二维麦克斯韦方程的算子分裂高阶格式
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摘要
本文主要研究麦克斯韦方程的带有分裂算子的有限差分方法和数值模拟首先将对称方法与高阶分裂算子差分方法[31]相结合,在前人的基础上研究了二维麦克斯韦方程的高阶对称分裂时域有限差分(高阶ss—FDTD)方法,构造了数值格式,用Fourier方法正明了格式的无条件稳定性,分析了数值弥散误差并通过数值算例进行验证然后对三维麦克斯韦方程对称分裂时域有限差分方法(ss—FDTD)给出了新的能量模分析,推导出了能量恒等式,并通过数值算例进一步证明了这种格式在离散的H1模下是能量守恒的:
全文共分为三章
第一章引言部分介绍了研究课题的背景和意义,给出了研究问题的模型,介绍了这类方程的常用的数值方法和论文中的研究方法
第二章利用分裂技巧和电磁场的对称性结合四阶中心差分方法,提出了高阶对称分裂时域有限差分格式(H0一ss—FDTD),分析了格式的可解性给出了应用格式的求解步骤通过推导这种格式的等价格式,发现H0一ss—FDTD格式与关于时问足二阶的,关于空间足四阶的,因此,H0一ss—FDTD格式是一种(2,4)格式然后,用R=Jurier方法分析了H0一ss—FDTD格式的数值弥散性质,推导出了数值弥散关系式,证明了这种格式是无条件稳定的通过对增长因子的分析,我们发现高阶ss—FDTD格式是非耗散的(n。n—dissipative)为了更加直观的了解H0一ss—FDTD格式的增长因子,我们用M砒lab画出在不同情况下增长因子模的变化趋势,验证了H0一ss—FDTD格式也是无条件稳定的,并与c—N格式的数值弥散误差进行比较最后详细列出了在边界附近点上方程的离散方法和格式,这部分是应用高阶差分解决实际问题比较麻烦的地方.
第三章考虑三维电导率为零的麦克斯韦方程的对称分裂时域有限差分(ss—FDTD)方法的能量守恒性通过新的能量方法与差分算子бX.бY.бZ作用后的格式,首次给出r数值逼近格式ss—FDTD在离散的H1模下的能量守恒式,并证明了格式在离散的日H1模下的守恒性,数值算例验证了格式解的能量守恒性
In this thesis, we study ?nite di?erence methods (FD) of Maxwell’s equations byusing operator-splitting and their application in computation. Firstly, we propose high-order symmetrical ?nite di?erence time domain methods (FDTD) for the 2D Maxwell’sequations by using operator-splitting and high-order (HO) ?nite di?erence methods.Then, theoretical analysis of HO-SS-FDTD on stability and numerical dispersion er-ror is given by Fourier methods. Numerical experiments are carried out and con?rm thetheoretical analysis. Thirdly, we consider further analysis of the symmetrical splitting?nite di?erence time domain methods (SS-FDTD) of 3D Maxwell’s equations. We drivenew energy conserved identities of this scheme and give stability analysis. Numericalexperiments are presented and computational results show that the SS-FDTD scheme isenergy conserved in the discrete H1 norm.
The study is divided into three parts:
Part I: We introduce the background and importance of selected research topic, in-cluding the model equations of the research problem and some usual numerical methodsof these equations.
Part II: We consider high-order ?nite di?erence time domain methods of 2D Maxwell’sequations by using operator splitting and high-order ?nite di?erence methods. A newkind of symmetric splitting high-order ?nite di?erence time domain method for the 2DMaxwell’s equations, called HO-SS-FDTD, is ?rstly proposed. This method using theYee’s staggered grids consists of four stages of equations, and can be solved e?ciently. By deriving its equivalent scheme we ?nd that HO-SS-FDTD is of second order in time,fourth order in space (denoted by (2,4)- scheme). By using the Fourier method, thismethods is proved to be unconditionally stable and has reasonable numerical dispersionerror. Numerical dispersion relation of HO-SS-FDTD is derived and compared with theCrank-Nicolson scheme, CN-FDTD. It is found that HO-SS-FDTD is non-dissipative.Numerical experiments are carried out and show that the variation of the module of thegrowth factor. Numerical dispersion error is illustrated in the di?erent cases. Finally, Weimplement the boundary conditions of HO-SS-FDTD for the two-dimensional Maxwell’sequation in a rectangular domain with the perfectly electrical conducting boundary condi-tions. This part is useful in processing the discretization of the points near the boundaryand in programming.
Part III: We study the conservation of energy of the symmetric splitting ?nite-di?erence time-domain(SS-FDTD) methods for the 3D Maxwell s equations with zeroconductivity(σ= 0). By using new energy methods and the operated SS-FDTD schemesby the di?erence operatorsδx,δy,δz, we derive the energy conserved identities SS-FTDTin the discrete H1-norm, which show that SS-FDTD is conserved in the discrete H1 norm.Numerical experiments are done and verify the energy conservation of the solution .
