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Toward the development of a threedimensional unconditionally stable finitedifference timedomain method
 IEEE Trans. Microw. Theory Tech
, 2000
"... Abstract—In this paper, an unconditionally stable threedimensional (3D) finitedifference timemethod (FDTD) is presented where the time step used is no longer restricted by stability but by accuracy. The principle of the alternating direction implicit (ADI) technique that has been used in formul ..."
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Cited by 44 (0 self)
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Abstract—In this paper, an unconditionally stable threedimensional (3D) finitedifference timemethod (FDTD) is presented where the time step used is no longer restricted by stability but by accuracy. The principle of the alternating direction implicit (ADI) technique that has been used in formulating an unconditionally stable twodimensional FDTD is applied. Unlike the conventional ADI algorithms, however, the alternation is performed in respect to mixed coordinates rather than to each respective coordinate direction. Consequently, only two alternations in solution marching are required in the 3D formulations. Theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness and efficiency of the method. It is found that the number of iterations with the proposed FDTD can be at least four times less than that with the conventional FDTD at the same level of accuracy. Index Terms—Alternating direct implicit (ADI) technique, FDTD method, instability, unconditional stable. I.
Principles of mimetic discretizations of differential operators
 Compatible Spatial Discretizations, volume 142 of The IMA Volumes in Mathematics and its Applications
, 2006
"... Abstract. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all a ..."
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Cited by 42 (3 self)
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Abstract. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic mappings between differential forms and cochains. The key concept of the framework is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and preserves the invariants of the De Rham cohomology groups. This allows, among other things, for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume, and finite difference methods. We describe how these methods result from a choice of the reconstruction operator and explain when they are equivalent. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian. Key words. Mimetic discretizations, compatible spatial discretizations, finite element
Finite formulation of Electromagnetic field”,
 ICS Newsletter,
, 2001
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Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients
 SIAM J. Scient. Comput
, 2001
"... Abstract. We consider solving threedimensional electromagnetic problems in parameter regimes where the quasistatic approximation applies and the permeability, permittivity, and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interf ..."
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Cited by 39 (22 self)
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Abstract. We consider solving threedimensional electromagnetic problems in parameter regimes where the quasistatic approximation applies and the permeability, permittivity, and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing Maxwell’s equations in the frequency domain. The present article extends methods we proposed earlier for constant permeability [E. Haber, U. Ascher, D. Aruliah, and D. Oldenburg, J. Comput. Phys., 163 (2000), pp. 150–171; D. Aruliah, U. Ascher, E. Haber, and D. Oldenburg, Math. Models Methods Appl. Sci., to appear.] to handle also problems in which the permeability is variable and may contain significant jump discontinuities. In order to address the problem of slow convergence we reformulate Maxwell’s equations in terms of potentials, applying a Helmholtz decomposition to either the electric field or the magnetic field. The null space of the curl operators can then be annihilated by adding a stabilizing term, using a gauge condition, and thus obtaining a strongly elliptic differential operator. A staggered grid finite volume discretization is subsequently applied to the reformulated PDE system. This scheme
A finitedifference timedomain method without the Courant stability conditions
 IEEE Microwave Guided Wave Lett
, 1999
"... Abstract—In this paper, a finitedifference timedomain method that is free of the constraint of the Courant stability condition is presented for solving electromagnetic problems. In it, the alternating direction implicit (ADI) technique is applied in formulating the finitedifference timedomain ( ..."
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Cited by 37 (1 self)
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Abstract—In this paper, a finitedifference timedomain method that is free of the constraint of the Courant stability condition is presented for solving electromagnetic problems. In it, the alternating direction implicit (ADI) technique is applied in formulating the finitedifference timedomain (FDTD) algorithm. Although the resulting formulations are computationally more complicate than the conventional FDTD, the proposed FDTD is very appealing since the time step used in the simulation is no longer restricted by stability but by accuracy. As a result, computation speed can be improved. It is found that the number of iterations with the proposed FDTD can be at least three times less than that with the conventional FDTD with the same numerical accuracy. Index Terms — Alternating direct implicit technique (ADI), FDTD method, instability, unconditional stable. I.
A NONDISSIPATIVE STAGGERED FOURTHORDER ACCURATE EXPLICIT FINITE DIFFERENCE SCHEME FOR THE TIMEDOMAIN MAXWELL'S EQUATIONS
, 1999
"... We consider a divergencefree nondissipative fourthorder explicit staggered nite di erence scheme for the hyperbolic Maxwell's equations. Special onesided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results sho ..."
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Cited by 35 (0 self)
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We consider a divergencefree nondissipative fourthorder explicit staggered nite di erence scheme for the hyperbolic Maxwell's equations. Special onesided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is longtime stable, and is fourthorder convergent over complex domains that include dielectric interfaces and perfectly conducting surfaces. We also examine the scheme's behavior near metal surfaces that are not alligned with the grid axes, and compare its accuracy to that obtained by the Yee scheme.
Fast Solution Methods in Electromagnetics
, 1997
"... Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either ..."
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Cited by 33 (0 self)
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Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements. 1. Introduction Computational electromagnetics is a fascinating discipline that has drawn the attention of mathematicians, engineers, physicists, and computer scientists alike. It is a discipline that creates a symbiotic marriage between mathematics, physics, computer science, and various applicatio...
Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations
, 1997
"... . The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackl ..."
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Cited by 32 (12 self)
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. The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on on N'ed'elec's H(curl;\Omega\Gamma7131/59948 edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the cu...
Design and fabrication of silicon photonic crystal optical waveguides
 Journal of Lightwave Technology
, 2000
"... Abstract—We have designed and fabricated waveguides that incorporate twodimensional (2D) photonic crystal geometry for lateral confinement of light, and total internal reflection for vertical confinement. Both square and triangular photonic crystal lattices were analyzed. A threedimensional (3D) ..."
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Cited by 32 (2 self)
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Abstract—We have designed and fabricated waveguides that incorporate twodimensional (2D) photonic crystal geometry for lateral confinement of light, and total internal reflection for vertical confinement. Both square and triangular photonic crystal lattices were analyzed. A threedimensional (3D) finitedifference timedomain (FDTD) analysis was used to find design parameters of the photonic crystal and to calculate dispersion relations for the guided modes in the waveguide structure. We have developed a new fabrication technique to define these waveguides into silicononinsulator material. The waveguides are suspended in air in order to improve confinement in the vertical direction and symmetry properties of the structure. Highresolution fabrication allowed us to include different types of bends and optical cavities within the waveguides. Index Terms—Finitedifference timedomain (FDTD) methods, nanooptics, optical device fabrication, photonic bandgap (PBG) materials, photonic crystals (PCS), photonic crystal waveguides. I.
Discrete Hodge Operators
 NUMER. MATH
, 1999
"... Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, ..."
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Cited by 32 (3 self)
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Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.