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  • 51.
    Engquist, Björn
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Variable velocity: wave extrapolation and reflection1977Konferensbidrag (Refereegranskat)
  • 52.
    Engquist, Björn
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Wavelet based numerical homogenization1998Konferensbidrag (Refereegranskat)
    Abstract [en]

    Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.

  • 53.
    Engquist, Björn
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Well-posedness of one-way wave equations1976Konferensbidrag (Refereegranskat)
  • 54.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Fatemi, E
    Osher, S
    Numerical solution of high frequency expansion for hyperbolic equations1994Konferensbidrag (Refereegranskat)
  • 55.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Fokas, Antonios
    Heirer, E.
    Iserles, A.
    Highly Oscillatory Problems2009Bok (Refereegranskat)
  • 56.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Fornberg, B
    Johansson, J
    Studiematerial till Numerisk analys1970Bok (Refereegranskat)
  • 57.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Fornberg, B
    Johansson, J
    Studiematerial till Numeriska metoder1970Bok (Refereegranskat)
  • 58.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Golub, G.
    From numerical analysis to computational science2000Ingår i: Mathematics Unlimited: 2001 and Beyond, Springer, 2000, s. 433-448Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    Introduction 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. The electronic computer originated from the intense research and development done during the second world war. In the early applications of these computers the computational techniques that were designed for calculation by pencil and paper or tables and mechanical machines were directly implemented on the new devices. Together with a deeper understanding of the computational processes new algorithms soon emerged. The foundation of modern numerical analysis was built in the period from the late forties to the late fifties. It became justi

  • 59.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Greenbaum, A
    Murphy, W.D.
    Global boundary conditions and fast Helmholtz solvers1989Ingår i: IEEE transactions on magnetics, ISSN 0018-9464, E-ISSN 1941-0069, Vol. 25, nr 4, s. 2804-2806Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Electromagnetic scattering from a conducting two-dimensional cylinder is modeled by solving Helmholtz's equation with the far-field radiation boundary condition replaced by a global boundary condition allowing. This allows the boundary condition to be applied very near the scatterer. The discrete problem is solved by a biconjugate gradient algorithm. IncompleteLUdecomposition is used as a preconditioning strategy, resulting in very fast convergence. The numerical solution is compared with known solutions and found to converge fasters forka⩽10, wherekis the wave number andais the radius of the cylinder

  • 60.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Gustafsson, B
    Steady state computations for wave propagation problems1987Ingår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 49, s. 39-64Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The behavior of difference approximations of hyperbolic partial differential equations as time t → ∞  is studied. The rate of convergence to steady state is analyzed theoretically and expe imentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties

  • 61.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Gustafsson, B
    Third International Conference on Hyperbolic Problems: Proceedings of the conference dedicated to Professor Heinz-Otto Kreiss on his 60th birthday1991 (uppl. 1-2)Bok (Refereegranskat)
    Abstract [en]

    These volumes contain papers from the third International Conference on Hyperbolic Problems, which was held on June 11-15, 1990 in Uppsala, Sweden. The conference reflected the current vitality of research in hyperbolic problems and the interaction between theory, numerical methods and applications. Most of the papers deal with non-linear problems. This is particularly true for the applications where fluid mechanics dominates.

  • 62.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Gustafsson, B
    Vreeburg, J
    Numerical solution of a PDE system describing a catalytic converter1978Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 27, nr 3, s. 295-314Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 63.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Halpern, L
    Far field boundary conditions for computation over long time1988Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 4, nr 1, s. 21-45Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 64.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Halpern, L
    Long-time behavior of absorbing boundary conditions1990Ingår i: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, ISSN 0170-4214, Vol. 13, nr 3, s. 189-203Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 65.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Harten, A
    Osher, S
    A high order essentially non-oscillatory shock capturing method1987Konferensbidrag (Refereegranskat)
  • 66.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time2011Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation following the framework of the heterogeneous multiscale method. The numerical methods couple simulations on macro- and microscales for problems with rapidly fluctuating material coefficients. The computational complexity of the new method is significantly lower than that of traditional techniques. We focus on HMM approximation applied to long time integration of one-dimensional wave propagation problems in both periodic and non-periodic medium and show that the dispersive effect that appear after long time is fully captured.

