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335
Numerical Solution Of Problems On Unbounded Domains. A Review
 A review, Appl. Numer. Math
, 1998
"... While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many areas of ..."
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Cited by 61 (13 self)
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While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computa...
The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems
 Journal of Computational Physics
, 2000
"... The present work describes an alternative to the classical partial differential equationsbased approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possi ..."
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Cited by 47 (1 self)
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The present work describes an alternative to the classical partial differential equationsbased approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometrical content of the original problem, and is made possible by the existence and properties of a common mathematical structure of physical field theories. The goal is to maintain the focus, both in the modeling and in the discretizati on step, on the physics of the problem, thinking in terms of numerical methods for physical field problems, and not for a particular mathematical form (for example, a partial differential equation) into which the original physical problem happens to be translated.
Fourthorder time stepping for stiff PDEs
 SIAM J. Sci. Comput
, 2005
"... Abstract. A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison ..."
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Cited by 37 (3 self)
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Abstract. A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential timedifferencing (ETD) scheme against the competing methods of implicitexplicit differencing, integrating factors, timesplitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
A fast spectral algorithm for nonlinear wave equations with linear dispersion
 J. Comput. Phys
, 1999
"... Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including wellknown cases such as the Korteweg–de Vries and nonlinear Schrödinger equations. For the best computational efficiency, one needs also to use highorder methods in ti ..."
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Cited by 29 (5 self)
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Spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains, including wellknown cases such as the Korteweg–de Vries and nonlinear Schrödinger equations. For the best computational efficiency, one needs also to use highorder methods in time while somehow bypassing the usual severe stability restrictions. We use linearly implicit multistep methods, with the innovation of choosing different methods for different ranges in Fourier space—high accuracy at low wavenumbers and Astability at high wavenumbers. This new approach compares favorably to alternatives such as splitstep and integrating factor (or linearly exact) methods. c ○ 1999 Academic Press Key Words: spectral methods; nonlinear waves; KdV; NLS; linearly implicit. 1.
Mimetic Discretizations for Maxwell's Equations
 J. Comput. Phys
, 1999
"... This paper is a part of our attempt to develop a discrete analog of vector and tensor calculus that can be used to accurately approximate continuum models for a wide range of physical processes on logically rectangular, nonorthogonal, nonsmooth grids. These mimetic FDMs mimic fundamental properties ..."
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Cited by 26 (5 self)
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This paper is a part of our attempt to develop a discrete analog of vector and tensor calculus that can be used to accurately approximate continuum models for a wide range of physical processes on logically rectangular, nonorthogonal, nonsmooth grids. These mimetic FDMs mimic fundamental properties of the original continuum differential operators and allow the discrete approximations of partial differential equations (PDEs) to preserve critical properties including conservation laws and symmetries in the solution of the underlying physical problem. In particular, we have constructed discrete analogs of firstorder differential 881 00219991/99 operators, such as div, grad, and curl, that satisfy the discrete analogs of theorems of vector and tensor calculus [1013]. This approach has also been used to construct highquality mimetic FDMs for the divergence and gradient in approximating the diffusion equation [15, 38, 39]
Mimetic Discretizations for Maxwell's Equations and the Equations of Magnetic Diffusion
"... We construct reliable finitedifference methods for approximating the solutions to Maxwell's equations and equations of magnetic field diffusion using discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form. These methods mim ..."
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Cited by 22 (12 self)
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We construct reliable finitedifference methods for approximating the solutions to Maxwell's equations and equations of magnetic field diffusion using discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form. These methods mimic many fundamental properties of the underlying physical problem, including the conservation laws, the symmetries in the solution, the nondivergence of particular vector fields and they do not allow spurious modes. The constructed method can be applied when there are strongly discontinuous properties of the media and nonorthogonal, nonsmooth computational grids. In this paper we apply discrete vector analysis techniques [1][4] to construct mimetic finitedifference methods to Maxwell's firstorder curl equations (hyperbolic type) and to the equations of magnetic diffusion (parabolic type). The system of firstorder Maxwell's curl equations can be written as follows: @ ~ B=@t = \Gammacur...
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 21 (14 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
Microwave imaging via spacetime beamforming for early detection of breast cancer
 IEEE Transactions on Antennas and Propagation
, 2003
"... Abstract—A method of microwave imaging via spacetime (MIST) beamforming is proposed for detecting earlystage breast cancer. An array of antennas is located near the surface of the breast and an ultrawideband (UWB) signal is transmitted sequentially from each antenna. The received backscattered sig ..."
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Cited by 21 (2 self)
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Abstract—A method of microwave imaging via spacetime (MIST) beamforming is proposed for detecting earlystage breast cancer. An array of antennas is located near the surface of the breast and an ultrawideband (UWB) signal is transmitted sequentially from each antenna. The received backscattered signals are passed through a spacetime beamformer that is designed to image backscattered signal energy as a function of location. The beamformer spatially focuses the backscattered signals to discriminate against clutter and noise while compensating for frequencydependent propagation effects. As a consequence of the significant dielectricproperties contrast between normal and malignant tissue, localized regions of large backscatter energy levels in the image correspond to malignant tumors. A dataadaptive algorithm for removing artifacts in the received signals due to backscatter from the skinbreast interface is also presented. The effectiveness of these algorithms is demonstrated using a variety of numerical breast phantoms based on anatomically realistic MRIderived FDTD models of the breast. Very small (2 mm) malignant tumors embedded within the complex fibroglandular structure of the breast are easily detected above the background clutter. The MIST approach is shown to offer significant improvement in performance over previous UWB microwave breast cancer detection techniques based on simpler focusing schemes. Index Terms—Breast cancer detection, finitedifference timedomain (FDTD), microwave imaging, spacetime beamforming, ultrawideband radar. I.
Passivity Enforcement via Perturbation of Hamiltonian Matrices
 IEEE TRANS. CASI
, 2004
"... This paper presents a new technique for the passivity enforcement of linear timeinvariant multiport systems in statespace form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system inputoutput transfer func ..."
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Cited by 18 (4 self)
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This paper presents a new technique for the passivity enforcement of linear timeinvariant multiport systems in statespace form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system inputoutput transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on firstorder perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that firstorder perturbation is applicable. Several examples illustrate and validate the procedure.
Efficient Electrostatic and Electromagnetic Simulation Using IES³
 IEEE Computational Science and Engineering
, 1998
"... Integral equation techniques are often used to extract models of integrated circuit structures. This extraction involves solving a dense system of linear equations, and using direct solution methods is prohibitive for large problems. In this paper, we present IES 3 (pronounced "ice cube"), a fast ..."
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Cited by 17 (2 self)
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Integral equation techniques are often used to extract models of integrated circuit structures. This extraction involves solving a dense system of linear equations, and using direct solution methods is prohibitive for large problems. In this paper, we present IES 3 (pronounced "ice cube"), a fast Integral Equation Solver for threedimensional problems with arbitrary kernels. We apply our method to solving electrostatic problems and electromagnetic problems in the electrically small regime (i.e., when circuit structures are at most a wavelength or so in size). The overall approach gives O(N log N) complexity, where N is the number of panels in a discretization of the conductor surfaces. 1 Introduction Extracting compact, accurate linear models for packages, interconnect, and components plays a significant role in modern Radio Frequency designs. Models can be extracted in a variety of ways, but for the high accuracy that critical sections of RF designs demand, only direct numeric sim...