Communications in Applied Mathematics and Computational Science

A third order finite volume WENO scheme for Maxwell's equations on tetrahedral meshes

Marina Kotovshchikova, Dmitry K. Firsov, and Shiu Hong Lui

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/camcos.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A third order type II WENO finite volume scheme for tetrahedral unstructured meshes is applied to the numerical solution of Maxwell’s equations. Stability and accuracy of the scheme are severely affected by mesh distortions, domain geometries, and material inhomogeneities. The accuracy of the scheme is enhanced by a clever choice of a small parameter in the WENO weights. Also, hybridization with a polynomial scheme is proposed to eliminate unnecessary and costly WENO reconstructions in regions where the solution is smooth. The proposed implementation is applied to several test problems to demonstrate the accuracy and efficiency, as well as usefulness of the scheme to problems with singularities.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 13, Number 1 (2018), 87-106.

Dates
Received: 9 March 2017
Revised: 3 January 2018
Accepted: 7 January 2018
First available in Project Euclid: 28 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1522202440

Digital Object Identifier
doi:10.2140/camcos.2018.13.87

Mathematical Reviews number (MathSciNet)
MR3778321

Zentralblatt MATH identifier
06864867

Subjects
Primary: 65M08: Finite volume methods 78M12: Finite volume methods, finite integration techniques

Keywords
weighted essentially nonoscillatory (WENO) schemes finite volume schemes Maxwell's equations tetrahedral meshes

Citation

Kotovshchikova, Marina; Firsov, Dmitry K.; Lui, Shiu Hong. A third order finite volume WENO scheme for Maxwell's equations on tetrahedral meshes. Commun. Appl. Math. Comput. Sci. 13 (2018), no. 1, 87--106. doi:10.2140/camcos.2018.13.87. https://projecteuclid.org/euclid.camcos/1522202440


Export citation

References

  • R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys. 114 (1994), no. 1, 45–58.
  • F. Aràndiga, A. Baeza, A. M. Belda, and P. Mulet, Analysis of WENO schemes for full and global accuracy, SIAM J. Numer. Anal. 49 (2011), no. 2, 893–915.
  • C. A. Balanis, Advanced engineering electromagnetics, Wiley, 1989.
  • T. J. Barth and P. O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, 28th Aerospace Sciences Meeting, no. 90-0013, AIAA, 1990.
  • P. Batten, C. Lambert, and D. M. Causon, Positively conservative high-resolution convection schemes for unstructured elements, Internat. J. Numer. Methods Engrg. 39 (1996), no. 11, 1821–1838.
  • D. Baumann, C. Fumeaux, and R. Vahldieck, Field-based scattering-matrix extraction scheme for the FVTD method exploiting a flux-splitting algorithm, IEEE T. Microw. Theory 53 (2005), no. 11, 3595–3605.
  • P. Bonnet, X. Ferrieres, F. Issac, F. Paladian, J. Grando, J. C. Alliot, and J. Fontaine, Numerical modeling of scattering problems using a time domain finite volume method, J. Electromagnet. Wave 11 (1997), no. 8, 1165–1189.
  • R. Borges, M. Carmona, B. Costa, and W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008), no. 6, 3191–3211.
  • A. Chatterjee and R.-S. Myong, Efficient implementation of higher-order finite volume time-domain method for electrically large scatterers, Prog. Electromagn. Res. B 17 (2009), 233–254.
  • M. Dumbser and M. Käser, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys. 221 (2007), no. 2, 693–723.
  • M. Dumbser, M. Käser, V. A. Titarev, and E. F. Toro, Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys. 226 (2007), no. 1, 204–243.
  • L. J. Durlofsky, B. Engquist, and S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comput. Phys. 98 (1992), no. 1, 64–73.
  • D. Firsov, J. LoVetri, I. Jeffrey, V. Okhmatovski, C. Gilmore, and W. Chamma, High-order FVTD on unstructured grids using an object-oriented computational engine, ACES J. 22 (2007), no. 1, 71–82.
  • R. F. Harrington, Time-harmonic electromagnetic fields, McGraw-Hill, 1961.
  • A. Harten and S. R. Chakravarthy, Multi-dimensional ENO schemes for general geometries, Tech. Report 187637, NASA, 1991.
  • A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes, III, J. Comput. Phys. 71 (1987), no. 2, 231–303.
  • A. K. Henrick, T. D. Aslam, and J. M. Powers, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys. 207 (2005), no. 2, 542–567.
  • P. Hillion, Numerical integration on a triangle, Internat. J. Numer. Methods Engrg. 11 (1977), no. 5, 797–815.
  • G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228.
  • X.-D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994), no. 1, 200–212.
  • Y. Liu and Y.-T. Zhang, A robust reconstruction for unstructured WENO schemes, J. Sci. Comput. 54 (2013), no. 2–3, 603–621.
  • C. Ollivier-Gooch and M. Van Altena, A high-order-accurate unstructured mesh finite-volume scheme for the advection–diffusion equation, J. Comput. Phys. 181 (2002), no. 2, 729–752.
  • S. Piperno, M. Remaki, and L. Fezoui, A nondiffusive finite volume scheme for the three-dimensional Maxwell's equations on unstructured meshes, SIAM J. Numer. Anal. 39 (2002), no. 6, 2089–2108.
  • T. Pringuey and R. S. Cant, High order schemes on three-dimensional general polyhedral meshes - application to the level set method, Commun. Comput. Phys. 12 (2012), no. 1, 1–41.
  • M. Remaki, A new finite volume scheme for solving Maxwell's system, COMPEL 19 (2000), no. 3, 913–932.
  • V. Shankar, W. F. Hall, and A. H. Mohammadian, A time-domain differential solver for electromagnetic scattering problems, Proc. IEEE 77 (1989), no. 5, 709–721.
  • J. Shi, C. Hu, and C.-W. Shu, A technique of treating negative weights in WENO schemes, J. Comput. Phys. 175 (2002), no. 1, 108–127.
  • C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471.
  • T. Sonar, On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection, Comput. Methods Appl. Mech. Engrg. 140 (1997), no. 1–2, 157–181.
  • J. L. Steger and R. F. Warming, Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods, J. Comput. Phys. 40 (1981), no. 2, 263–293.
  • P. Tsoutsanis, V. A. Titarev, and D. Drikakis, WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys. 230 (2011), no. 4, 1585–1601.
  • M. Wirianto, W. A. Mulder, and E. C. Slob, Applying essentially non-oscillatory interpolation to controlled-source electromagnetic modelling, Geophys. Prospect. 59 (2011), no. 1, 161–175.
  • Y.-T. Zhang and C.-W. Shu, Third order WENO scheme on three dimensional tetrahedral meshes, Commun. Comput. Phys. 5 (2009), no. 2–4, 836–848.