Vol. 96
Latest Volume
All Volumes
PIERB 97 [2022] PIERB 96 [2022] PIERB 95 [2022] PIERB 94 [2021] PIERB 93 [2021] PIERB 92 [2021] PIERB 91 [2021] PIERB 90 [2021] PIERB 89 [2020] PIERB 88 [2020] PIERB 87 [2020] PIERB 86 [2020] PIERB 85 [2019] PIERB 84 [2019] PIERB 83 [2019] PIERB 82 [2018] PIERB 81 [2018] PIERB 80 [2018] PIERB 79 [2017] PIERB 78 [2017] PIERB 77 [2017] PIERB 76 [2017] PIERB 75 [2017] PIERB 74 [2017] PIERB 73 [2017] PIERB 72 [2017] PIERB 71 [2016] PIERB 70 [2016] PIERB 69 [2016] PIERB 68 [2016] PIERB 67 [2016] PIERB 66 [2016] PIERB 65 [2016] PIERB 64 [2015] PIERB 63 [2015] PIERB 62 [2015] PIERB 61 [2014] PIERB 60 [2014] PIERB 59 [2014] PIERB 58 [2014] PIERB 57 [2014] PIERB 56 [2013] PIERB 55 [2013] PIERB 54 [2013] PIERB 53 [2013] PIERB 52 [2013] PIERB 51 [2013] PIERB 50 [2013] PIERB 49 [2013] PIERB 48 [2013] PIERB 47 [2013] PIERB 46 [2013] PIERB 45 [2012] PIERB 44 [2012] PIERB 43 [2012] PIERB 42 [2012] PIERB 41 [2012] PIERB 40 [2012] PIERB 39 [2012] PIERB 38 [2012] PIERB 37 [2012] PIERB 36 [2012] PIERB 35 [2011] PIERB 34 [2011] PIERB 33 [2011] PIERB 32 [2011] PIERB 31 [2011] PIERB 30 [2011] PIERB 29 [2011] PIERB 28 [2011] PIERB 27 [2011] PIERB 26 [2010] PIERB 25 [2010] PIERB 24 [2010] PIERB 23 [2010] PIERB 22 [2010] PIERB 21 [2010] PIERB 20 [2010] PIERB 19 [2010] PIERB 18 [2009] PIERB 17 [2009] PIERB 16 [2009] PIERB 15 [2009] PIERB 14 [2009] PIERB 13 [2009] PIERB 12 [2009] PIERB 11 [2009] PIERB 10 [2008] PIERB 9 [2008] PIERB 8 [2008] PIERB 7 [2008] PIERB 6 [2008] PIERB 5 [2008] PIERB 4 [2008] PIERB 3 [2008] PIERB 2 [2008] PIERB 1 [2008]
2022-09-25
Divergence Error Based p-Adaptive Discontinuous Galerkin Solution of Time-Domain Maxwell's Equations
By
Progress In Electromagnetics Research B, Vol. 96, 153-172, 2022
Abstract
⋅A p-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the p-adaptive method. The nature of divergence error is explored, and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost compared to a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout. Numerical results are presented to show performance of the proposed divergence error based p-adaptive solutions. It is shown that an accuracy similar to that of uniformly higher order solutions is obtained in terms of the scattering width, using fewer degrees of freedom.
Citation
Apurva Tiwari Avijit Chatterjee , "Divergence Error Based p-Adaptive Discontinuous Galerkin Solution of Time-Domain Maxwell's Equations," Progress In Electromagnetics Research B, Vol. 96, 153-172, 2022.
doi:10.2528/PIERB22080403
http://www.jpier.org/PIERB/pier.php?paper=22080403
References

1. Yee, K., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, 302-307, 1966.

2. Hagness, S. C., A. Taflove, and S. D. Gedney, "Finite-difference time-domain methods," Handbook of Numerical Analysis, Vol. 13, No. 4, 199-315, 2005.

3. Shankar, V., A. H. Mohammadian, and W. F. Hall, "A time-domain, finite-volume treatment for the Maxwell equations," Electromagnetics, Vol. 10, No. 1-2, 127-145, 1990.

4. Hesthaven, J. S. and T. Warburton, Nodal Discontinuous Galerkin Methods, Texts in Applied Mathematics, Vol. 54, Springer, New York, NY, 2008.

5. Assous, F., et al., "On a finite-element method for solving the three-dimensional Maxwell equations," Journal of Computational Physics, Vol. 109, No. 2, 222-237, 1993.

6. Munz, C.-D., et al., "Divergence correction techniques for Maxwell solvers based on a hyperbolic model," Journal of Computational Physics, Vol. 161, No. 2, 484-511, 2000.

7. Cockburn, B., F. Li, and C.-W. Shu, "Locally divergence-free discontinuous Galerkin methods for the Maxwell equations," Journal of Computational Physics, Vol. 194, No. 2, 588-610, 2004.

