61 results on '"Björn Engquist"'
Search Results
2. Iterated averaging of three-scale oscillatory systems
- Author
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Seong Jun Kim, Gil Ariel, Björn Engquist, and Richard Tsai
- Subjects
Scale (ratio) ,Iterated function ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Applied mathematics ,Mathematics ,Method of averaging - Published
- 2014
3. Application of Optimal Transport and the Quadratic Wasserstein Metric to Full-Waveform Inversion
- Author
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Yunan Yang, Björn Engquist, Brittany F. Hamfeldt, and Junzhe Sun
- Subjects
Mathematical analysis ,FOS: Physical sciences ,Inversion (meteorology) ,Numerical Analysis (math.NA) ,010502 geochemistry & geophysics ,01 natural sciences ,Convexity ,Geophysics (physics.geo-ph) ,010101 applied mathematics ,Physics - Geophysics ,Geophysics ,Amplitude ,Quadratic equation ,Geochemistry and Petrology ,Wasserstein metric ,Norm (mathematics) ,FOS: Mathematics ,Time domain ,Mathematics - Numerical Analysis ,0101 mathematics ,Full waveform ,0105 earth and related environmental sciences ,Mathematics - Abstract
Conventional full-waveform inversion (FWI) using the least-squares norm ($L^2$) as a misfit function is known to suffer from cycle skipping. This increases the risk of computing a local rather than the global minimum of the misfit. In our previous work, we proposed the quadratic Wasserstein metric ($W_2$) as a new misfit function for FWI. The $W_2$ metric has been proved to have many ideal properties with regards to convexity and insensitivity to noise. When the observed and predicted seismic data are regarded as two density functions, the quadratic Wasserstein metric corresponds to the optimal cost of rearranging one density into the other, where the transportation cost is quadratic in distance. The difficulty of transforming seismic signals into nonnegative density functions is discussed. Unlike the $L^2$ norm, $W_2$ measures not only amplitude differences, but also global phase shifts, which helps to avoid cycle skipping issues. In this work, we build on our earlier method to cover more realistic high-resolution applications by embedding the $W_2$ technique into the framework of the adjoint-state method and applying it to seismic relevant 2D examples: the Camembert, the Marmousi, and the 2004 BP models. We propose a new way of using the $W_2$ metric trace-by-trace in FWI and compare it to global $W_2$ via the solution of the Monge-Amp\`ere equation. With corresponding adjoint source, the velocity model can be updated using the l-BFGS method. Numerical results show the effectiveness of $W_2$ for alleviating cycle skipping issues and sensitivity to noise. Both mathematical theory and numerical examples demonstrate that the quadratic Wasserstein metric is a good candidate for a misfit function in seismic inversion., Comment: 30 pages, 51 figures, 2nd version
- Published
- 2016
4. Convexity of the quadratic Wasserstein metric as a misfit function for full-waveform inversion
- Author
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Björn Engquist, Junzhe Sun, and Yunan Yang
- Subjects
Mathematical analysis ,Finite difference ,Inversion (meteorology) ,010502 geochemistry & geophysics ,01 natural sciences ,Convexity ,010101 applied mathematics ,Nonlinear system ,Quadratic equation ,Wasserstein metric ,Time domain ,0101 mathematics ,Full waveform ,0105 earth and related environmental sciences ,Mathematics - Published
- 2016
5. A Multiscale Method for Highly Oscillatory Dynamical Systems Using a Poincaré Map Type Technique
- Author
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Gil Ariel, Richard Tsai, Björn Engquist, Seong Jun Kim, and Yoonsang Lee
- Subjects
Numerical Analysis ,Dynamical systems theory ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Ode ,Projection (linear algebra) ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Ordinary differential equation ,State space ,Vector field ,Software ,Subspace topology ,Poincaré map ,Mathematics - Abstract
We propose a new heterogeneous multiscale method (HMM) for computing the effective behavior of a class of highly oscillatory ordinary differential equations (ODEs). Without the need for identifying hidden slow variables, the proposed method is constructed based on the following ideas: a nonstandard splitting of the vector field (the right hand side of the ODEs); comparison of the solutions of the split equations; construction of effective paths in the state space whose projection onto the slow subspace has the correct dynamics; and a novel on-the-fly filtering technique for achieving a high order accuracy. Numerical examples are given.
