Nonreflecting Boundary Conditions for Time-Dependent Scattering

Grote, Marcus J.; Keller, Joseph B.

doi:10.1006/jcph.1996.0157

Cited by 151 publications

(153 citation statements)

References 5 publications

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“…Because the atomistic displacement includes various wave numbers, the governing equations of motion lead to dispersion during wave propagation. Therefore, interfacial conditions derived for scattering problems like non-reflecting boundary conditions and Dirichlet-to-Neumann methods cannot be applied here [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

A mathematical framework of the bridging scale method

Tang

Hou

Liu

2005

Numerical Meth Engineering

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SUMMARYIn this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we identify two error sources in this method. First, we use a linear finite element interpolation, and derive the coarse grid equations directly from Newton's second law. Numerical error in this length scale exists mainly due to inadequate approximation for the effects of the fine scale fluctuations. An modified linear element (MLE) scheme is developed to improve the accuracy. Secondly, we derive an exact multiscale interfacial condition to treat the interfaces between the molecular dynamics region D and the complementary domain C , using a time history kernel technique. The interfacial condition proposed in the original BSM may be regarded as a leading order approximation to the exact one (with respect to the coarsening ratio). This approximation is responsible for minor reflections across the interfaces, with a dependency on the choice of D . We further illustrate the framework and analysis with linear and non-linear lattices in one-dimensional space.

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Section: Introductionmentioning

confidence: 99%

A mathematical framework of the bridging scale method

Tang

Hou

Liu

2005

Numerical Meth Engineering

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“…The area of ABC's has been the subject of massive research, see for example [8,12,10,16,2], where the approach in [16,2] yields local boundary conditions that can be made arbitrarily accurate. When it comes to the implementation of exact NRBC's it is, for special geometries, possible to localize the boundary conditions in time while still keeping them exact.…”

Section: Introductionmentioning

confidence: 99%

Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

Eriksson

Nordström

2016

Found Comput Math

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Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

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“…Its construction requires the solution of the onedimensional wave equation on B. Grote and Keller [11,12] developed an exact NRBC (which can be also regarded as a high-order converging NRBC) for the three-dimensional timedependent wave equation, based on spherical harmonic transformations. They extended this NRBC for the case of elastic waves in Reference [13].…”

Section: Introductionmentioning

confidence: 99%

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Joolen

Neta

Givoli

2005

Numerical Meth Engineering

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SUMMARYRecently developed non-reflecting boundary conditions are applied for exterior time-dependent wave problems in unbounded domains. The linear time-dependent 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 high-order non-reflecting 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 high-order 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 two-dimensional problem, with a rectangular boundary.

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Nonreflecting Boundary Conditions for Time-Dependent Scattering

Cited by 151 publications

References 5 publications

A mathematical framework of the bridging scale method

A mathematical framework of the bridging scale method

Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

High‐order Higdon‐like boundary conditions for exterior transient wave problems

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