| 1. |
- Engquist, Björn, et al.
(författare)
-
Projection shock capturing algorithms
- 1990
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Ingår i: Twelfth International Conference on Numerical Methods in Fluid Dynamics. - Berlin : Springer-Verlag. ; , s. 335-336
-
Konferensbidrag (refereegranskat)
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| 2. |
- Häggblad, Jon, 1981-
(författare)
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Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
- 2012
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length. The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme. We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.The stability properties of coefficient modifications are essential for practical usability. We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.The implementations and analysis of the developed methods are validated through systematic numerical tests.
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| 3. |
- Abdulle, A., et al.
(författare)
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Finite element heterogeneous multiscale methods with near optimal computational complexity
- 2007
-
Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 6:4, s. 1059-1084
-
Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the heterogeneous multiscale methods (HMM), we propose a micro-macro approach which combines the finite element method (FEM) for the macroscopic solver and the pseudospectral method for the microsolver. Unlike the micro-macromethods based on the standard FEM proposed so far, in the HMM we obtain, for periodic homogenization problems, a method that (slow) variable.
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| 4. |
- Andersson, U, et al.
(författare)
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A contribution to wavelet-based subgrid modeling
- 1999
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Ingår i: Applied and Computational Harmonic Analysis. - 1063-5203 .- 1096-603X. ; 7:2, s. 151-164
-
Tidskriftsartikel (refereegranskat)abstract
- A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations is described. The technique is based on Haar wavelet projections of the discrete operator followed by a sparse approximation. As numerical testing suggests, the resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented.
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| 5. |
- Ariel, Gil, et al.
(författare)
-
A multiscale method for stiff ordinary differential equations with resonance
- 2009
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 78:266, s. 929-956
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.
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| 6. |
- Ariel, Gil, et al.
(författare)
-
A reversible multiscale integration method
- 2009
-
Ingår i: Communications in Mathematical Sciences. - : Duke University Press. - 1539-6746 .- 1945-0796. ; 7:3, s. 595-610
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale, time reversible method for computing the effective slow behavior of systems of highly oscillatory ordinary differential equations is presented. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the system. The algorithm follows the framework of the heterogeneous multiscale method. The notion of time reversibility in the multiple time-scale setting is discussed. The algorithm requires nontrivial matching between the microscopic state variables and the macroscopic slow ones. Numerical examples show the efficiency of the multiscale method and the advantages of time reversibility.
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| 7. |
- Ariel, Gil, et al.
(författare)
-
Gaussian Beam Decomposition of High Frequency Wave Fields Using Expectation-Maximization
- 2011
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 230:6, s. 2303-2321
-
Tidskriftsartikel (refereegranskat)abstract
- A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.
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| 8. |
- Ariel, G., et al.
(författare)
-
Multiscale computations for highly oscillatory problems
- 2009
-
Ingår i: Multiscale Modeling and Simulation in Science. - Berlin, Heidelberg : Springer Berlin/Heidelberg. ; , s. 237-287
-
Konferensbidrag (refereegranskat)abstract
- We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.
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| 9. |
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| 10. |
- Ariel, Gil, et al.
(författare)
-
Numerical multiscale methods for coupled oscillators
- 2009
-
Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 7:3, s. 1387-1404
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale method for computing the effective slow behavior of a system of weakly coupled nonlinear planar oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the ordinary differential equation. The technique is more efficient than existing methods, and its advantages are demonstrated with examples. The algorithm follows the framework of the heterogeneous multiscale method.
