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126
Radiation Boundary Condition for the Numerical Simulation of Waves
 Acta Numerica
, 1999
"... We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of ..."
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Cited by 91 (3 self)
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We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient timedependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to
A New Discrete Transparent Boundary Condition for Standard and Wide Angle "Parabolic" Equations in Underwater Acoustics
"... This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nic ..."
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Cited by 48 (16 self)
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This paper is concerned with transparent boundary conditions (TBCs) for standard and wide angle “parabolic” equations (SPE, WAPE) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC is derived from the discrete whole–space problem that yields an unconditionally stable scheme. The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems.
Perfectly matched layers for hyperbolic systems: General formulation, wellposedness and stability
 SIAM J. Appl. Math
"... Abstract. Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limted to special cases. In particular, the basic question of whether or not ..."
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Cited by 41 (5 self)
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Abstract. Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limted to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is wellposed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell’s equations, the linearized Euler equations, as well as arbitrary 2×2 systems in (2 + 1) dimensions. Key words. Perfectly matched layers, stability. AMS subject classifications. 35L45, 35B35 1. Introduction. Many
A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
, 2008
"... ..."
Highorder nonreflecting boundary scheme for timedependent waves
 Journal of Computational Physics
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Highorder nonreflecting boundary conditions for dispersive waves
, 2003
"... Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the fin ..."
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Cited by 24 (5 self)
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Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the finite domain bounded by B. The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original loworder implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdontype NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via numerical examples.
2007a): A modified Trefftz method for twodimensional Laplace equation considering the domain’s characteristic length
 CMES: Computer Modeling in Engineering & Sciences
"... Abstract: A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for twodimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by th ..."
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Cited by 20 (9 self)
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Abstract: A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for twodimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equation, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accurate as one desired. Then, we use the collocation method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to explore why the conventional Trefftz method is failure and the modified one still survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numerical results with absolute errors about in the orders from 10−10 to 10−16. The new method is powerful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary.
A Perfectly Matched Layer Approach to the Linearized Shallow Water Equations Models
, 2002
"... A limitedarea model of linearized shallow water equations (SWE) on an fplane for a rectangular domain is considered. The rectangular domain is extended to include the socalled perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the eq ..."
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Cited by 19 (2 self)
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A limitedarea model of linearized shallow water equations (SWE) on an fplane for a rectangular domain is considered. The rectangular domain is extended to include the socalled perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the equations are obtained in this layer by splitting the shallow water equations in the coordinate directions and introducing the absorption coefficients. The performance of the PML as an absorbing boundary treatment is demonstrated using a commonly employed bellshaped Gaussian initially introduced at the center of the rectangular physical domain. Three
A vortex particle method for twodimensional compressible flow
 J. COMPUT. PHYS
, 2002
"... A vortex particle method is developed for simulating twodimensional, unsteady compressible flow. The method uses the Helmholtz decomposition of the velocity field to separately treat the irrotational and solenoidal portions of the flow, and the particles are allowed to change volume to conserve mas ..."
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Cited by 18 (3 self)
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A vortex particle method is developed for simulating twodimensional, unsteady compressible flow. The method uses the Helmholtz decomposition of the velocity field to separately treat the irrotational and solenoidal portions of the flow, and the particles are allowed to change volume to conserve mass. In addition to having vorticity and dilatation properties, the particles also carry density, enthalpy, and entropy. The resulting evolution equations contain terms that are computed with techniques used in some incompressible methods. Truncation of unbounded domains via a nonreflecting boundary condition is also considered. The fast multipole method is adapted to compressible particles in order to make the method computationally efficient. The new method is applied to several problems, including sound generation by corotating vortices and generation of vorticity by baroclinic torque.
Quadratic Optimization in the Problems of Active Control of Sound
, 2002
"... We analyze the problem of suppressing the unwanted component of a timeharmonic acoustic #eld #noise# on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate antisound. Pre ..."
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Cited by 17 (5 self)
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We analyze the problem of suppressing the unwanted component of a timeharmonic acoustic #eld #noise# on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate antisound. Previously,wehave obtained general solutions for active controls in both continuous and discrete formulations of the problem. Wehave also obtained optimal solutions that minimize the overall absolute acoustic source strength of active control sources. These optimal solutions happen to be particular layers of monopoles on the perimeter of the protected region. Mathematically, minimization of acoustic source strength is equivalent to minimization in the sense of L 1 .