- Published
- 2012
6. Consistent modeling of boundaries in acoustic finite-difference time-domain simulations
- Author
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Björn Engquist and Jon Häggblad
- Subjects
Physics ,Time Factors ,Acoustics and Ultrasonics ,Wave propagation ,Mathematical analysis ,Finite-difference time-domain method ,Computational mathematics ,Numerical Analysis, Computer-Assisted ,Acoustics ,Acoustic wave ,Models, Theoretical ,Grid ,Motion ,Sound ,Classical mechanics ,Arts and Humanities (miscellaneous) ,Pressure ,Scattering, Radiation ,Acoustic wave equation ,Computer Simulation ,Boundary value problem ,Time domain - Abstract
The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback, however, is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material. In this paper zero boundary data are considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively. A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in Tornberg and Engquist [J. Comput. Phys. 227, 6922-6943 (2008)]. Here this treatment is generalized to three dimensions. Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced. The structure of the standard method is preserved, making it easy to use in existing solvers. The effectiveness is demonstrated in several numerical tests.
- Published
- 2012
7. A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements
- Author
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Lexing Ying, Björn Engquist, and Paul Tsuji
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Preconditioner ,Applied Mathematics ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference ,Mixed finite element method ,Computer Science::Numerical Analysis ,Finite element method ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Maxwell's equations ,Electromagnetic field solver ,Modeling and Simulation ,symbols ,Mathematics ,Extended finite element method ,Stiffness matrix - Abstract
This paper is concerned with preconditioning the stiffness matrix resulting from finite element discretizations of Maxwell's equations in the high frequency regime. The moving PML sweeping preconditioner, first introduced for the Helmholtz equation on a Cartesian finite difference grid, is generalized to an unstructured mesh with finite elements. The method dramatically reduces the number of GMRES iterations necessary for convergence, resulting in an almost linear complexity solver. Numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method.
- Published
- 2012
8. Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
- Author
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Björn Engquist and Lexing Ying
- Subjects
Variable coefficient ,Helmholtz equation ,Preconditioner ,Ecological Modeling ,Mathematical analysis ,65F08, 65N22, 65N80 ,MathematicsofComputing_NUMERICALANALYSIS ,General Physics and Astronomy ,General Chemistry ,High Frequency Waves ,Mathematics::Numerical Analysis ,Computer Science Applications ,symbols.namesake ,Modeling and Simulation ,Green's function ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Mathematics - Numerical Analysis ,Mathematics - Abstract
This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate $LDL^t$ factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost and the preconditioned iterative solver converges in a number of iterations that is essentially indefinite of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner., Comment: 25 pages
- Published
- 2011
9. Gaussian beam decomposition of high frequency wave fields using expectation–maximization
- Author
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Richard Tsai, Björn Engquist, Gil Ariel, and Nicolay Tanushev
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Numerical analysis ,Gaussian ,Mathematical analysis ,Wave equation ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Superposition principle ,Fourier transform ,Modeling and Simulation ,Expectation–maximization algorithm ,symbols ,Physics::Accelerator Physics ,Beam (structure) ,Mathematics ,Gaussian beam - 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.
- Published
- 2011
10. Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions
- Author
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Eric T. Chung and Björn Engquist
- Subjects
Numerical Analysis ,Computational Mathematics ,Finite volume method ,Discontinuous Galerkin method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Dissipative system ,Acoustic wave equation ,Galerkin method ,Finite element method ,Mathematics ,Numerical stability - Abstract
In this paper, we develop and analyze 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 apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
- Published
- 2009
11. A multiscale method for highly oscillatory ordinary differential equations with resonance
- Author
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Richard Tsai, Gil Ariel, and Björn Engquist
- Subjects
Backward differentiation formula ,Algebra and Number Theory ,Differential equation ,Oscillation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Ode ,Geometry ,Resonance (particle physics) ,L-stability ,Computational Mathematics ,Ordinary differential equation ,Mathematics - 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 on ...
