47 results on '"Björn Engquist"'
Search Results
2. Fast algorithm for computing nonlocal operators with finite interaction distance
- Author
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Xiaochuan Tian and Björn Engquist
- Subjects
Peridynamics ,Computational complexity theory ,Computer science ,Applied Mathematics ,General Mathematics ,Hierarchical matrix ,Fast multipole method ,Numerical analysis ,Numerical Analysis (math.NA) ,Differential operator ,01 natural sciences ,Fractional calculus ,Functional Analysis (math.FA) ,010101 applied mathematics ,Mathematics - Functional Analysis ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Multipole expansion - Abstract
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.
- Published
- 2019
- Full Text
- View/download PDF
3. Multiscale numerical methods for passive advection–diffusion in incompressible turbulent flow fields
- Author
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Björn Engquist and Yoonsang Lee
- Subjects
Numerical Analysis ,010504 meteorology & atmospheric sciences ,Scale (ratio) ,Physics and Astronomy (miscellaneous) ,Advection ,Turbulence ,Applied Mathematics ,Numerical analysis ,Computation ,Numerical Analysis (math.NA) ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Multigrid method ,Flow (mathematics) ,Modeling and Simulation ,FOS: Mathematics ,65Mxx, 76F25, 76M50 ,Statistical physics ,Mathematics - Numerical Analysis ,0101 mathematics ,Diffusion (business) ,0105 earth and related environmental sciences ,Mathematics - Abstract
We propose a seamless multiscale method which approximates the macroscopic behavior of the passive advection-diffusion equations with steady incompressible velocity fields with multi-spatial scales. The method uses decompositions of the velocity fields in the Fourier space, which are similar to the decomposition in large eddy simulations. It also uses a hierarchy of local domains with different resolutions as in multigrid methods. The effective diffusivity from finer scale is used for the next coarser level computation and this process is repeated up to the coarsest scale of interest. The grids are only in local domains whose sizes decrease depending on the resolution level so that the overall computational complexity increases linearly as the number of different resolution grids increases. The method captures interactions between finer and coarser scales but has to sacrifice some of the interaction between the fine scales. The proposed method is numerically tested with 2D examples including a successful approximation to a continuous spectrum flow., 20 pages, 8 figures, submitted to Journal of Computational Physics
- Published
- 2016
- Full Text
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4. Sweeping preconditioners for elastic wave propagation with spectral element methods
- Author
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Lexing Ying, Paul Tsuji, Björn Engquist, and Jack Poulson
- Subjects
Numerical Analysis ,Mathematical optimization ,Applied Mathematics ,Wave equation ,Computational Mathematics ,Perfectly matched layer ,Factorization ,Modeling and Simulation ,Frequency domain ,Schur complement ,Applied mathematics ,Boundary value problem ,Element (category theory) ,Analysis ,Block (data storage) ,Mathematics - Abstract
We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LD L T factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.
- Published
- 2014
5. Fast sweeping methods for hyperbolic systems of conservation laws at steady state
- Author
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Brittany D. Froese, Yen-Hsi Richard Tsai, and Björn Engquist
- Subjects
Mathematical optimization ,Steady state (electronics) ,Physics and Astronomy (miscellaneous) ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematical Physics ,Mathematics ,Numerical Analysis ,Conservation law ,Applied Mathematics ,Numerical analysis ,Numerical Analysis (math.NA) ,Mathematical Physics (math-ph) ,Hyperbolic systems ,Computer Science Applications ,Shock (mechanics) ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Hyperbolic partial differential equation - Abstract
Fast sweeping methods have become a useful tool for computing the solutions of static Hamilton-Jacobi equations. By adapting the main idea behind these methods, we describe a new approach for computing steady state solutions to systems of conservation laws. By exploiting the flow of information along characteristics, these fast sweeping methods can compute solutions very efficiently. Furthermore, the methods capture shocks sharply by directly imposing the Rankine-Hugoniot shock conditions. We present convergence analysis and numerics for several one- and two-dimensional examples to illustrate the use and advantages of this approach.
- Published
- 2013
6. Seafloor identification in sonar imagery via simulations of Helmholtz equations and discrete optimization
- Author
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Quyen Q. Huynh, Haomin Zhou, Björn Engquist, and Christina Frederick
- Subjects
Physics and Astronomy (miscellaneous) ,Helmholtz equation ,Acoustics ,Microlocal analysis ,FOS: Physical sciences ,02 engineering and technology ,01 natural sciences ,Sonar ,Physics::Geophysics ,Physics - Geophysics ,Discrete optimization ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,14. Life underwater ,Mathematics - Numerical Analysis ,0101 mathematics ,Numerical Analysis ,Geometrical optics ,Applied Mathematics ,020206 networking & telecommunications ,Ranging ,Numerical Analysis (math.NA) ,Inverse problem ,Seafloor spreading ,Geophysics (physics.geo-ph) ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,00-01, 99-00 ,Modeling and Simulation ,Geology - Abstract
We present a multiscale approach for identifying features in ocean beds by solving inverse problems in high frequency seafloor acoustics. The setting is based on Sound Navigation And Ranging (SONAR) imaging used in scientific, commercial, and military applications. The forward model incorporates multiscale simulations, by coupling Helmholtz equations and geometrical optics for a wide range of spatial scales in the seafloor geometry. This allows for detailed recovery of seafloor parameters including material type. Simulated backscattered data is generated using numerical microlocal analysis techniques. In order to lower the computational cost of the large-scale simulations in the inversion process, we take advantage of a pre-computed library of representative acoustic responses from various seafloor parameterizations.