全文共分为三章
第一章引言部分介绍了研究课题的背景和意义,给出了研究问题的模型,介绍了这类方程的常用的数值方法和论文中的研究方法
第二章利用分裂技巧和电磁场的对称性结合四阶中心差分方法,提出了高阶对称分裂时域有限差分格式(H0一ss—FDTD),分析了格式的可解性给出了应用格式的求解步骤通过推导这种格式的等价格式,发现H0一ss—FDTD格式与关于时问足二阶的,关于空间足四阶的,因此,H0一ss—FDTD格式是一种(2,4)格式然后,用R=Jurier方法分析了H0一ss—FDTD格式的数值弥散性质,推导出了数值弥散关系式,证明了这种格式是无条件稳定的通过对增长因子的分析,我们发现高阶ss—FDTD格式是非耗散的(n。n—dissipative)为了更加直观的了解H0一ss—FDTD格式的增长因子,我们用M砒lab画出在不同情况下增长因子模的变化趋势,验证了H0一ss—FDTD格式也是无条件稳定的,并与c—N格式的数值弥散误差进行比较最后详细列出了在边界附近点上方程的离散方法和格式,这部分是应用高阶差分解决实际问题比较麻烦的地方.
第三章考虑三维电导率为零的麦克斯韦方程的对称分裂时域有限差分(ss—FDTD)方法的能量守恒性通过新的能量方法与差分算子бX.бY.бZ作用后的格式,首次给出r数值逼近格式ss—FDTD在离散的H1模下的能量守恒式,并证明了格式在离散的日H1模下的守恒性,数值算例验证了格式解的能量守恒性
In this thesis, we study ?nite di?erence methods (FD) of Maxwell’s equations byusing operator-splitting and their application in computation. Firstly, we propose high-order symmetrical ?nite di?erence time domain methods (FDTD) for the 2D Maxwell’sequations by using operator-splitting and high-order (HO) ?nite di?erence methods.Then, theoretical analysis of HO-SS-FDTD on stability and numerical dispersion er-ror is given by Fourier methods. Numerical experiments are carried out and con?rm thetheoretical analysis. Thirdly, we consider further analysis of the symmetrical splitting?nite di?erence time domain methods (SS-FDTD) of 3D Maxwell’s equations. We drivenew energy conserved identities of this scheme and give stability analysis. Numericalexperiments are presented and computational results show that the SS-FDTD scheme isenergy conserved in the discrete H1 norm.
The study is divided into three parts:
Part I: We introduce the background and importance of selected research topic, in-cluding the model equations of the research problem and some usual numerical methodsof these equations.
Part II: We consider high-order ?nite di?erence time domain methods of 2D Maxwell’sequations by using operator splitting and high-order ?nite di?erence methods. A newkind of symmetric splitting high-order ?nite di?erence time domain method for the 2DMaxwell’s equations, called HO-SS-FDTD, is ?rstly proposed. This method using theYee’s staggered grids consists of four stages of equations, and can be solved e?ciently. By deriving its equivalent scheme we ?nd that HO-SS-FDTD is of second order in time,fourth order in space (denoted by (2,4)- scheme). By using the Fourier method, thismethods is proved to be unconditionally stable and has reasonable numerical dispersionerror. Numerical dispersion relation of HO-SS-FDTD is derived and compared with theCrank-Nicolson scheme, CN-FDTD. It is found that HO-SS-FDTD is non-dissipative.Numerical experiments are carried out and show that the variation of the module of thegrowth factor. Numerical dispersion error is illustrated in the di?erent cases. Finally, Weimplement the boundary conditions of HO-SS-FDTD for the two-dimensional Maxwell’sequation in a rectangular domain with the perfectly electrical conducting boundary condi-tions. This part is useful in processing the discretization of the points near the boundaryand in programming.
Part III: We study the conservation of energy of the symmetric splitting ?nite-di?erence time-domain(SS-FDTD) methods for the 3D Maxwell s equations with zeroconductivity(σ= 0). By using new energy methods and the operated SS-FDTD schemesby the di?erence operatorsδx,δy,δz, we derive the energy conserved identities SS-FTDTin the discrete H1-norm, which show that SS-FDTD is conserved in the discrete H1 norm.Numerical experiments are done and verify the energy conservation of the solution .