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| 11. |
- Atle, Andreas, 1972-
(författare)
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Approximations of Integral Equations for WaveScattering
- 2006
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Wave scattering is the phenomenon in which a wave field interacts with physical objects. An incoming wave is scattered at the surface of the object and a scattered wave is produced. Common practical cases are acoustic, electromagnetic and elastic wave scattering. The numerical simulation of the scattering process is important, for example, in noise control, antenna design, prediction of radar cross sections and nondestructive testing.Important classes of numerical methods for accurate simulation of scattering are based on integral representations of the wave fields and theses representations require the knowledge of potentials on the surfaces of the scattering objects. The potential is typically computed by a numerical approximation of an integral equation that is defined on the surface. We first develop such numerical methods in time domain for the scalar wave equation. The efficiency of the techniques are improved by analytic quadrature and in some cases by local approximation of the potential.Most scattering simulations are done for harmonic or single frequency waves. In the electromagnetic case the corresponding integral equation method is called the method of moments. This numerical approximation is computationally very costly for high frequency waves. A simplification is suggested by physical optics, which directly gives an approximation of the potential without the solution of an integral equation. Physical optics is however only accurate for very high frequencies.In this thesis we improve the accuracy in the physical optics approximation of scalar waves by basing the computation of the potential on the theory of radiation boundary conditions. This theory describes the local coupling of derivatives in the wave field and if it is applied at the surface of the scattering object it generates an expression for the unknown potential. The full wave field is then computed as for other integral equation methods.The new numerical techniques are analyzed mathematically and their efficiency is established in a sequence of numerical experiments. The new on surface radiation conditions give, for example, substantial improvement in the estimation of the scattered waves in the acoustic case. This numerical experiment corresponds to radar cross-section estimation in the electromagnetic case.
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| 12. |
- Atle, Andreas, et al.
(författare)
-
On surface radiation conditions for high-frequency wave scattering
- 2007
-
Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 204:2, s. 306-316
-
Tidskriftsartikel (refereegranskat)abstract
- A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist-Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.
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| 13. |
|
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| 14. |
- Bamberger, A, et al.
(författare)
-
Higher order paraxial wave equation approximations in heterogeneous media
- 1988
-
Ingår i: SIAM Journal on Applied Mathematics. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1399 .- 1095-712X. ; 48, s. 129-154
-
Tidskriftsartikel (refereegranskat)abstract
- A new family of paraxial wave equation approximations is derived. These approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computational problems, e.g., in seismology, underwater acoustics and as artificial boundary conditions. The equations are written as systems which simplify computations. The support and singular support are studied; energy estimates are given which prove the well-posedness. The reflection and transmission are shown to be continuously dependent on material interfaces in heterogeneous media
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| 15. |
- Bamberger, A, et al.
(författare)
-
Parabolic wave equation approximations in heterogenous media
- 1988
-
Ingår i: SIAM Journal on Applied Mathematics. - 0036-1399 .- 1095-712X. ; 48, s. 99-128
-
Tidskriftsartikel (refereegranskat)abstract
- The properties of different variants of parabolic approximations of scalar wave equations are analyzed. These equations are of general form which includes those used in seismology, underwater acoustics and other applications. A new version of the parabolic approximation is derived for heterogeneous media. It has optimal properties with respect to wave reflection at material interfaces. The amplitud of the reflected and transmitted waves depend continuously on the interface. Existence, uniqueness and energy estimates are proved.
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| 16. |
- Brenan, K.E., et al.
(författare)
-
Backward differentiation approximations of nonlinear differential/algebraic systems
- 1988
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 51:84, s. 659-676
-
Tidskriftsartikel (refereegranskat)abstract
- Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.
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| 17. |
- Carstensen, Carsten, et al.
(författare)
-
Computational Multiscale Methods
- 2009
-
Ingår i: Oberwolfach Reports. - : European Mathematical Society Publishing House. - 1660-8933 .- 1660-8941. ; 6:2, s. 1597-1599
-
Tidskriftsartikel (refereegranskat)abstract
- Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientific communities.
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| 18. |
- Chung, E., et al.
(författare)
-
Convergence analysis of fully discrete finite volume methods for Maxwell's equations in nonhomogeneous media
- 2005
-
Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 43:1, s. 303-317
-
Tidskriftsartikel (refereegranskat)abstract
- We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi-Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates.