- Published
- 2016
7. 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
8. 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
9. The heterogeneous multiscale method
- Author
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Assyr Abdulle, Eric Vanden-Eijnden, Björn Engquist, and Weinan E
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Numerical Analysis ,Dynamical systems theory ,Computer science ,General Mathematics ,Finite difference ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,010103 numerical & computational mathematics ,Spall ,Thermal conduction ,01 natural sciences ,Finite element method ,010101 applied mathematics ,ComputingMethodologies_PATTERNRECOGNITION ,Stochastic simulation ,Fracture (geology) ,Statistical physics ,0101 mathematics ,Hidden Markov model - Abstract
The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.
- Published
- 2012
10. 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
11. Gaussian beam decomposition of high frequency wave fields
- Author
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Nicolay Tanushev, Björn Engquist, and Richard Tsai
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Field (physics) ,Applied Mathematics ,Gaussian ,Computer Science Applications ,Computational physics ,Gaussian random field ,Gaussian filter ,Computational Mathematics ,symbols.namesake ,Superposition principle ,Modeling and Simulation ,Quantum mechanics ,symbols ,Gaussian function ,Physics::Accelerator Physics ,Beam (structure) ,Mathematics ,Gaussian beam - Abstract
In this paper, we present a method of decomposing a highly oscillatory wave field into a sparse superposition of Gaussian beams. The goal is to extract the necessary parameters for a Gaussian beam superposition from this wave field, so that further evolution of the high frequency waves can be computed by the method of Gaussian beams. The methodology is described for R^d with numerical examples for d=2. In the first example, a field generated by an interface reflection of Gaussian beams is decomposed into a superposition of Gaussian beams. The beam parameters are reconstructed to a very high accuracy. The data in the second example is not a superposition of a finite number of Gaussian beams. The wave field to be approximated is generated by a finite difference method for a geometry with two slits. The accuracy in the decomposition increases monotonically with the number of beams.
- Published
- 2009
12. 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
13. 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
14. 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
15. 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
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
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24. A finite element based level-set method for multiphase flow applications
- Author
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Anna-Karin Tornberg and Björn Engquist
- Subjects
Level set method ,Discretization ,Numerical analysis ,Multiphase flow ,General Engineering ,Mechanics ,Weak formulation ,Finite element method ,Theoretical Computer Science ,Physics::Fluid Dynamics ,Classical mechanics ,Computational Theory and Mathematics ,Incompressible flow ,Modeling and Simulation ,Computer Vision and Pattern Recognition ,Navier–Stokes equations ,Software ,Mathematics - Abstract
A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects.
- Published
- 2000
25. 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
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26. 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
27. 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
28. 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
29. 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
30. Application of the Wasserstein metric to seismic signals
- Author
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Brittany D. Froese and Björn Engquist
- Subjects
Computer science ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Computation ,Numerical analysis ,Fidelity ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Measure (mathematics) ,Physics::Geophysics ,Simple (abstract algebra) ,Wasserstein metric ,Development (differential geometry) ,Waveform inversion ,Algorithm ,Mathematical Physics ,media_common - Abstract
Seismic signals are typically compared using travel time difference or $L_2$ difference. We propose the Wasserstein metric as an alternative measure of fidelity or misfit in seismology. It exhibits properties from both of the traditional measures mentioned above. The numerical computation is based on the recent development of fast numerical methods for the Monge-Ampere equation and optimal transport. Applications to waveform inversion and registration are discussed and simple numerical examples are presented.
- Published
- 2013
- Full Text
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31. 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
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. Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time
- Author
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Björn Engquist, Olof Runborg, and Henrik Holst
- Subjects
Computational Mathematics ,Scale (ratio) ,Wave propagation ,Beräkningsmatematik ,Numerical analysis ,Direct numerical simulation ,Computational mathematics ,Stability proof ,Statistical physics ,Macro ,Hidden Markov model ,Computational science ,Mathematics - Abstract
Multiscale wave propagation problems are computationally costly to solve by traditional techniques 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 in the framework of the heterogeneous multiscale method (HMM). The numerical methods couple simulations on macro- and microscales for problems with rapidly oscillating coefficients. The complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the smallest scale, when computing solutions at a fixed time and accuracy. We show numerical examples of the HMM applied to long time integration of wave propagation problems in both periodic and non-periodic medium. In both cases our HMM accurately captures the dispersive effects that occur. We also give a stability proof for the HMM, when it is applied to long time wave propagation problems. QC 20111116
- Published
- 2012
34. Numerical Analysis of Multiscale Computations
- Author
-
Yen-Hsi Richard Tsai, Björn Engquist, and Olof Runborg
- Subjects
010101 applied mathematics ,Computer science ,Numerical analysis ,Computation ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences - Published
- 2012
35. 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
36. 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
37. 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
38. 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
39. Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem
- Author
-
Richard Sharp, Björn Engquist, and Yen-Hsi Richard Tsai
- Subjects
Double inverted pendulum ,Scale (ratio) ,Double pendulum ,Control theory ,Ordinary differential equation ,Numerical analysis ,Path (graph theory) ,Ode ,Applied mathematics ,Inverted pendulum ,Mathematics - Abstract
In this article, we study a class of numerical ODE schemes that use a time filtering strategy and operate in two time scales. The algorithms follow the framework of the heterogeneous multiscale methods (HMM) [1]. We apply the methods to compute the averaged path of the inverted pendulum under a highly oscillatory vertical forcing on the pivot. The averaged equation for related problems has been studied analytically in [9]. We prove and show numerically that the proposed methods approximate the averaged equation and thus compute the average path of the inverted pendulum.