- Published
- 2008
12. Consistent boundary conditions for the Yee scheme
- Author
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Björn Engquist and Anna-Karin Tornberg
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Structure (category theory) ,Boundary (topology) ,Wave equation ,Space (mathematics) ,Stability (probability) ,Computer Science Applications ,Computational Mathematics ,Two-dimensional space ,Modeling and Simulation ,Convergence (routing) ,Boundary value problem ,Mathematics - Abstract
A new set of consistent boundary conditions for Yee scheme approximations of wave equations in two space dimensions are developed and analyzed. We show how the classical staircase boundary conditions for hard reflections or, in the electromagnetic case, conducting surfaces in certain cases give O(1) errors. The proposed conditions keep the structure of the Yee scheme and are thus well suited for high performance computing. The higher accuracy is achieved by modifying the coefficients in the difference stencils near the boundary. This generalizes our earlier results with Gustafsson and Wahlund in one space dimension. We study stability and convergence and we present numerical examples.
- Published
- 2008
13. Asymptotic and numerical homogenization
- Author
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Panagiotis E. Souganidis and Björn Engquist
- Subjects
Numerical Analysis ,Nonlinear system ,Partial differential equation ,Differential equation ,General Mathematics ,Computation ,Numerical analysis ,Mathematical analysis ,Homogenization (chemistry) ,Numerical partial differential equations ,Mathematics ,Numerical stability - 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.
- Published
- 2008
14. On surface radiation conditions for high-frequency wave scattering
- Author
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Björn Engquist and Andreas Atle
- Subjects
Electromagnetics ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Acoustics ,Physical optics ,Computational Mathematics ,symbols.namesake ,symbols ,Neumann boundary condition ,Logarithmic derivative ,Boundary value problem ,Perfect conductor ,On surface radiation condition ,Bessel function ,Mathematics - 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.
- Published
- 2007
15. A new type of boundary treatment for wave propagation
- Author
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Bertil Gustafsson, Per Wahlund, Anna-Karin Tornberg, and Björn Engquist
- Subjects
Finite volume method ,Computer Networks and Communications ,Wave propagation ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Computational mathematics ,Type (model theory) ,Minimax approximation algorithm ,Computational Mathematics ,Rate of convergence ,Boundary value problem ,Computer Science::Databases ,Computer Science::Distributed, Parallel, and Cluster Computing ,Software ,Mathematics - Abstract
Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary co ...
- Published
- 2006
16. Discretization of Dirac delta functions in level set methods
- Author
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Anna-Karin Tornberg, Björn Engquist, and Richard Tsai
- Subjects
Numerical Analysis ,Dirac measure ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Mathematical analysis ,Regularization perspectives on support vector machines ,Dirac delta function ,Regularization (mathematics) ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Level set ,Singular function ,Modeling and Simulation ,symbols ,Piecewise ,Mathematics - 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.
- Published
- 2005
17. Heterogeneous multiscale methods for stiff ordinary differential equations
- Author
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Björn Engquist and Yen-Hsi Richard Tsai
- Subjects
Backward differentiation formula ,Computational Mathematics ,Runge–Kutta methods ,Algebra and Number Theory ,Differential equation ,Applied Mathematics ,Numerical analysis ,Ordinary differential equation ,Mathematical analysis ,Numerical methods for ordinary differential equations ,Explicit and implicit methods ,Numerical stability ,Mathematics - Abstract
The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing differe ...
- Published
- 2005
18. Convergence Analysis of Fully Discrete Finite Volume Methods for Maxwell's Equations in Nonhomogeneous Media
- Author
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Björn Engquist and Eric T. Chung
- Subjects
Numerical Analysis ,Finite volume method ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Space (mathematics) ,Stability (probability) ,Computational Mathematics ,symbols.namesake ,Maxwell's equations ,Convergence (routing) ,symbols ,Order (group theory) ,Voronoi diagram ,Mathematics - 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.
- Published
- 2005
19. Numerical approximations of singular source terms in differential equations
- Author
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Björn Engquist and Anna-Karin Tornberg
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Differential equation ,Applied Mathematics ,Mathematical analysis ,Immersed boundary method ,Grid ,Regularization (mathematics) ,Computer Science Applications ,Numerical integration ,Computational Mathematics ,Dimensional regularization ,Modeling and Simulation ,Numerical stability ,Mathematics ,Numerical partial differential equations - Abstract
Singular terms in differential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi-dimensions. New high order multi-dimensional techniques for differential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions in level-set methods to extend one dimensional regularization to higher dimensions may produce O(1) errors.
