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Radiation Boundary Conditions for Maxwell's Equations: A Review of Accurate Time{domain Formulations
 J. Comp. Math
"... We review timedomain formulations of radiation boundary conditions for Maxwell’s equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on iden ..."
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We review timedomain formulations of radiation boundary conditions for Maxwell’s equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.
HighOrder TwoWay Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering
 J. Comput. Phys
, 2001
"... this paper we focus on the critical case, which corresponds to the physical selffocusing (# 1 and D 2). In that case, solutions of the NLS can become singular (i.e., blow up) after finite propagation distance, provided that their initial power (L norm) is above a certain threshold N c , which is ca ..."
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Cited by 10 (8 self)
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this paper we focus on the critical case, which corresponds to the physical selffocusing (# 1 and D 2). In that case, solutions of the NLS can become singular (i.e., blow up) after finite propagation distance, provided that their initial power (L norm) is above a certain threshold N c , which is called the critical power
Dynamics of steps along a martensitic phase boundary I: semianalytical solution
"... We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase transforming material with a stressstrain law that is piecewise linear with respect to one component of shear strain and ..."
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Cited by 6 (0 self)
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We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase transforming material with a stressstrain law that is piecewise linear with respect to one component of shear strain and linear with respect to another. Under these assumptions we derive a semianalytical solution describing a steady sequential motion of the steps under an external loading. Our analysis yields kinetic relations between the driving force, the velocity of the steps and other characteristic parameters of the motion. These are studied in detail for the twostep and threestep configurations. We show that the kinetic relations are significantly affected by the material anisotropy. Our results indicate the existence of multiple solutions exhibiting sequential step motion. Key words: lattice model, phase boundary, interphase step, sequential motion 1
Artificial boundary conditions for the numerical simulation of unsteady acoustic waves
 Author information Frédéric Nataf, Laboratory J.L. Lions, UPMC and CNRS
"... A central characteristic feature of an important class of hyperbolic PDEs in odddimension spaces is the presence of lacunae, i.e., sharp aft fronts of the waves, in their solutions. This feature, which is often associated with the Huygens ’ principle, is employed to construct accurate nonlocal arti ..."
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A central characteristic feature of an important class of hyperbolic PDEs in odddimension spaces is the presence of lacunae, i.e., sharp aft fronts of the waves, in their solutions. This feature, which is often associated with the Huygens ’ principle, is employed to construct accurate nonlocal artificial boundary conditions (ABCs) for the Maxwell equations. The setup includes the propagation of electromagnetic waves from moving sources over unbounded domains. For the purpose of obtaining a finite numerical approximation the domain is truncated, and the ABCs guarantee that the outer boundary be completely transparent for all the outgoing waves. The lacunaebased approach has earlier been used for the scalar wave equation, as well as for acoustics. In the current paper, we emphasize the key distinctions between those previously studied models and the Maxwell equations of electrodynamics, as they manifest themselves in the context of lacunaebased integration. The extent of temporal nonlocality of the proposed ABCs is fixed and limited, and this is not a result of any approximation, it is rather an immediate implication of the existence of lacunae. The ABCs can be applied to any numerical scheme that is used to integrate the Maxwell equations. They do not require any geometric adaptation, and they guarantee that the accuracy of the boundary treatment will not deteriorate even on long time intervals. The paper presents a number of numerical demonstrations that corroborate the theoretical design features of the boundary conditions.
PERFECTLY MATCHED LAYERS WITH HIGH RATE DAMPING FOR HYPERBOLIC SYSTEMS
"... We propose a simple method for constructing nonreflecting boundary conditions via Perfectly Matched Layer approach. The basic idea of the method is to build a layer with high rate damping properties with are provided by adding the stiff relaxation source terms to all equations of the system. No com ..."
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We propose a simple method for constructing nonreflecting boundary conditions via Perfectly Matched Layer approach. The basic idea of the method is to build a layer with high rate damping properties with are provided by adding the stiff relaxation source terms to all equations of the system. No complicated modification of the system to be solved is then required. 1
Solving the Helmholtz Equation for General Geometry Using Simple Grids∗
"... The method of difference potentials was originally proposed by Ryaben’kii, and is a generalized discrete version of the method of Calderon’s operators. It handles nonconforming curvilinear boundaries, variable coefficients, and nonstandard boundary conditions while keeping the complexity of the s ..."