- Published
- 2005
40. 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
41. From Numerical Analysis to Computational Science
- Author
-
Björn Engquist and Gene H. Golub
- Subjects
Development (topology) ,Exponential growth ,Computer performance ,Computer science ,Computation ,Numerical analysis ,Gauss ,Numerical computing ,Arithmetic ,Algorithm - Abstract
The modern development of numerical computing is driven by the rapid increase in computer performance. The present exponential growth approximately follows Moore’s law, doubling in capacity every eighteen months. Numerical computing has, of course, been part of mathematics for a very long time. Algorithms by the names of Euclid, Newton and Gauss, originally designed for computation ‘by hand’, are still used today in computer simulations.
- Published
- 2001
42. Wavelet-based algorithms for linear initial-value problems
- Author
-
A. Jiang, Björn Engquist, Stanley Osher, and Sifen Zhong
- Subjects
Theoretical computer science ,Wavelet ,Dimension (vector space) ,Computer science ,Numerical analysis ,Initial value problem ,Applied mathematics ,Hyperbolic systems - Abstract
the complexity decreases tremendously, e.g. for one dimensional parabolicequation it becomes O(log4N) as opposed to O(N(log3N)).We expect to do multidimension hyperbolic problems using the new ideaand get a significant speed-up.For a general multidimensional parabolic equation, the complexity is O(log4 N)).For a d dimensional hyperbolic system the complexity is O(N'22 log3 N). Thisis advantageous for dimension d =
- Published
- 1993
43. 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
44. Selected topics in the theory and practice of computational fluid dynamics
- Author
-
Arthur Rizzi and Björn Engquist
- Subjects
Shock wave ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Explosive material ,business.industry ,Applied Mathematics ,Fluid mechanics ,Mechanics ,Tourbillon ,Computational fluid dynamics ,Computer Science Applications ,Vortex ,Computational Mathematics ,Inviscid flow ,Modeling and Simulation ,Calculus ,business ,Mathematics - Abstract
Computational fluid dynamics (CFD) is a large branch of scientific computing that lately has undergone explosive growth. It draws upon elements from related disciplines: fluid mechanics, numerical ...
- Published
- 1987
45. Some results on uniformly high-order accurate essentially nonoscillatory schemes
- Author
-
Stanley Osher, Sukumar Chakravarthy, Ami Harten, and Björn Engquist
- Subjects
Computational Mathematics ,Numerical Analysis ,Conservation law ,Truncation error (numerical integration) ,Applied Mathematics ,Bounded function ,Shock capturing method ,Total variation diminishing ,Mathematical analysis ,Piecewise ,Order of accuracy ,Hyperbolic partial differential equation ,Mathematics - Abstract
We continue the construction and the analysis of essentially nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy in the sense of truncation error, at extrema of the solution. In this paper we construct an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory. This means that, for piecewise smooth solutions, the variation of the numerical approximation is bounded by that of the true solution up to O(h^R^ ^-^ ^1), for 0
- Published
- 1986
46. Far field boundary conditions for computation over long time
- Author
-
Björn Engquist and Laurence Halpern
- Subjects
Computational Mathematics ,Numerical Analysis ,Boundary conditions in CFD ,Applied Mathematics ,Mathematical analysis ,Free boundary problem ,Neumann boundary condition ,Cauchy boundary condition ,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 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.
- Published
- 1988
47. Particle Method Approximation of Oscillatory Solutions to Hyperbolic Differential Equations
- Author
-
Björn Engquist and Thomas Y. Hou
- Subjects
Numerical Analysis ,Computational Mathematics ,Partial differential equation ,Differential equation ,Continuous solution ,Approximations of π ,Applied Mathematics ,Mathematical analysis ,Particle method ,Grid ,Hyperbolic partial differential equation ,Homogenization (chemistry) ,Mathematics - Abstract
Particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied. Convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid. Highly oscillatory solutions to the Broadwell and variable coefficient Carleman models are considered. Homogenization results are given and the approximations of more general systems are discussed. Numerical examples are presented.
- Published
- 1989
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