms 45:345–368MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Forsythe GE, Wasow WR (1960) Finite difference method for partial differential equations. Wiley, USAMATHGoogle Scholar
  20. 20.
    Gao L, Zhang B, Liang D (2007) The splitting finite-difference time-domain methods for Maxwell equations in two dimensions. J Comput Appl Math 205:207–230MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gustafson KE, Sethian JA (1991) Vortex methods and vortex motion. SIAM, USACrossRefMATHGoogle Scholar
  22. 22.
    Harrington RF (1993) Field computation by moment method. Oxford University Press, USACrossRefGoogle Scholar
  23. 23.
    Hayat T, Abbas Z, Sajid M (2006) Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys Lett A 358(56):396–403CrossRefMATHGoogle Scholar
  24. 24.
    Hosseinzadeh H, Dehghan M, Mirzaei D (2013) The boundary elements method for magneto-hydrodynamic (MHD) channel flows at high Hartmann number. Appl Math Modell 37:2337–2351MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ilati M, Dehghan M (2016) Remediation of contaminated groundwater by meshless local weak forms. Comput Math Appl 72:2408–2416MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Jung JH (2007) A note on the gibbs phenomenon with multiquadric radial basis functions. Appl Numer Math 57:213–229MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kaufmann T, Fumeaux C, Vahldieck R (2008) The meshless radial point interpolation method for time-domain electromagnetics. Dig IEEE MTT-S Int Microw Symp 61:15–20Google Scholar
  28. 28.
    Kaufmann T, Yu Y, Engström C, Chen Z, Fumeaux C (2012) Recent developments of the meshless radial point interpolation method for time-domain electromagnetics. Int J Numer Modell Electron Netw Dev Fields 25:468–489CrossRefGoogle Scholar
  29. 29.
    Kopriva DA (2009) Implementing Spectral methods for partial differential equations: algorithms for scientists and engineers. Springer, The NetherlandsCrossRefMATHGoogle Scholar
  30. 30.
    Lebedev AS, Fedoruk MP, Shtyrina OV (2006) Finite-volume algorithm for solving the time-dependent Maxwell equations on unstructured meshes. Comput Math Math Phys 46:1219–1233MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lee J, Fornberg B (2004) Some unconditionally stable time stepping methods for the 3D Maxwell’s equa tions. J Comput Appl Math 166:497–523MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, The NetherlandsGoogle Scholar
  33. 33.
    Liu GR, Zhang GY, Gu YT, Wang YY (2005) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech 36:421–430MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Li X (2016) Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl Numer Math 99:77–97MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Liu GR (2009) Meshfree methods: moving beyond the finite element method, vol 2. CRC Press, SingaporeCrossRefGoogle Scholar
  36. 36.
    Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific, SingaporeCrossRefMATHGoogle Scholar
  37. 37.
    Mastryukov AF, Mikhailenko BG (2010) Solving the 2D Maxwell equations by a Laguerre spectral method. Numer Anal Appl 3:118–132CrossRefGoogle Scholar
  38. 38.
    Maxwell JC (1952) A dynamical theory of the electromagnetic field. Sci Pap James Clerk Maxwell 1:528–567Google Scholar
  39. 39.
    Monk P (2003) Finite element methods for Maxwell’s equations. Clarendon Press, USACrossRefMATHGoogle Scholar
  40. 40.
    Movahhedi M, Abdipour A, Nentchev A, Dehghan M, Selberherr S (2007) Alternating-direction implicit formulation of the finite-element time-domain method. IEEE Microw Theory Tech 55:1322–1331CrossRefGoogle Scholar
  41. 41.
    Mukherjee S (2002) The boundary node method. Springer, IthacaCrossRefMATHGoogle Scholar
  42. 42.
    Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318CrossRefMATHGoogle Scholar
  43. 43.
    Onate E, Perazzo F, Miquel J (2001) A finite point method for elasticity problems. Comput Struct 79:2151–2163CrossRefGoogle Scholar
  44. 44.
    Peaceman D, Rachford H (1955) The numerical solution of elliptic and parabolic differential equations. J SIAM 3:28–41MATHGoogle Scholar
  45. 45.
    Sabouri M, Dehghan M (2015) An efficient implicit spectral element method for time-dependent nonlinear diffusion equations by evaluating integrals at one quadrature point. Comput Math Appl 70:2513–2541MathSciNetCrossRefGoogle Scholar
  46. 46.
    Sarabadan S, Shahrezaee M, Rad JA, Parand K (2014) Numerical solution of Maxwell equations using local weak form meshless techniques. J Math Comput Sci 13:168–185Google Scholar
  47. 47.
    Sengupta DL, Sarkar TK (2003) Maxwell, Hertz, the Maxwellians, and the early history of electromagnetic waves. IEEE Antennas Propag Mag 45:13–19CrossRefGoogle Scholar
  48. 48.
    Shakeri F, Dehghan M (2013) A high order finite volume element method for solving elliptic partial integro-differential equations. Appl Numer Math 65:105–118MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Shakeri F, Dehghan M (2008) The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. Comput Math Appl 56:2175–2188MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Shashkov M (1995) Conservative Finite-difference methods on general grids. CRC Press, USAGoogle Scholar
  51. 51.
    Stuben K, Trottenberg U (1982) Multigrid methods: fundamental algorithms, model problem analysis and applications. Springer, BerlinMATHGoogle Scholar
  52. 52.
    Xu J, Belytschko T Discontinuous radial basis function approximations for meshfree methods. Meshfree methods for partial differential equations II, volume 43 of the series lecture notes in computational science and engineering, pp 231–253Google Scholar
  53. 53.
    Yee K (1966) Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. Antennas Propag 14:302–307CrossRefMATHGoogle Scholar
  54. 54.
    Yu Y, Chen Z (2009) A 3-D radial point interpolation method for meshless time-domain modeling. Microw Theory Tech 57:2015–2020CrossRefGoogle Scholar
  55. 55.
    Yu Y, Chen Z (2009) Towards the development of unconditionally stable time-domain meshless numerical methods. Microw Symp Dig, pp 7–12Google Scholar
  56. 56.
    Yu Y, Chen Z (2010) Towards the development of an unconditionally stable time-domain meshless method. Microw Theory Tech 58:578–586CrossRefGoogle Scholar
  57. 57.
    Zeng F, Ma H, Liang D (2014) Energy-conserved splitting spectral method for two-dimensional Maxwell’s equations. J Comput Appl Math 265:301–321MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran

Personalised recommendations