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| 19. |
- Chung, Eric T., et al.
(författare)
-
Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions
- 2009
-
Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 47:5, s. 3820-3848
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we developed and analyzed a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we proposed a new way to handle the second order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
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| 20. |
- Chung, Eric T., et al.
(författare)
-
Optimal discontinuous Galerkin methods for wave propagation
- 2006
-
Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 44:5, s. 2131-2158
-
Tidskriftsartikel (refereegranskat)abstract
- We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.
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| 21. |
- Clayton, R, et al.
(författare)
-
Absorbing boundary conditions for acoustic and elastic wave equations
- 1977
-
Ingår i: Bulletin of The Seismological Society of America (BSSA). - : Elsevier BV. - 0037-1106 .- 1943-3573. ; 67:6, s. 1529-1540
-
Tidskriftsartikel (refereegranskat)abstract
- The two dimensional elastic wave equations are used to model wave propagation in mediums with large or unbounded domains. In order to numerically simulate those problems the equations have to be put on the computer and artificial boundary conditions that allow minimal wave reflection must be introduced. These boundary conditions are well known in the literature as absorbing or radiating boundary conditions. They have been of interest to physicists and meteorologists for some time. In this paper three different methods for deriving radiating boundary conditions for the elastic wave equations are presented. One of these techniques gives exact absorbing boundary conditions for both P (longitudinal) and S (transverse) waves generated from a surface source. From this, approximate absorbing boundary conditions are derived. These conditions are easy to employ in finite difference codes.
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| 22. |
- Clayton, R.W., et al.
(författare)
-
Absorbing boundary conditions for wave-equation migration
- 1980
-
Ingår i: Geophysics. - : Society of Exploration Geophysicists. - 0016-8033 .- 1942-2156. ; 45:5, s. 895-904
-
Tidskriftsartikel (refereegranskat)abstract
- The standard boundary conditions used at the sides of’s seismic section in wave-equation migration generateartificial reflections. These reflections from the edges of the computational grid appear as artifacts in the finalsection. Padding the section with zero traces on either 5ide adds to the cost of’migration and simply delays theine\ itable reflections.We develop stable absorbing boundary condition5 that annihilate almost all of the artificial reflections. Thisih demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented canbe used with any ofthe different type\ of finite-diftcrence wa\e-equation migration, at essentially no extra cost.
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| 23. |
- Deuflhard, P, et al.
(författare)
-
Large scale scientific computing : Proceedings of workshop held at the Oberwolfach Mathematical Institute
- 1987. - 7
-
Bok (refereegranskat)abstract
- The series Lecture Notes in Computer Science (LNCS), including its subseries Lecture Notes in Artificial Intelligence (LNAI) and Lecture Notes in Bioinformatics (LNBI), has established itself as a medium for the publication of new developments in computer science and information technology research and teaching - quickly, informally, and at a high level. The cornerstone of LNCS's editorial policy is its unwavering commitment to report the latest results from all areas of computer science and information technology research, development, and education. LNCS has always enjoyed close cooperation with the computer science R & D community, with numerous renowned academics, and with prestigious institutes and learned societies. Our mission is to serve this community by providing a most valuable publication service. LNCS commenced publication in 1973 and quite rapidly attracted attention, not least because of its thus far unprecedented publication turnaround times. The 1980s and 1990s witnessed a substantial growth in the series, particularly in terms of volumes published. In the late 1990s we developed a systematic approach to providing LNCS in a full-text electronic version, in parallel to the printed books. Another new feature introduced in the late 1990s was the conceptualization of a couple of color-cover sublines. Still, original research results reported in proceedings and postproceedings remain the core of LNCS.
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| 24. |
- Dorobantu, M, et al.
(författare)
-
Wavelet-based numerical homogenization
- 1998
-
Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 35:2, s. 540-559
-
Tidskriftsartikel (refereegranskat)abstract
- A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in find and coarse scale components followed by the elemination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory is proved. The procedure can be applied to problems that are not cell-periodic.
