基于二维TE波常用时域有限差分算法的分析
|
摘要
针对传统的时域有限差分法受Courant稳定条件的限制,且存在交替方向隐式时域有限差分法数值色散较大的问题,本文以TE波为例,研究了Crank-Nicoloson差分方式的近似去耦时域有限差分法基本原理,并对其稳定性进行分析,证明该方法是无条件稳定。通过数值仿真,从运行时间和吸收效果方面与传统的时域有限差分法和交替方向隐式时域有限差分法进行对比。仿真结果表明,近似去耦时域有限差分法比交替方向时域有限差分法的吸收效果好,但比传统的时域有限差分法吸收效果差;近似去耦时域有限差分法比交替方向时域有限差分法运行时间长,但比传统时域有限差分法运行时间短,说明近似去耦时域有限差分法突破了Courant稳定条件的限制,且在吸收效果方面比交替方向时域有限差分法好。该研究具有广阔的应用前景。
Because the traditional time-domain finite difference method is restricted by Courant stable condition,the numerical dispersion of the finite difference method is very important.In this paper,the stability of the approximate decoupling time domain finite difference method(CNAD-FDTD)is analyzed,and it is proved that the method is unconditional and stable.Then,by numerical simulation,from two aspects of running time and absorption effect are compared with those of the conventional finite difference time domain method(FDTD)and based on alternating implicit finite difference time domain method(ADI-FDTD),it is known that the CNADFDTD has a better absorption effect than ADI-FDTD,but has a gap compared with the traditional FDTD,the CNAD-FDTD runs longer than the ADI-FDTD,but shorter than the traditional FDTD.It can be concluded that the approximate decoupling time domain finite difference method(CNAD-FDTD)breaks the restriction of Courant stable condition and is better than the ADI-FDTD in the absorption effect.
Because the traditional time-domain finite difference method is restricted by Courant stable condition,the numerical dispersion of the finite difference method is very important.In this paper,the stability of the approximate decoupling time domain finite difference method(CNAD-FDTD)is analyzed,and it is proved that the method is unconditional and stable.Then,by numerical simulation,from two aspects of running time and absorption effect are compared with those of the conventional finite difference time domain method(FDTD)and based on alternating implicit finite difference time domain method(ADI-FDTD),it is known that the CNADFDTD has a better absorption effect than ADI-FDTD,but has a gap compared with the traditional FDTD,the CNAD-FDTD runs longer than the ADI-FDTD,but shorter than the traditional FDTD.It can be concluded that the approximate decoupling time domain finite difference method(CNAD-FDTD)breaks the restriction of Courant stable condition and is better than the ADI-FDTD in the absorption effect.
引文
[1]葛德彪,闫玉波.电磁波时域有限差分方法[M].西安:西安电子科技大学出版社,2011.
[2]Yee K S.Numerical Solution of Initial Boundary Value Problems Involving Maxwell’S Equation in Isotropic Media[J].IEEE Transactions on Antennas and Propagation,1966,14(3):302-307.
[3]Schneider J B.A Selective Survey of the Finite-Difference Time-Domain Literature[J].IEEE Antennas and propagation,1995,37(4):39-57.
[4]Namiki T.A New FDTD Algorithm Based on Alternating-Direction Implicit Method[J].IEEE Transaction on Microwave Theory and Techniques,1999,47(10):2003-2007.
[5]Namiki T.3-D ADI-FDTD Method-Unconditionally Stable Time-Domain Algorithm for Solving Full Vector Maxwell’s Equations[J]IEEE Transaction on Microwave Theory and Techniques,2000,48(10):1743-1748.
[6]Zheng F,Chen Z,Zhang J.Toward the Development of a Three-Dimensional Unconditionally Stable Finite-Difference TimeDomain Method[J].IEEE Transaction on Microwave Theory and Techniques,2000,48(9):1550-1558.
[7]Zheng Hongxing,Leung K W.A Nonorthogonal ADI-FDTD Algorithm for Solving Two Dimensional Scattering Problems[J].IEEE Transaction on Antennas and Propagation,2009,57(12):3891-3902.
[8]Sun G L,Trueman C W.Approximate Crank-Nicolson Schemes for the Finite-Difference Time-Domain Method for TEz Waves[J].IEEE Transaction on Antennas and Propagation,2004,52(11):2963-2972.
[9]Sun G L,Trueman C W.Unconditionally Stable Crank-Nicolson Scheme for Solving the Two-Dimensional Maxwell’s Equations[J].Electronics Letters,2003,39(7):595-597.
[10]Xie X,Pan G,Hall S.A Crank-Nicholson-Base Unconditionally Stable Time-Domain Algorithm for 2Dand 3Dproblem[J].Microwave and Optical Technology Letters,2007,49(2):261-265.
[11]Li J X,Jiang H L,Zhao X M,et al.Effective CNAD-and ADE-Based CFS-PML Formulations for Truncating the Dispersive FDTD Domains[J].IEEE Antennas and Wireless Propagation Letters,2015,14:1267-1270.
[12]王秉中.计算电磁学[M].北京:科学出版社,2002.
[13]Berenger J P.A Perfectly Matched Layer for the Absorption of Electromagnetic Waves[J].Journal of Computational Physics,1994,114(9):185-200.