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The method of difference potentials was originally proposed by Ryaben’kii, and is a generalized discrete version of the method of Calderon’s operators. It handles nonconforming curvilinear boundaries, variable coefficients, and nonstandard boundary conditions while keeping the complexity of the solver at the level of a finitedifference scheme on a regular structured grid. Compact finite difference schemes enable high order accuracy on small stencils and so require no additional boundary conditions beyond those needed for the differential equation itself. Previously, we have used difference potentials combined with compact schemes for solving transmission/scattering problems in regions of a simple shape. In this paper, we generalize our previous work to incorporate general shaped boundaries and interfaces, including a formulation that involves multiple scattering.
solutions driven by moving sources ✩
"... We propose a methodology for calculating the solution of an initialvalue problem for the threedimensional wave equation over arbitrarily long time intervals. The solution is driven by moving sources that are compactly supported in space for any particular moment of time; the extent of the support ..."
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We propose a methodology for calculating the solution of an initialvalue problem for the threedimensional wave equation over arbitrarily long time intervals. The solution is driven by moving sources that are compactly supported in space for any particular moment of time; the extent of the support is assumed bounded for all times. By a simple change of variables the aforementioned formulation obviously translates into the problem of propagation of waves across a medium in motion, which has multiple applications in unsteady aerodynamics, advective acoustics, and other areas. The algorithm constructed in the paper can employ any appropriate (i.e., consistent and stable) explicit finitedifference scheme for the wave equation. This scheme is used as a core computational technique and modified so that to allow for a nondeteriorating calculation of the solution for as long as necessary. Provided that the original underlying scheme converges in some sense, i.e., in suitable norms with a particular rate, we prove the grid convergence of the new algorithm in the same sense uniformly in time on arbitrarily long intervals. Thus, the new algorithm obviously does not accumulate error in the course of time; besides, it requires only a fixed nongrowing amount of computer resources (memory and processor time) per one time step; these amounts are linear with respect to the grid dimension and thus optimal. The algorithm is inherently threedimensional; it relies on the
unknown title
, 2003
"... Artificial boundary conditions for the numerical simulation accuracy can always be made as high as that of the interior approximation, and it will not deteriorate even when integrating over long time intervals. Besides, the ABCs are most flexible from the standpoint of geometry and can handle Artif ..."
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Artificial boundary conditions for the numerical simulation accuracy can always be made as high as that of the interior approximation, and it will not deteriorate even when integrating over long time intervals. Besides, the ABCs are most flexible from the standpoint of geometry and can handle Artificial boundary conditions (ABCs) is a common name for a group of methods employed for solving infinitedomain problems on a computer. ABCs facilitate truncation of the original unbounded domain and Journal of Computational Physics 189 (2003) 626–650 *Tel.: +19195151877; fax: +19195153798.Email address: tsynkov@math.ncsu.edu.irregular boundaries on regular grids with no fitting/adaptation needed and no accuracy loss induced. Finally, they allow for a wide range of model settings. In particular, not only one can analyze the simplest advective acoustics case with the
duced by Bérenger [3,4]. He proposed to surround the computational domain by a layer of artificial material * Corresponding author.
"... Numerical solution of infinitedomain problems requires truncation of the unbounded domain for the purpose of constructing a finitedimensional discretization. In doing so, one clearly needs to set some artificial boundary conditions (ABCs) at the outer boundary of the finite computational domain [ ..."
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Numerical solution of infinitedomain problems requires truncation of the unbounded domain for the purpose of constructing a finitedimensional discretization. In doing so, one clearly needs to set some artificial boundary conditions (ABCs) at the outer boundary of the finite computational domain [1,2]. The ABCs shall provide a closure for the truncated formulation and guarantee that its solution will not differ much from the
LacunaeBased Artificial Boundary Conditions for the Numerical Simulation of Unsteady Waves Governed by Vector Models?
"... Abstract. Artificial boundary conditions (ABCs) are constructed for the computation of unsteady acoustic and electromagnetic waves. The waves propagate from a source or a scatterer toward infinity, and are simulated numerically on a truncated domain, while the ABCs provide the required closure at th ..."
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Abstract. Artificial boundary conditions (ABCs) are constructed for the computation of unsteady acoustic and electromagnetic waves. The waves propagate from a source or a scatterer toward infinity, and are simulated numerically on a truncated domain, while the ABCs provide the required closure at the external artificial boundary. They guarantee the complete transparency of this boundary for all the outgoing waves. They are nonlocal in both space and time but can be implemented efficiently because their temporal nonlocality is fixed and limited. The restriction of temporal nonlocality of the proposed ABCs does not come as a result of any model simplification or approximation, but rather as a consequence of a fundamental property of the solutions — the presence of lacunae, or in other words, sharp aft fronts of the waves, in odddimension spaces. 1 Outline of the Algorithm Properties Two major wellrecognized difficulties encountered when computing the propagation of waves over unbounded domains are the accumulation of error during long time intervals and the necessity to truncate the domain and subsequently set the artificial boundary conditions (ABCs) as a closure for the resulting finite