引文
[1]c.C—P Chen T Lee N Murugesan and S C Hagness Generalized FDTD—ADI：An uncondi-tionally stable full一wave Maxwell’sequations solver for VLSI interconnect modeling IEEEICCAD一2000 IEEE／ACM Int Conf on Computer Aided Design 2000 156—163
[2]Chen W Li X Liang D Energy—conserved splitting finite difierence time—domain methods for Maxwell’s equations in three dimensions SIAM J Numer Anal 2010 108：445—485
[3]R Courant K niedrichs and H Lewy．On the partial difierence equations of mathenlaticalphysics IBM Journal 11f19671 215—234
[4]Craddock IJ Railton CJ A new technique for the stable of static field solution in theFDTD method for the analysis of thin wire and narrow IEEE Transactions on MicrowaveTheory and Techniques，46f81f19981，1091—1096
[5]J E Dendy Jr G Fairweather Alternating-direction Galerkin methods for parabolic andhyperbolic problems on rectangular polygons，SIAMJ Nulner Anal，12f19751，144-163
[6]J Douglas H H Rachford On the nulnerical solution of heat conduction problems in twoand three space variables，Trans Amer Math Soc，82f19561，421-439
[7]L Gao，B Zhang，D Liang Splitting finite difierence methods on staggered grids forthe three—dimensional time—dependent Maxwell，s eduations Coiiiiilun Comput Phys4（2）（2008），405—432
[8]L Gao，B Zhang，dependent Maxwelltion．Science PressD Liang Analysis of an ADI finite difierence methods for the time—s equations in 3-D in：Advances in Scientific Computing and Applica-beijing，2004，171—180
[9]L Gao，B Zhang，D Liang The splitting finite—difference time-domain methods for Maxwell，sequations in two dimensions，J Comput Appl Math，205(2007)，207—230
[10]R Holland Implicit three—dimensional finite differencing of Maxwell，s equations，IEEETrans Nucl Sci，vol NS-31f19841，1322—1326
[11]T Namiki A new FDTD algorithm based on alternating direction implicit method，IEEETrans Microwave Theory Tech 47f19991 2003—2007
[12]T Namiki k Ito Investigation of numerical errors of the two—dimensions ADLFDTD method，IEEE Trans Microwave Theory Tech，48f20001，1950—1956
[13]P G Petropoulosary condition forference schemes．L Zhao，A C Cangellaris A reflection less sponge layer absorbing boundthe solution of Maxwell，s equations with high—order staggered finite difJ Comput Phys，139(1998)，18&208
[14]A Tflove M E Brodwin Numerical solutionproblem using the time—dependent Maxwell，stech 23f19751 623—630
[15]A Taflove S Hagnessof steady-state electric magnetic scatteringequations IEEE Trans Microwave TheoryComputational Electrodynamics：Artech House Boston MA 2000
[16]J．W Thomas Numerical Partial Differential equations，世界图书出版公司，1995
[17]Thoma P，Weiland T Numerical stability of finite difierence time domain methods，IEEE Transactions on Magnetics，34(51f19981，2740—2743
[18]Xie Z Chan C H and Zhang B An explicit fourth—order staggered finite—difierence time—domain method for Maxwell，s equations，J Comput Appl Math，147f20021，75—98
[19]Xie z，Chan C H and Zhang B An explicit fourth—order orthogonal curvilinear staggeredgrid FDTD method fnr Maxwell，s equations J Comput Appl Math 175(20021 739—763
[20]K S Yee Numerical solution of initial boundary value problems involving Maxwell，s equa- tions in isotropic media，IEEE Trans Antennas and Propagation，14f19961，302—307
[21]A Yefet and P G Petropoulos A non—dissipative staggered fourth accurate explicit finite difference scheme fnr the time—domain Maxwell，s equations，J Comput Phs，(2001)，168—286
[22]A P Zhao Analysis of the numerical dispersion of the 2一D alternating—direction implicit FDTD method，IEEE Trans Microwave Theory tech，50f20021，1156—1164
[23]Zheng Fenghua，Chen Zhizhang Numerical dispersion analysis of the unconditionally stable3一D ADI—FDTD Method『J]，IEEETrans on Microwave Theory Tech，49f5)2001，1006—1009
[24]F Zheng Z Chen J Zhang Toward the development of a three—dimensional uncondition—ally stable finite—difierence time—domain methods IEEE Trans Microwave Theory Tech 48(2000)，1550—1558
[25]葛德彪，闩玉波电磁波时域有限差分方法(第二版)，西安电子科技大学出版社，2005，6
[26]李荣华，冯果忱微分方程数值解法，高等教育出版社，1989
[27]徐利军，袁乃昌高阶ADI—FDTD算法数值色散分析，电子与信息学报27（10）（2005），1662—1665
[28]王长清现代计算电磁学基础，北京大学出版社，北京：2005