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| 25. |
- Durlofsky, L. J., et al.
(författare)
-
Triangle based adaptive stencils for the solution of hyperbolic conservation laws
- 1992
-
Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 98:1, s. 64-73
-
Tidskriftsartikel (refereegranskat)abstract
- A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.
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| 26. |
- E., Weinan, et al.
(författare)
-
Heterogeneous multiscale methods : A review
- 2007
-
Ingår i: Communications in Computational Physics. - 1815-2406 .- 1991-7120. ; 2:3, s. 367-450
-
Forskningsöversikt (refereegranskat)abstract
- This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, micro-fluidics, solids, interface problems, stochastic problems, and statistically self-similar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.
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| 27. |
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| 28. |
- Eliasson, P, et al.
(författare)
-
The effect of dissipation and coarse grid resolution for multigrid in flow problem
- 1996
-
Konferensbidrag (refereegranskat)abstract
- This paper is to investigate the effects of the numerical dissipation and the resolution of the solution on coarser grids for multigrid with the Euler equation approximations. The convergence is accomplished by multi-stage explicit time-stepping to steady state accelerated by FAS multigrid. A theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy. Numerical results give grid independent convergence in one dimension. For twodimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density
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| 29. |
- Engquist, Björn, et al.
(författare)
-
A fast directional algorithm for high frequency acoustic scattering in two dimensions
- 2009
-
Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 7:2, s. 327-345
-
Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with fast solution of high frequency acoustic scattering problems in two dimensions. We introduce a directional multiscale algorithm for the $N$-body problem of the two dimensional Helmholtz kernel. The algorithm follows the approach developed in Engquist and Ying, SIAM J. Sci. Comput., 29 (4), 2007, where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric conguration, the interaction between these two regions through the 2D Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank separated representation for the interaction between these regions. Based on this representation, the computation of the far field interaction is organized in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(NlogN)$ complexity for problems from two dimensional scattering applications. Finally, we combine this fast directional algorithm with standard boundary integral formulations to solve acoustic scattering problems that are of thousands of wavelengths in size.
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| 30. |
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| 31. |
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| 32. |
- Engquist, Björn, et al.
(författare)
-
A system for automatic code generation for hyperbolic problems
- 1975
-
Ingår i: Signum Newsletter. - New York, New York, USA : ACM Press. ; 10, s. 20-22
-
Tidskriftsartikel (refereegranskat)abstract
- DCG is a program which generates FORTRAN code for difference approximations from a mathematical specification of a hyperbolic system. In our system the user presents his problem in a mathematical notation. DCG analyses the problem, chooses algorithms and generates a complete program for the numerical integration. The main parts of DCG are a program for symbolic analysis, “the analyser”, a library of FORTRAN-code and a program, “the synthesiser” that composes a complete FORTRAN program using code from the library.
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| 33. |
- Engquist, Björn, et al.
(författare)
-
Absorbing boundary conditions for domain decomposition
- 1998
-
Ingår i: Applied Numerical Mathematics. - 0168-9274 .- 1873-5460. ; 27:4, s. 315-324
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we would like to point out some similarities between two artificial boundary conditions. One is the far field or absorbing boundary conditions for computations over unbounded domain. The other is the boundary conditions used at the boundary between subdomains in domain decomposition. We show some convergence result for the generalized Schwarz alternating method (GSAM), in which a convex combination of Dirichlet data and Neumann data is exchanged at the artificial boundary. We can see clearly how the mixed boundary condition and the relative size of overlap will affect the convergence rate. These results can be extended to more general coercive elliptic partial differential equations using the equivalence of elliptic operators. Numerically first- and second-order approximations of the Dirichlet-to-Neumann operator are constructed using local operators, where information tangential to the boundary is included. Some other possible extensions and applications are pointed out. Finally numerical results are presented.
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| 34. |
- Engquist, Björn, et al.