- Published
- 2004
20. Computational high frequency wave propagation
- Author
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Björn Engquist and Olof Runborg
- Subjects
Physics ,Wavefront ,Ray tracing (physics) ,Numerical Analysis ,Mathematical optimization ,Partial differential equation ,Computer simulation ,Geometrical optics ,Wave propagation ,Eikonal equation ,General Mathematics ,Mathematical analysis ,Vlasov equation - 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.
- Published
- 2003
21. [Untitled]
- Author
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Björn Engquist and Anna-Karin Tornberg
- Subjects
Numerical Analysis ,Differential equation ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,General Engineering ,Immersed boundary method ,Regularization (mathematics) ,Theoretical Computer Science ,Computational Mathematics ,Discontinuity (linguistics) ,Computational Theory and Mathematics ,Numerical approximation ,Rate of convergence ,Gravitational singularity ,Software ,Mathematics - Abstract
The rate of convergence for numerical methods approximating differential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coefficients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious deficiencies in existing regularization methods are also pointed out.
- Published
- 2003
22. High-Frequency Wave Propagation by the Segment Projection Method
- Author
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Olof Runborg, Björn Engquist, and Anna-Karin Tornberg
- Subjects
Numerical Analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Geometrical optics ,Wave propagation ,Eikonal equation ,Applied Mathematics ,Mathematical analysis ,Geometry ,Wave equation ,Computer Science Applications ,Ray tracing (physics) ,Computational Mathematics ,Superposition principle ,Modeling and Simulation ,Projection method ,Mathematics - Abstract
Geometrical optics is a standard technique used for the approximation of high-frequency wave propagation. Computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics. These new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves. In this paper we introduce a new partial differential technique based on the segment projection method in phase space. The superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics. The computational complexity is of the same order as that of ray tracing. The new algorithm is described and a number of computational examples are given, including a simulation of waveguides.
- Published
- 2002
23. Numerical methods for multiscale inverse problems
- Author
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Christina Frederick and Björn Engquist
- Subjects
Physics ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Mathematical analysis ,Mathematics::Analysis of PDEs ,65N21, 35R25, 65N30, 35B27 ,02 engineering and technology ,Numerical Analysis (math.NA) ,Inverse problem ,010502 geochemistry & geophysics ,01 natural sciences ,Multiscale modeling ,Stability (probability) ,Wavelength ,020303 mechanical engineering & transports ,0203 mechanical engineering ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Nabla symbol ,0105 earth and related environmental sciences - Abstract
We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0 0$, and exploration seismology, $b < 0$.
- Published
- 2014
- Full Text
- View/download PDF
24. A Contribution to Wavelet-Based Subgrid Modeling
- Author
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Gunnar Ledfelt, Olof Runborg, Ulf Andersson, and Björn Engquist
- Subjects
Numerical analysis ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Sparse approximation ,01 natural sciences ,Homogenization (chemistry) ,Haar wavelet ,Physics::Fluid Dynamics ,010101 applied mathematics ,Wavelet ,Applied mathematics ,0101 mathematics ,Scale model ,Mathematics ,Numerical stability ,Numerical partial differential equations - 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.
- Published
- 1999
- Full Text
- View/download PDF
25. The Convergence Rate of Finite Difference Schemes in the Presence of Shocks
- Author
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Björn Engquist and Björn Sjögreen
- Subjects
Shock wave ,Pointwise ,Numerical Analysis ,Computational Mathematics ,Runge–Kutta methods ,Rate of convergence ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Finite difference method ,Order of accuracy ,Mathematics ,Shock (mechanics) - Abstract
Finite difference approximations generically have ${\cal O}(1)$ pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.
- Published
- 1998
26. Absorbing boundary conditions for domain decomposition
- Author
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Hongkai Zhao and Björn Engquist
- Subjects
Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Mixed boundary condition ,Robin boundary condition ,Poincaré–Steklov operator ,Computational Mathematics ,symbols.namesake ,Dirichlet boundary condition ,Neumann boundary condition ,Free boundary problem ,symbols ,Cauchy boundary condition ,Boundary value problem ,Mathematics - 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.
- Published
- 1998
27. Wavelet-Based Numerical Homogenization
- Author
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Mihai Dorobantu and Björn Engquist
- Subjects
Operator splitting ,Numerical Analysis ,Computational Mathematics ,Elliptic operator ,Elliptic curve ,Wavelet ,Operator (computer programming) ,Differential equation ,Applied Mathematics ,Mathematical analysis ,Homogenization (chemistry) ,Mathematics - Abstract
A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination 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 are proved. The procedure can be applied to problems that are not cell-periodic.