[29]盛新庆计算电磁学要论，中国科学技术大学出版社，合~E：2008
[30]黄懿三维Maxwell方程并行时域有限差分方法研究，北京：中科院应用数学研究所，2007
[31]谷秀芹，高理平二维麦克斯韦方程的高阶s—FDTD方法，硕士学位毕业论文
[2]Chen W Li X Liang D Energy—conserved splitting finite difierence time—domain methods for Maxwell’s equations in three dimensions SIAM J Numer Anal 2010 108：445—485
[3]R Courant K niedrichs and H Lewy．On the partial difierence equations of mathenlaticalphysics IBM Journal 11f19671 215—234
[4]Craddock IJ Railton CJ A new technique for the stable of static field solution in theFDTD method for the analysis of thin wire and narrow IEEE Transactions on MicrowaveTheory and Techniques，46f81f19981，1091—1096
[5]J E Dendy Jr G Fairweather Alternating-direction Galerkin methods for parabolic andhyperbolic problems on rectangular polygons，SIAMJ Nulner Anal，12f19751，144-163
[6]J Douglas H H Rachford On the nulnerical solution of heat conduction problems in twoand three space variables，Trans Amer Math Soc，82f19561，421-439
[7]L Gao，B Zhang，D Liang Splitting finite difierence methods on staggered grids forthe three—dimensional time—dependent Maxwell，s eduations Coiiiiilun Comput Phys4（2）（2008），405—432
[8]L Gao，B Zhang，dependent Maxwelltion．Science PressD Liang Analysis of an ADI finite difierence methods for the time—s equations in 3-D in：Advances in Scientific Computing and Applica-beijing，2004，171—180
[9]L Gao，B Zhang，D Liang The splitting finite—difference time-domain methods for Maxwell，sequations in two dimensions，J Comput Appl Math，205(2007)，207—230
[10]R Holland Implicit three—dimensional finite differencing of Maxwell，s equations，IEEETrans Nucl Sci，vol NS-31f19841，1322—1326
[11]T Namiki A new FDTD algorithm based on alternating direction implicit method，IEEETrans Microwave Theory Tech 47f19991 2003—2007
[12]T Namiki k Ito Investigation of numerical errors of the two—dimensions ADLFDTD method，IEEE Trans Microwave Theory Tech，48f20001，1950—1956
[13]P G Petropoulosary condition forference schemes．L Zhao，A C Cangellaris A reflection less sponge layer absorbing boundthe solution of Maxwell，s equations with high—order staggered finite difJ Comput Phys，139(1998)，18&208
[14]A Tflove M E Brodwin Numerical solutionproblem using the time—dependent Maxwell，stech 23f19751 623—630
[15]A Taflove S Hagnessof steady-state electric magnetic scatteringequations IEEE Trans Microwave TheoryComputational Electrodynamics：Artech House Boston MA 2000
[16]J．W Thomas Numerical Partial Differential equations，世界图书出版公司，1995
[17]Thoma P，Weiland T Numerical stability of finite difierence time domain methods，IEEE Transactions on Magnetics，34(51f19981，2740—2743
[18]Xie Z Chan C H and Zhang B An explicit fourth—order staggered finite—difierence time—domain method for Maxwell，s equations，J Comput Appl Math，147f20021，75—98
[19]Xie z，Chan C H and Zhang B An explicit fourth—order orthogonal curvilinear staggeredgrid FDTD method fnr Maxwell，s equations J Comput Appl Math 175(20021 739—763
[20]K S Yee Numerical solution of initial boundary value problems involving Maxwell，s equa- tions in isotropic media，IEEE Trans Antennas and Propagation，14f19961，302—307
[21]A Yefet and P G Petropoulos A non—dissipative staggered fourth accurate explicit finite difference scheme fnr the time—domain Maxwell，s equations，J Comput Phs，(2001)，168—286
[22]A P Zhao Analysis of the numerical dispersion of the 2一D alternating—direction implicit FDTD method，IEEE Trans Microwave Theory tech，50f20021，1156—1164
[23]Zheng Fenghua，Chen Zhizhang Numerical dispersion analysis of the unconditionally stable3一D ADI—FDTD Method『J]，IEEETrans on Microwave Theory Tech，49f5)2001，1006—1009
[24]F Zheng Z Chen J Zhang Toward the development of a three—dimensional uncondition—ally stable finite—difierence time—domain methods IEEE Trans Microwave Theory Tech 48(2000)，1550—1558
[25]葛德彪，闩玉波电磁波时域有限差分方法(第二版)，西安电子科技大学出版社，2005，6
[26]李荣华，冯果忱微分方程数值解法，高等教育出版社，1989
[27]徐利军，袁乃昌高阶ADI—FDTD算法数值色散分析，电子与信息学报27（10）（2005），1662—1665
[28]王长清现代计算电磁学基础，北京大学出版社，北京：2005
[29]盛新庆计算电磁学要论，中国科学技术大学出版社，合~E：2008
[30]黄懿三维Maxwell方程并行时域有限差分方法研究，北京：中科院应用数学研究所，2007
[31]谷秀芹，高理平二维麦克斯韦方程的高阶s—FDTD方法，硕士学位毕业论文