volume | PIER Journals

Abstract

The time domain Maxwell's equations are numerically solved using a multigrid method in a scattered field formulation and a cell-vertex based finite volume time domain framework. The multilevel method is an adaptation of Ni's [9] cell-vertex based multigrid technique, proposed for accelerating steady state convergence of nonlinear Euler equations of gas dynamics. Accelerated convergence to steady state of the time domain Maxwell's equations, for problems involving electromagnetic scattering, is obtained using multiple grids without the use of additional numerical damping usually required in nonlinear problems. The linear nature of the Maxwell's system also allows for a more accurate representation of the fine-grid problem on the coarse grid.

1. Shankar, V., "A gigaflop performance algorithm for solving Maxwell's equations of electromagnetics," AIAA Paper, 584-590, 91--1578, June 1991. Google Scholar

2. 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.
doi:10.2528/PIERB09073102 Google Scholar

3. Shang, J. S., "Characteristic-based algorithms for solving the Maxwell equations in the time domain," IEEE Antennas and Propagation Society Magazine, Vol. 37, No. 3, 15-25, 1995.
doi:10.1109/74.388807 Google Scholar

4. Chatterjee, A. and A. Shrimal, "Essentially nonoscillatory finite volume scheme for electromagnetic scattering by thin dielectric coatings," AIAA Journal, Vol. 42, No. 2, 361-365, 2004.
doi:10.2514/1.553 Google Scholar

5. Yee, K., "Numerical solutions of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966. Google Scholar

6. Brandt, A., "Multi-level adaptive solutions to boundary-value problems," Mathematics of Computation, Vol. 31, No. 138, 333-390, 1977.
doi:10.1090/S0025-5718-1977-0431719-X Google Scholar

7. Turkel, E., "Progress in computational physics," Computers and Fluids, Vol. 11, No. 2, 121-144, 1983.
doi:10.1016/0045-7930(83)90006-3 Google Scholar

8. Chatterjee, A. and G. Shevare, "A time-accurate multigrid algorithm for Euler equations," Fourteenth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, 453, Springer-Verlag, 1995. Google Scholar

9. Ni, R.-H., "A multiple-grid scheme for solving the Euler equations," AIAA Journal, Vol. 20, 1565-1571, 1982. Google Scholar

10. Jameson, A., "Solution of the Euler equations for two-dimensional flow by a multigrid methods," Applied Mathematics and Computation, Vol. 13, 327-356, 1983.
doi:10.1016/0096-3003(83)90019-X Google Scholar

11. Deore, N. and A. Chatterjee, "A cell-vertex finite volume time domain method for electromagnetic scattering," Progress In Electromagnetics Research M, Vol. 12, 1-15, 2010.
doi:10.2528/PIERM10022003 Google Scholar

12. Koeck, C., "Computation of three-dimensional flow using the Euler equations and a multiple-grid scheme," International Journal for Numerical Methods in Fluids, Vol. 5, 483-500, 1985.
doi:10.1002/fld.1650050507 Google Scholar

13. Ni, R. H. and J. C. Bogoian, "Prediction of 3-D multistage turbine flow field using a multiple-grid Euler solver," AIAA Paper, 1-9, 89--0203, Jan. 1989. Google Scholar

14. Roe, P. L., "Fluctuations and signals --- A framework for numerical evolution problems," Numerical Methods in Fluid Dynamics, 219-257, K. W. Morton and M. J. Baines (eds.), Academic Press, 1982. Google Scholar

15. Hall, M. G., "Cell-vertex multigrid schemes for solution of the Euler equations," Numerical Methods for Fluid Dynamics II, 303-345, K. W. Morton and M. J. Baines (eds.), Clarendon Press, Oxford, 1985. Google Scholar

16. Radespiel, R., C. Rossow, and R. C. Swanson, "Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations," AIAA Journal, Vol. 28, No. 8, 1464-1472, 1990.
doi:10.2514/3.25239 Google Scholar

17. Swanson, R. C. and R. Radespiel, "Cell centered and cell vertex multigrid schemes for the Navier-Stokes equations," AIAA Journal, Vol. 29, No. 5, 697-703, 1991.
doi:10.2514/3.10643 Google Scholar

18. French, A. D., "Solution of the Euler equations on cartesian grids," Applied Numerical Mathematics, Vol. 49, 367-379, 2004.
doi:10.1016/j.apnum.2003.12.014 Google Scholar

19. Shankar, V., W. F. Hall, and A. H. Mohammadin, "A time-domain differential solver for electromagnetic scattering problems," Proceedings of the IEEE, Vol. 77, No. 5, 709-721, 1989.
doi:10.1109/5.32061 Google Scholar

Dr. Narendra Deore

Prof. Avijit Chatterjee