- Published
- 1998
28. Convergence of a Multigrid Method for Elliptic Equations with Highly Oscillatory Coefficients
- Author
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Erding Luo and Björn Engquist
- Subjects
Dirichlet problem ,Numerical Analysis ,Computational Mathematics ,Partial differential equation ,Multigrid method ,Rate of convergence ,Discretization ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference method ,Finite difference ,Mathematics - 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.
- Published
- 1997
29. Blowup of solutions of the unsteady Prandtl's equation
- Author
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Weinan E and Björn Engquist
- Subjects
Physics::Fluid Dynamics ,symbols.namesake ,Class (set theory) ,Mathematics::General Mathematics ,Applied Mathematics ,General Mathematics ,Prandtl number ,Mathematical analysis ,symbols ,Mathematics - Abstract
We prove that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtl's equation blow up in nite time
- Published
- 1997
30. New Coarse Grid Operators for Highly Oscillatory Coefficient Elliptic Problems
- Author
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Erding Luo and Björn Engquist
- Subjects
Numerical Analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Iterative method ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Grid ,Computer Science Applications ,Mathematical Operators ,Computational Mathematics ,Multigrid method ,Mesh generation ,Modeling and Simulation ,Convergence (routing) ,Mathematics - 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.
- Published
- 1996
31. Multi-phase computations in geometrical optics
- Author
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Olof Runborg and Björn Engquist
- Subjects
Partial differential equation ,Independent equation ,Eikonal equation ,Applied Mathematics ,Mathematical analysis ,Euler equations ,Nonlinear system ,symbols.namesake ,Computational Mathematics ,Multigrid method ,Simultaneous equations ,symbols ,Numerical partial differential equations ,Mathematics - Abstract
In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited. In particular, it does not include solutions with multiple phases, corresponding to crossing waves. Our objective has been to generalize these equations to accommodate solutions containing more than one phase. The new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations. However, they also incorporate a finite superposition principle. The maximum allowed number of intersecting waves in the solution can be chosen arbitrarily, but a higher number means that a larger system of PDEs must be solved. The PDEs form a hyperbolic system of conservation laws with source terms.Although the equations are only weakly hyperbolic, and thus not well-posed in the strong sense, several examples show the viability of solving the equations numerically. The technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments.
- Published
- 1996
- Full Text
- View/download PDF
32. Numerical Solution of the High Frequency Asymptotic Expansion for the Scalar Wave Equation
- Author
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E Fatemi, Stanley Osher, and Björn Engquist
- Subjects
Numerical Analysis ,Asymptotic analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Differential equation ,Applied Mathematics ,Mathematical analysis ,First-order partial differential equation ,Wave equation ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Modeling and Simulation ,Asymptotic expansion ,Scalar field ,Mathematics - Abstract
New numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation. The nonlinear partial differential equations defining the terms in the expansio ...
- Published
- 1995
33. 1. Methods to Numerically Solve Wave Equations
- Author
-
Björn Engquist, Mrinal K. Sen, Matthew M. Haney, K. Sandberg, Stéphane Operto, Chris Chapman, I. Štekl, Martin Käser, Bernhard Hustedt, Anthony T. Patera, Jozef Kristek, Henrik Bernth, Lexing Ying, Martin Galis, Jian-Ping Wang, A. Tagliani, Jean Virieux, Josep de la Puente, Hafedh Ben Hadj Ali, J. Etgen, R.-E. Plessix, G. Beylkin, Heiner Igel, Yogi A. Erlangga, Géza Seriani, Peter Moczo, Peter Mora, William W. Symes, Bruno Riollet, Jean-Pierre Vilotte, Thomas Bohlen, Luc Giraud, Fabio Cavallini, Enrico Priolo, Eleuterio F. Toro, R. Brossier, Patrick Amestoy, Arthur C. H. Cheng, Jian-Fei Lu, Robert W. Graves, Andrzej Hanyga, José M. Carcione, Jean-Yves L'Excellent, Michael Dumbser, A. Quarteroni, E. Zampieri, Yang Liu, J. Dellinger, Dimitri Komatitsch, Erik H. Saenger, and R. G. Pratt
- Subjects
Physics ,Mathematical analysis ,Finite difference method ,Inhomogeneous electromagnetic wave equation ,Wave equation - Published
- 2012
34. Fast Wavelet Based Algorithms for Linear Evolution Equations
- Author
-
Stanley Osher, Björn Engquist, and Sifen Zhong
- Subjects
Speedup ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Type (model theory) ,Parabolic partial differential equation ,Integral equation ,Computational Mathematics ,Wavelet ,Boundary value problem ,Hyperbolic partial differential equation ,Algorithm ,Mathematics - Abstract
A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.