(författare)
-
Absorbing boundary conditions for the numerical simulation of waves
- 1997
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 1, s. 629-651
-
Tidskriftsartikel (refereegranskat)abstract
- In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artifical boundaries. These boundary conditions not only guarantee stable difference approximations, but also minimize the (unphysical) artificial reflections that occur at the boundaries.
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| 35. |
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| 36. |
- Engquist, Björn, et al.
(författare)
-
Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time
- 2011
-
Tidskriftsartikel (refereegranskat)abstract
- Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation following the framework of the heterogeneous multiscale method. The numerical methods couple simulations on macro- and microscales for problems with rapidly fluctuating material coefficients. The computational complexity of the new method is significantly lower than that of traditional techniques. We focus on HMM approximation applied to long time integration of one-dimensional wave propagation problems in both periodic and non-periodic medium and show that the dispersive effect that appear after long time is fully captured.
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| 37. |
- Engquist, Björn, et al.
(författare)
-
Asymptotic and numerical homogenization
- 2008
-
Ingår i: Acta Numerica. - 0962-4929 .- 1474-0508. ; 17, s. 147-190
-
Tidskriftsartikel (refereegranskat)abstract
- Homogenization is an important mathematical framework for developing effective models of differential equations with oscillations. We include in the presentation techniques for deriving effective equations, a brief discussion on analysis of related limit processes and numerical methods that are based on homogenization principles. We concentrate on first- and second-order partial differential equations and present results concerning both periodic and random media for linear as well as nonlinear problems. In the numerical sections, we comment on computations of multi-scale problems in general and then focus on projection-based numerical homogenization and the heterogeneous multi-scale method.
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| 38. |
- Engquist, Björn, et al.
(författare)
-
Automatic analysis in PDE software
- 1984
-
Ingår i: PDE Software. - Amsterdam, The Netherlands : Elsevier Science. - 0444876200 ; , s. 399-409
-
Konferensbidrag (refereegranskat)
|
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| 39. |
- Engquist, Björn, et al.
(författare)
-
Automatic computer code generation for hyperbolic and parabolic differential equations
- 1980
-
Ingår i: SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING. - : Society for Industrial & Applied Mathematics (SIAM). - 0196-5204 .- 2168-3417. ; 1:2, s. 249-259
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Tidskriftsartikel (refereegranskat)abstract
- A program system which generates computer code for the numerical solution of systems of hyperbolic and parabolic differential equations is described. The input to the program is a mathematical formulation of a hyperbolic or parabolic initial boundary value problem in one space dimension. The differential equations and boundary conditions are analyzed by the program system, and a finite difference algorithm is designed for the given problem. The output is an executable FORTRAN program.
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| 40. |
- Engquist, Björn, et al.
(författare)
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Computation of oscillatory solutions to hyperbolic differential equations using particle methods
- 1987
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Konferensbidrag (refereegranskat)abstract
- Numerical approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continous solution is not well resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models are considered. The continous and the corresponding numerical models converge to the same homogenized limit as the frequency in the oscillation increases.
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| 41. |
- Engquist, Björn, et al.
(författare)
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Computation of oscillatory solutions to partial differential equations
- 1988
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Ingår i: Lecture notes in mathematics. - 0075-8434 .- 1617-9692. ; 1270, s. 68-82
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Tidskriftsartikel (refereegranskat)abstract
- Numerical approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continous solution is not well resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models are considered. The continous and the corresponding numerical models converge to the same homogenized limit as the frequency in the oscillation increases.
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| 42. |
- Engquist, Björn, et al.