8. Li, F. and C.-W. Shu, "Locally divergence-free discontinuous galerkin methods for MHD equations," Journal of Scientific Computing, Vol. 22, 413-442, 2005.

9. Chandrashekar, P., "A global divergence conforming DG method for hyperbolic conservation laws with divergence constraint," Journal of Scientific Computing, Vol. 79, No. 1, 79-102, 2019.

10. Li, F. and L. Xu, "Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations," Journal of Computational Physics, Vol. 231, No. 6, 2655-2675, 2012.

11. Yakovlev, S., L. Xu, and F. Li, "Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations," Journal of Computational Science, Vol. 4, No. 1-2, 80-91, 2013.

12. Cioni, J. P., L. Fezoui, and H. Steve, "A parallel time-domain maxwell solver using upwind schemes and triangular meshes," IMPACT of Computing in Science and Engineering, Vol. 5, No. 3, 215-247, 1993.

13. Kompenhans, M., et al., "Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation," Journal of Computational Physics, Vol. 306, 216-236, December 2016.

14. Rueda-Ramírez, A. M., et al., "Truncation error estimation in the p-anisotropic discontinuous Galerkin spectral element method," Journal of Scientific Computing, Vol. 78, No. 1, 433-466, 2019.

15. Kompenhans, M., et al., "Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods," Computers & Fluids, Vol. 139, 36-46, 2016.

16. Chatterjee, A., "A multilevel numerical approach with application in time-domain electromagnetics," Communications in Computational Physics, Vol. 17, No. 3, 703-720, 2015.

17. Chatterjee, A. and R. S. Myong, "Efficient implementation of higher-order finite volume time-domain method for electrically large scatterers," Progress In Electromagnetics Research B, Vol. 17, 233-254, 2009.

18. Moxey, D., et al., "Towards p-adaptive spectral/hp element methods for modelling industrial flows," Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016, ed. by M. L. Bittencourt, N. A. Dumont, and J. S. Hesthaven, 63{79, Springer International Publishing, Cham, 2017.

19. Abarbanel, S. and D. Gottlieb, "On the construction and analysis of absorbing layers in CEM," Applied Numerical Mathematics, Vol. 27, No. 4, 331-340, 1998.

20. Tόth, G., "The ∇ ⋅ B = 0 constraint in shock-capturing magnetohydrodynamics codes," Journal of Computational Physics, Vol. 161, No. 2, 605-652, 2000.

21. Brackbill, J. U. and D. C. Barnes, "The effect of Nonzero ∇ ⋅ B on the numerical solution of the magnetohydrodynamic equations," Journal of Computational Physics, Vol. 35, No. 3, 426-430, 1980.

22. Naranjo Alvarez, S., et al., "The virtual element method for resistive magnetohydrodynamics," Computer Methods in Applied Mechanics and Engineering, Vol. 381, 113815, 2021.

23. Chris Fragile, P., et al., "Divergence-free magnetohydrodynamics on conformally moving, adaptive meshes using a vector potential method," Journal of Computational Physic: X,, Vol. 2, 100020, 2019.

24. Wheatley, V., H. Kumar, and P. Huguenot, "On the role of Riemann solvers in discontinuous Galerkin methods for magnetohydrodynamics," Journal of Computational Physics, Vol. 229, No. 3, 660-680, 2010.

25. Ramshaw, J. D., "A method for enforcing the solenoidal condition on magnetic field in numerical calculations," Journal of Computational Physics, Vol. 52, No. 3, 592-596, 1983.

26. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed., Artech House, 1995.

27. Fraysse, F., J. De Vicente, and E. Valero, "The estimation of truncation error by τ-estimation revisited," Journal of Computational Physics, Vol. 231, No. 9, 3457-3482, 2012.

28. Syrakos, A., et al., "Numerical experiments on the efficiency of local grid refinement based on truncation error estimates," Journal of Computational Physics, Vol. 231, No. 20, 6725-6753, 2012.

29. Zienkiewicz, O. C., R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, 2013.

30. Hernández, A., et al., "An adaptive meshing automatic scheme based on the strain energy density function," Engineering Computations, Vol. 14, No. 6, 604-629, 1997.

31. Davies, R. W., K. Morgan, and O. Hassan, "A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain," Computational Mechanics, Vol. 44, No. 3, 321-331, 2009.

32. Ledger, P. D., et al., "Arbitrary order edge elements for electromagnetic scattering simulations using hybrid meshes and a PML," International Journal for Numerical Methods in Engineering, Vol. 55, No. 3, 339-358, 2002.

33. Young, J. W. and J. C. Bertrand, "Multiple scattering by two cylinders," Journal of the Acoustical Society of America, Vol. 58, No. 6, 1190-1195, 1975.