- Published
- 1994
35. A Coupled Finite Difference – Gaussian Beam Method for High Frequency Wave Propagation
- Author
-
Yen-Hsi Richard Tsai, Björn Engquist, and Nicolay Tanushev
- Subjects
Physics ,Wavelength ,Geometrical optics ,Speed of sound ,Mathematical analysis ,Finite difference ,Finite difference method ,Domain decomposition methods ,M squared ,Gaussian beam - Abstract
Approximations of geometric optics type are commonly used in simulations of high frequency wave propagation. This form of technique fails when there is strong local variation in the wave speed on the scale of the wavelength or smaller. We propose a domain decomposition approach, coupling Gaussian beam methods where the wave speed is smooth with finite difference methods for the wave equations in domains with strong wave speed variation. In contrast to the standard domain decomposition algorithms, our finite difference domains follow the energy of the wave and change in time. A typical application in seismology presents a great simulation challenge involving the presence of irregularly located sharp inclusions on top of a smoothly varying background wave speed. These sharp inclusions are small compared to the domain size. Due to the scattering nature of the problem, these small inclusions will have a significant effect on the wave field. We present examples in two dimensions, but extensions to higher dimensions are straightforward.
- Published
- 2011
36. Gaussian beam decomposition for seismic migration
- Author
-
Richard Tsai, Nicolay Tanushev, Björn Engquist, and Sergey Fomel
- Subjects
Physics ,business.industry ,Wave propagation ,Gaussian ,Mathematical analysis ,Seismic migration ,Wave equation ,Seismic wave ,Physics::Geophysics ,Ray tracing (physics) ,symbols.namesake ,Optics ,symbols ,Reflection seismology ,business ,Gaussian beam - Abstract
In reflection seismology, seismic waves are recorded at the surface of the earth and migrated to image the subsurface. In mathematical terms, this is a boundary-value problem for the wave equation, where the boundary data are the recorded seismic wavefields. Gaussian beams, which are localized solutions of the wave equation, can be used for seismic migration. However, before Gaussian beams can be used for a solution of the boundary-value problem, the recorded data have to be represented in a compatible form. We present a new method for decomposing seismic data as a sum of Gaussian beams, based on transforming boundary data to a propagating wavefield. A simple synthetic seismic migration example illustrates our method and shows how different classes of Gaussian-beam parameters (such as curvature and extent) control the accuracy and sparseness of data representation.
- Published
- 2011
37. Sweeping preconditioner for the 3D Helmholtz equation
- Author
-
Björn Engquist, Lexing Ying, and Jack Poulson
- Subjects
Matrix (mathematics) ,Helmholtz equation ,Factorization ,Preconditioner ,Frequency domain ,Mathematical analysis ,Schur complement ,Boundary value problem ,Wave equation ,Mathematics - Abstract
TheHelmholtz equation describes wave propagation in the frequency domain and,as such, can be used for seismic imaging and fullwaveform inversion. We present two novel preconditioners for the efficientsolution of the Helmholtz equation in three dimensions. Both methodsfollow the general structure of constructing an approximate LDLt factorizationby eliminating the unknowns layer by layer starting from anabsorbing layer or boundary condition. In the first approach, werepresent the Schur complement matrices of the factorization in thehierarchical matrix framework. In the second approach, applying each Schurcomplement matrix is equivalent to solving a quasi-2D problem withthe multifrontal method. These preconditioners have linear application cost, andthe preconditioned iterative solvers converge in a number of iterationsthat is essentially independent of the number of unknowns orthe frequency. Numerical results on realistic 3D seismic models toconfirm the efficiency of these methods.
