Skeletonization Accelerated Solution of Crank-Nicolson Method for Solving Three-Dimensional Parabolic Equation
关键词:
Alternating direction implicit method, Crank-Nicolson method, hierarchical interpolative factorization, interpolative decomposition, Shur complement摘要
Parabolic equation models discretized with the finite difference method have been extensively studied for a long time. However, several explicit and implicit schemes exist in the literature. The advantage in explicit schemes is its simplicity, while its disadvantage is conditional stability. On the other hand, implicit schemes are unconditionally stable but require special treatment for a fast and accurate solution such as the Crank-Nicolson (CN) method. This method becomes computationally intensive for problems with dense meshes. The resulting matrix from the CN in two and three-dimensional cases requires high computational resources. This paper applies hierarchical interpolative factorization (HIF) to reduce the computational cost of the CN method. Numerical experiments are conducted to validate the proposed HIF acceleration.
##plugins.generic.usageStats.downloads##
参考
M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. London, IEE Institution of Electrical Engineers, 2000.
G. Apaydin and L. Sevgi, Radio Wave Propagation, and Parabolic Equation Modeling. Wiley-IEEE Press, 2017.
Ö. Ö. M. Kuzuoğlu, MATLAB-based Finite Element Programming in Electromagnetic Modeling. CRC Press, 2019.
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. 2nd ed., Philadelphia, PA: SIAM, 2004.
A. Hrovat, G. Kandus, and T. Javornik, “A survey of radio propagation modeling for tunnels,” IEEE Commun. Surveys Tuts., vol. 16, no. 2, pp. 658-669, 2nd Quarter, 2014. 1010 ACES JOURNAL, Vol. 35, No. 9, September 2020
H. F. Rasool, X. M. Pan, and X. Q. Sheng, “A fourier split-step based wide-angle threedimensional vector parabolic wave equation algorithm predicting the field strength over flat and irregular forest environments,” Applied Computational Electromagnetic Society Journal, vol. 34, no. 6, pp. 874-881, June 2019.
C. A. Zelly and C. C. Constantinou, “A threedimensional parabolic equation applied to VHF/ UHF propagation over irregular terrain,” IEEE Trans. Antennas Propag., vol. 47, pp. 1586-1596, Oct. 1999.
Z. He, T. Su, H. C. Yin, and R. S. Chen, “Wave propagation modeling of tunnels in complex meteorological environments with parabolic equation,” IEEE Trans. Antennas Propag., vol. 66, no. 12, pp. 6629-6634, 2018.
P. Angot, J. Keating, and P. D. Minev, “A direction splitting algorithm for incompressible flow in complex geometries,” Computer Methods in Applied Mechanics and Engineering, vol. 217-220, pp. 111-120, 2012.
T. A. Dauzhenka and I. A. Gishkeluk, “Quasilinear heat equation in three dimensions and Stefan problem in permafrost soils in the frame of alternating directions finite difference scheme,” Proc. WCE-London, UK, 2013.
E. L. Tan, “Efficient algorithms for Crank-Nicolsonbased finite-difference time-domain methods,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 2, pp. 408-413, Feb. 2008.
A. V. Londersele, D. D. Zutter, and D. V. Ginste, “Provably stable local application of CrankNicolson time integration to the FDTD method with nonuniform gridding and subgridding,” 2018 Intl. App. Comp. Electromagnetics Society Symposium (ACES), Denver, CO, USA, Mar. 25- 29, 2018.
N. Feng, Y. Zhang, Q. Sun, J. Zhu, W. T. Joines, and Q. H. Liu, “An accurate 3-D CFS-PML based Crank–Nicolson FDTD method and its applications in low-frequency subsurface sensing,” IEEE Trans. Antennas Propag., vol. 66, no. 6, pp. 2967- 2975, June 2018.
S. G. Garcia, T.-W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas and Wave Propag. Letter, vol. 1, pp. 31- 34, 2002.
R. Martelly and R. Janaswamy, “An ADI-PE approach for modeling radio transmission loss in tunnels,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1759-1770, June 2009.
G. Apaydin and L. Sevgi, “Calibration of threedimensional parabolic equation propagation models with the rectangular waveguide problem,” IEEE Antennas Propagat. Mag., vol. 54, no. 6, pp. 102-116, Dec. 2012.
X. Zhang, N. Sood, and C. D. Sarris, “Radio-wave propagation modeling in tunnels with a hybrid vector parabolic equation/waveguide mode theory method,” IEEE Trans. Antennas Propag., vol. 66, no. 12, pp. 6540-6551, June 2018.
K. L. Ho and L. Ying, “Hierarchical interpolative factorization for elliptic operators: Differential equations,” Comm. Pure Appl. Math., vol. 69, no. 8, 2016.
K. L. Ho and L. Ying, “Hierarchical interpolative factorization for elliptic operators: Integral equations,” Comm. Pure Appl. Math., vol. 69, pp. 1314-1353, no. 7, 2016.
Y. N. Liu, X. M. Pan, and X. Q. Sheng, “Skeletonization accelerated MLFMA solution of volume integral equation for plasmonic structures,” IEEE Trans. Antennas Propag., vol. 66, no. 3, pp. 1590-1594, Mar. 2018.
S. L. Huang, W. Song, Y. Z. Wang, Y. M. Wu, X. M. Pan, and X. Q. Sheng, “Efficient and accurate electromagnetic angular sweeping of rough surfaces by MPI parallel randomized low-rank decomposition,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 13, no. 1, pp. 1752-1760, 2020.
D. Wu, Y. N. Liu, Y. M. W. Wu, X. M. Pan, and X.-Q. Sheng, “Skeletonization improved calculation of electric fields by the impedance matrix of MoM,” IEEE Antennas and Wireless Propag. Lett., Accepted, 2020.
