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An optimal perfectly matched layer with unbounded absorbing function for timeharmonic . . .
, 2006
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Highorder higdonlike boundary conditions for exterior transient wave problems
 International Journal for Numerical Methods in Engineering 63 (7
, 2005
"... Recently developed nonreflecting boundary conditions are applied for exterior timedependent wave problems in unbounded domains. The linear timedependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial bounda ..."
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Recently developed nonreflecting boundary conditions are applied for exterior timedependent wave problems in unbounded domains. The linear timedependent wave equation, with or without a dispersive term, is considered in an infinite domain. 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 numerically in the finite domain bounded by B. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves highorder derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on B. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior twodimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliaryvariable formulation longtime corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in
A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations
 International Journal for Numerical Methods in Fluids 2011; DOI:10.1002/fld.2456
"... systems of partial differential equations ..."
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Continuum and Discrete InitialBoundaryValue Problems and Einstein’s Field Equations
, 2012
"... Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black hole problem within Einstein’s theory of gravitation, in which ..."
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Cited by 3 (2 self)
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Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initialboundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being
Perfectly matched layers
 in: S. Marburg and B. Nolte (Eds.), Computational Acoustics of Noise Propagation in Fluids
, 2008
"... One problem to be tackled for the numerical solution of any scattering problem in an unbounded domain is truncating the computational domain without perturbing too much the solution of the original problem. In an ideal framework, this truncation should satisfy, at least, three properties: efficienc ..."
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One problem to be tackled for the numerical solution of any scattering problem in an unbounded domain is truncating the computational domain without perturbing too much the solution of the original problem. In an ideal framework, this truncation should satisfy, at least, three properties: efficiency, easiness of
Perfectly matched layers for timeharmonic second order elliptic problems
 ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING
"... The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for timeharmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the H ..."
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The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for timeharmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the twodimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using nonintegrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.
Unstructured HighOrder GalerkinTemporalBoundary Methods for the KleinGordon Equation with NonReflecting Boundary Conditions
, 2010
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Absorbing Boundaries for the Nonlinear Schrödinger Equation
, 2008
"... We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF) which is used to solve time dependent Nonlinear Schrodinger Equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time, we decompose the solut ..."
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We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF) which is used to solve time dependent Nonlinear Schrodinger Equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time, we decompose the solution into a family of coherent states. Those coherent states which are outgoing are deleted, while those which are not are kept, thus minimizing the problem of reflected (wrapped) waves. Numerical results are given, and rigorous error estimates are described. 1