- Published
- 2011
38. Numerical methods for oscillatory solutions to hyperbolic problems
- Author
-
Björn Engquist and Jian-Guo Liu
- Subjects
FTCS scheme ,Partial differential equation ,Discretization ,Weak convergence ,Approximations of π ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Mathematical analysis ,First-order partial differential equation ,Hyperbolic partial differential equation ,Mathematics - Abstract
Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths. 01993 John Wiley & Sons, Inc.
- Published
- 1993
39. Sweeping Preconditioner for the Helmholtz Equation: Hierarchical Matrix Representation
- Author
-
Lexing Ying and Björn Engquist
- Subjects
Discretization ,Helmholtz equation ,Iterative method ,Preconditioner ,Applied Mathematics ,General Mathematics ,Hierarchical matrix ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,65F08, 65N22, 65N80 ,Solver ,Generalized minimal residual method ,Factorization ,Mathematics - Numerical Analysis ,Mathematics - Abstract
The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach., Comment: 34 pages
- Published
- 2010
40. Triangle based adaptive stencils for the solution of hyperbolic conservation laws
- Author
-
Stanley Osher, Björn Engquist, and Louis J. Durlofsky
- Subjects
Numerical Analysis ,Conservation law ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Linear interpolation ,Computer Science Applications ,Burgers' equation ,Computational Mathematics ,Modeling and Simulation ,Total variation diminishing ,Hyperbolic partial differential equation ,Linear equation ,Mathematics ,Interpolation - 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.
- Published
- 1992
41. Fast Hybrid Algorithms for High Frequency Scattering
- Author
-
Björn Engquist, Khoa Tran, Lexing Ying, Börje Nilsson, Louis Fishman, Anders Karlsson, and Sven Nordebo
- Subjects
Field (physics) ,Helmholtz equation ,Scattering ,Numerical analysis ,Fast multipole method ,Mathematical analysis ,Boundary (topology) ,Scattering theory ,Integral equation ,Mathematics - Abstract
This paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of a scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. On the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem. Numerical examples are given for the solution of the Helmholtz equation in two space dimensions.
- Published
- 2009
42. Projection shock capturing algorithms
- Author
-
Björn Engquist and Björn Sjögreen
- Subjects
Discretization ,Computer science ,Mathematical analysis ,Finite difference method ,Computational mathematics ,Point (geometry) ,Function (mathematics) ,Classification of discontinuities ,Projection (linear algebra) ,Computational physics ,Shock (mechanics) - Abstract
discretized in time and space with step sizes At, £xx and Ay, t~ = n a t , xj = j£xx, Yk = kay. The numerical solution in the point (t~, xj, Yk) is denoted by uj~ and the entire solution at t ime t~ is u ~ and a function of j, k. We consider the solution of the above problem with finite difference methods. The problem is difficult because of the occurrence of discontinuities in the solution. One of the more widely used ways to derive a second order method with good shock capturing properties is to advance the solution with a first order method containing sufficient numerical damping ~, = a ( ~ ~)
- Published
- 2008
43. Long-time behaviour of absorbing boundary conditions
- Author
-
Laurence Halpern and Björn Engquist
- Subjects
Boundary conditions in CFD ,General Mathematics ,Mathematical analysis ,General Engineering ,Neumann boundary condition ,Mason–Weaver equation ,Free boundary problem ,Boundary value problem ,Mixed boundary condition ,Different types of boundary conditions in fluid dynamics ,Robin boundary condition ,Mathematics - Abstract
A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.