(författare)
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Computational high frequency wave propagation
- 2003
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Ingår i: Acta Numerica. - 0962-4929 .- 1474-0508. ; 12, s. 181-266
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Tidskriftsartikel (refereegranskat)abstract
- Numerical simulation of high frequency acoustic, elastic or electro-magnetic wave propagation is important in many applications. Recently the traditional techniques of ray tracing based on geometrical optics have been augmented by numerical procedures based on partial differential equations. Direct simulations of solutions to the eikonal equation have been used in seismology, and lately approximations of the Liouville or Vlasov equation formulations of geometrical optics have generated impressive results. There are basically two techniques that follow from this latter approach: one is wave front methods and the other moment methods. We shall develop these methods in some detail after a brief review of more traditional algorithms for simulating high frequency wave propagation.
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| 43. |
- Engquist, Björn, et al.
(författare)
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Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients
- 1997
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications. - 0036-1429 .- 1095-7170. ; 34:6, s. 2254-2273
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Tidskriftsartikel (refereegranskat)abstract
- Standard multigrid methods are not so effective for equations with highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.
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| 44. |
- Engquist, Björn, et al.
(författare)
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Difference and finite element methods for hyperbolic differential equations
- 1979
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 17-18:3, s. 581-596
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Tidskriftsartikel (refereegranskat)abstract
- In recent years finite element methods have started to be applied to hyperbolic equations. Since modern finite element and finite difference methods for hyperbolic equations look very much alike, new results in the analysis of difference methods are also applicable to element methods. We shall discuss propagation of sharp signals, problems with different time scales and the effect of boundaries on stability and accuracy.
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| 45. |
- Engquist, Björn, et al.
(författare)
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Discretization of Dirac delta functions in level set methods
- 2005
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 207:1, s. 28-51
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Tidskriftsartikel (refereegranskat)abstract
- Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions. It is independent of the irregularity relative to the grid. In the second method, the regularization is constructed from a one-dimensional regularization that is extended to multi-dimensions with a variable support depending on the orientation of the singularity relative to the computational grid. Convergence analysis and numerical results are given. © 2005 Published by Elsevier Inc.
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| 46. |
- Engquist, Björn, et al.
(författare)
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Erding New coarse grid operators for highly oscillatory coefficient elliptic problems
- 1996
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Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 2, s. 296-306
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Tidskriftsartikel (refereegranskat)abstract
- New coarse grid operators are developed for elliptic problems with highly oscillatory coefficients. The new coarse grid operators are constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. A detailed description of this construction is provided. Numerical calculations for a two dimensional elliptic model problem show that the homogenized form of the equations is very useful in the design of coarse grid operators for the multigrid method. A more realistic problem of heat conduction in a composite structure is also considered.
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| 47. |
- Engquist, Björn, et al.
(författare)
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Far field boundary conditions for computation over long time
- 1988
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Ingår i: Applied Numerical Mathematics. - 0168-9274 .- 1873-5460. ; 4:1, s. 21-45
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Tidskriftsartikel (refereegranskat)abstract
- A new class of computational far field boundary conditions for hyperbolic partial differential equations is developed. These boundary conditions combine properties of absorbing boundary conditions for transient solutions and properties of far field boundary conditions for steady-state problems. The conditions can be used to limit the computational domain when both traveling waves and evanescent waves are present. Boundary conditions for scalar wave equations are derived and analyzed. Extensions to systems of equations are discussed and results from numerical experiments are presented.
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| 48. |
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| 49. |
- Engquist, Björn, et al.
(författare)
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Fast directional multilevel algorithms for oscillatory kernels
- 2007
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 29:4, s. 1710-1737
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Tidskriftsartikel (refereegranskat)abstract
- This paper introduces a new directional multilevel algorithm for solving N-body or N-point problems with highly oscillatory kernels. These systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non-uniformity of the particle distribution. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O(1/r) and is at a distance which is at least O(r(2)) away from from the ball. We then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated, low rank representation. Based on the resulting representations, our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency. The algorithm performs well on a large group of highly oscillatory kernels. Our algorithm is proved to have O(N log N) computational complexity for any given accuracy when the points are sampled from a two dimensional surface. We also provide numerical results to demonstrate these properties.
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| 50. |
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