[14]Berenger J P.Perfectly Matched Layer for the FDTD Solution of Wave-Structure Interaction Problem[J].IEEE Transaction on Antennas and Propagation,1996,44(1):110-117.
[15]Berenger J P.Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves[J].Journal of Computational Physics,1996,127(2):363-379.
[16]Stanislav O,Pan G W.An Updated Review of General Dispersion Relation for Conditionally and Unconditionally Stable FDTD Algorithms[J].IEEE Transaction on Antennas and Propagation,2008,56(8):2572-2583.
[17]Kusaf M,Oztoprak A Y.An Unco Nditionally Stable Split-Step FDTD Method for Low Anisotropy[J].IEEE Microwave and Wireless Components Letters,2008,18(4):224-226.
[18]Lu Yijun,Shen C Y.A Domain Decomposition Finite-Difference Method for Parallel Numerical Implementation of Time-Dependent Maxwell’s Equations[J].IEEE Transactions on Antennas and Propagation,1997,45(3):556-562.
[19]Park J H,Strikwerda J C.The Domain Decomposition Method for Maxwell’s Equations in Time Domain Simulations with Dispersive Metallic Media[J].Society for Industrial and Applied Mathematics,2014,32(2):684-702.
[20]Taflove A,ME Brodwin.Computation of the Electromagnetic Fields and Induced Temperatures Within a Model of the Microwave-Irradiated Human Eye[J].IEEE Transactions on Microwave Theory&Techniques,1975,23(11):888-896.
[2]Yee K S.Numerical Solution of Initial Boundary Value Problems Involving Maxwell’S Equation in Isotropic Media[J].IEEE Transactions on Antennas and Propagation,1966,14(3):302-307.
[3]Schneider J B.A Selective Survey of the Finite-Difference Time-Domain Literature[J].IEEE Antennas and propagation,1995,37(4):39-57.
[4]Namiki T.A New FDTD Algorithm Based on Alternating-Direction Implicit Method[J].IEEE Transaction on Microwave Theory and Techniques,1999,47(10):2003-2007.
[5]Namiki T.3-D ADI-FDTD Method-Unconditionally Stable Time-Domain Algorithm for Solving Full Vector Maxwell’s Equations[J]IEEE Transaction on Microwave Theory and Techniques,2000,48(10):1743-1748.
[6]Zheng F,Chen Z,Zhang J.Toward the Development of a Three-Dimensional Unconditionally Stable Finite-Difference TimeDomain Method[J].IEEE Transaction on Microwave Theory and Techniques,2000,48(9):1550-1558.
[7]Zheng Hongxing,Leung K W.A Nonorthogonal ADI-FDTD Algorithm for Solving Two Dimensional Scattering Problems[J].IEEE Transaction on Antennas and Propagation,2009,57(12):3891-3902.
[8]Sun G L,Trueman C W.Approximate Crank-Nicolson Schemes for the Finite-Difference Time-Domain Method for TEz Waves[J].IEEE Transaction on Antennas and Propagation,2004,52(11):2963-2972.
[9]Sun G L,Trueman C W.Unconditionally Stable Crank-Nicolson Scheme for Solving the Two-Dimensional Maxwell’s Equations[J].Electronics Letters,2003,39(7):595-597.
[10]Xie X,Pan G,Hall S.A Crank-Nicholson-Base Unconditionally Stable Time-Domain Algorithm for 2Dand 3Dproblem[J].Microwave and Optical Technology Letters,2007,49(2):261-265.
[11]Li J X,Jiang H L,Zhao X M,et al.Effective CNAD-and ADE-Based CFS-PML Formulations for Truncating the Dispersive FDTD Domains[J].IEEE Antennas and Wireless Propagation Letters,2015,14:1267-1270.
[12]王秉中.计算电磁学[M].北京:科学出版社,2002.
[13]Berenger J P.A Perfectly Matched Layer for the Absorption of Electromagnetic Waves[J].Journal of Computational Physics,1994,114(9):185-200.
[14]Berenger J P.Perfectly Matched Layer for the FDTD Solution of Wave-Structure Interaction Problem[J].IEEE Transaction on Antennas and Propagation,1996,44(1):110-117.
[15]Berenger J P.Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves[J].Journal of Computational Physics,1996,127(2):363-379.
[16]Stanislav O,Pan G W.An Updated Review of General Dispersion Relation for Conditionally and Unconditionally Stable FDTD Algorithms[J].IEEE Transaction on Antennas and Propagation,2008,56(8):2572-2583.
[17]Kusaf M,Oztoprak A Y.An Unco Nditionally Stable Split-Step FDTD Method for Low Anisotropy[J].IEEE Microwave and Wireless Components Letters,2008,18(4):224-226.
[18]Lu Yijun,Shen C Y.A Domain Decomposition Finite-Difference Method for Parallel Numerical Implementation of Time-Dependent Maxwell’s Equations[J].IEEE Transactions on Antennas and Propagation,1997,45(3):556-562.
[19]Park J H,Strikwerda J C.The Domain Decomposition Method for Maxwell’s Equations in Time Domain Simulations with Dispersive Metallic Media[J].Society for Industrial and Applied Mathematics,2014,32(2):684-702.
[20]Taflove A,ME Brodwin.Computation of the Electromagnetic Fields and Induced Temperatures Within a Model of the Microwave-Irradiated Human Eye[J].IEEE Transactions on Microwave Theory&Techniques,1975,23(11):888-896.