- Published
- 1990
44. Multiscale methods for the wave equation
- Author
-
Olof Runborg, Björn Engquist, and Henrik Holst
- Subjects
Computational Mathematics ,Beräkningsmatematik ,Mathematical analysis ,Computational mathematics ,Applied mathematics ,multiscale methods ,wave equation ,Construct (python library) ,HMM ,Wave equation ,Computer Science::Databases ,Mathematics - Abstract
We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale schemebased on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in themedium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoreticalresults and numerical examples. QC 20111117
- Published
- 2007
45. Multiscale Methods in Science and Engineering
- Author
-
Björn Engquist, Olof Runborg, and Per Lötstedt
- Subjects
Multigrid method ,Discontinuous Galerkin method ,Numerical analysis ,Mathematical analysis ,Monte Carlo method ,Linear elasticity ,Computational electromagnetics ,Homogenization (chemistry) ,Inverted pendulum ,Mathematics - Abstract
Multiscale Discontinuous Galerkin Methods for Elliptic Problems with Multiple Scales.- Discrete Network Approximation for Highly-Packed Composites with Irregular Geometry in Three Dimensions.- Adaptive Monte Carlo Algorithms for Stopped Diffusion.- The Heterogeneous Multi-Scale Method for Homogenization Problems.- A Coarsening Multigrid Method for Flow in Heterogeneous Porous Media.- On the Modeling of Small Geometric Features in Computational Electromagnetics.- Coupling PDEs and SDEs: The Illustrative Example of the Multiscale Simulation of Viscoelastic Flows.- Adaptive Submodeling for Linear Elasticity Problems with Multiscale Geometric Features.- Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems.- Multipole Solution of Electromagnetic Scattering Problems with Many, Parameter Dependent Incident Waves.- to Normal Multiresolution Approximation.- Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics.- Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem.- Multiscale Homogenization of the Navier-Stokes Equation.- Numerical Simulations of the Dynamics of Fiber Suspensions.
- Published
- 2005
46. Examples of Error Propagation from Discontinuities
- Author
-
Björn Sjögreen and Björn Engquist
- Subjects
Physics ,Propagation of uncertainty ,Partial differential equation ,Mathematical analysis ,Flux limiter ,Classification of discontinuities ,Mathematics::Geometric Topology - Abstract
We shall consider systems of hyperbolic partial differential equations, $${{u}_{t}} + f{{(u)}_{x}} = 0.$$
- Published
- 1998
47. Absorbing boundary conditions for acoustic and elastic wave equations
- Author
-
Björn Engquist and Robert W. Clayton
- Subjects
Physics ,Geophysics ,Geochemistry and Petrology ,Simple (abstract algebra) ,Computation ,Mathematical analysis ,Paraxial approximation ,Scalar (physics) ,Range (statistics) ,Boundary value problem ,Wave equation ,Domain (mathematical analysis) - Abstract
Boundary conditions are derived for numerical wave simulation that minimize artificial reflections from the edges of the domain of computation. In this way acoustic and elastic wave propagation in a limited area can be efficiently used to describe physical behavior in an unbounded domain. The boundary conditions are based on paraxial approximations of the scalar and elastic wave equations. They are computationally inexpensive and simple to apply, and they reduce reflections over a wide range of incident angles.
- Published
- 1977
48. Numerical solution of a PDE system describing a catalytic converter
- Author
-
Bertil Gustafsson, Joop Vreeburg, and Björn Engquist
- Subjects
Numerical Analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Order of accuracy ,Numerical diffusion ,Exponential integrator ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Modeling and Simulation ,Boundary value problem ,Numerical partial differential equations ,Numerical stability ,Mathematics - Abstract
Numerical approximations are studied for a large hyperbolic system coupled to a parabolic equation and a system of algebraic equations. The equations, which all are nonlinear, describe nonviscous compressible one-dimensional gas flow in a catalytic converter. Chemical reactions within the gas are included in the model. Well-posedness of the partial differential equations is analyzed together with stability of the numerical models. In particular an investigation is made of the effect of numerical dissipation and different boundary conditions. Numerical results are presented.
- Published
- 1978
49. Difference and finite element methods for hyperbolic differential equations
- Author
-
Heinz-Otto Kreiss and Björn Engquist
- Subjects
Mechanical Engineering ,Mathematical analysis ,Computational Mechanics ,Numerical methods for ordinary differential equations ,Finite difference ,General Physics and Astronomy ,Finite difference coefficient ,Mixed finite element method ,Computer Science Applications ,Mechanics of Materials ,Discontinuous Galerkin method ,Smoothed finite element method ,Mathematics ,Extended finite element method ,Numerical partial differential equations - 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.
- Published
- 1979
50. Absorbing boundary conditions for the numerical simulation of waves
- Author
-
Björn Engquist and Andrew J. Majda
- Subjects
Algebra and Number Theory ,Partial differential equation ,Computer simulation ,Applied Mathematics ,Computation ,Mathematical analysis ,Computational mathematics ,Wave equation ,law.invention ,Computational Mathematics ,Singularity ,law ,Cartesian coordinate system ,Boundary value problem ,Mathematics - 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 artificial boundaries. These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.
- Published
- 1977
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