The local radial point interpolation meshless method for solving Maxwell equations
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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.
KeywordsTime-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
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.
- 6.Binns KJ, Lawrenson PJ, Trowbridge CW (1993) The analytical and numerical solution of electric and magnetic fields. Wiley, USAGoogle Scholar
- 8.Burden RL, Faires JD (2010) Numerical analysis. Cengage LearnGoogle Scholar
- 9.Chari MVK, Silvester PP (1980) Finite element in electrical and magnetic field problems. Wiley, USAGoogle Scholar
- 10.Chatterjee R (2003) Antenna theory and practice, vol 1996(19). New Age International, IndiaGoogle Scholar
- 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
- 32.Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, The NetherlandsGoogle Scholar
- 38.Maxwell JC (1952) A dynamical theory of the electromagnetic field. Sci Pap James Clerk Maxwell 1:528–567Google Scholar
- 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
- 50.Shashkov M (1995) Conservative Finite-difference methods on general grids. CRC Press, USAGoogle Scholar
- 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
- 55.Yu Y, Chen Z (2009) Towards the development of unconditionally stable time-domain meshless numerical methods. Microw Symp Dig, pp 7–12Google Scholar