  • 67.
    Engquist, Björn
    et al.
    University of Texas Austin.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale Methods for One Dimensional Wave Propagation with High Frequency Initial Data2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    High frequency wave propagation problems are computationally costly to solve by traditional techniques because the short wavelength must be well represented over a domain determined by the largest scales of the problem. We have developed and analyzed a new numerical method for high frequency wave propagation in the framework of heterogeneous multiscale methods, closely related to the analytical method of geometrical optics. The numerical method couples simulations on macro- and micro-scales for problems with highly oscillatory initial data. The method has a computational complexity essentially independent of the wavelength. We give one numerical example with a sharp but regular jump in velocity on the microscopic scale for which geometrical optics fails but our HMM gives correct results. We briefly discuss how the method can be extended to higher dimensional problems.

  • 68.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale methods for the wave equation2007Ingår i: PAMM · Proc. Appl. Math. Mech. 7, 2007, s. 1140903-1140904Konferensbidrag (Övrigt vetenskapligt)
    Abstract [en]

    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.

  • 69.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multi-scale methods for wave propagation in heterogeneous media2011Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 9, nr 1, s. 33-56Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Multi-scale 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 multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couple simulations on macro-and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two, and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

  • 70.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Holst, Henrik
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time2012Ingår i: Numerical Analysis of Multiscale Computations / [ed] Björn Engquist, Olof Runborg, Yen-Hsi R. Tsai, Springer Verlag , 2012, s. 167-186Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    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.

  • 71.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Hou, T
    Particle method approximation of oscillatory solutions to hyperbolic differential equations1989Ingår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, ISSN 0036-1429, Vol. 26, nr 2, s. 289-319Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Particle methods approximating hyperbolic partial differential 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 coefficients Carleman models are considered. Homogenization results are given and the approximations of more general systems are discussed. Numercial exampels are presented

  • 72.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Hou, T.Y.
    Computation of oscillatory solutions to hyperbolic differential equations using particle methods1987Konferensbidrag (Refereegranskat)
    Abstract [en]

    Numerical approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continous solution is not well resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models are considered. The continous and the corresponding numerical models converge to the same homogenized limit as the frequency in the oscillation increases.

  • 73.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Hou, T.Y.
    Computation of oscillatory solutions to partial differential equations1988Ingår i: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 1270, s. 68-82Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Numerical approximations of hyperbolic partial differential equations with oscillatory solutions are studied. Convergence is analyzed in the practical case for which the continous solution is not well resolved on the computational grid. Averaged difference approximations of linear problems and particle method approximations of semilinear problems are presented. Highly oscillatory solutions to the Carleman and Broadwell models are considered. The continous and the corresponding numerical models converge to the same homogenized limit as the frequency in the oscillation increases.

  • 74.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Huynh, Q.Q.
    Iterative gradient-Newton type methods for steady shock computations1991Ingår i: SIAM, s. 60-75Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A class of modified Newton´s methods are applied to difference approximations of the two-dimensional steady Burgers´ equation and the transonic small disturbance equation. The solutions have sharp gradients which corresponds to boundare layers and shock waves in fluid dynamics. The nonlinear terms in the differential equations are approximated by modern shock capturing schemes. The regularity of the coefficients is analyzed theoretically and its effect on the convengence on the Newton´s method is studied numerically. Computational results from different types of gradient iterative methods and different types of preconditioners are presented. These methods are applied to the linear systems of the Newton iteration. The relative residuals in the Newton iterations are controlled such that a superlinear rate of convergence is preserved 

  • 75.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Numerical subgrid scale models for the Yee schemeManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The Yee scheme is a very common and practical algorithm for the simulation of wave propagation on uniform grids.  We develop numerical subgrid scale models in order to incorporate effects of obstacles and holes that are smaller than the grid spacing. The models are based on pre-computing at the microscale, and are thus including the effect of the detailed small scale shape.  Numerical examples in 1D, 2D and 3D are given.

  • 76.
    Engquist, Björn
    et al.
    Univ Texas Austin.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D2012Ingår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, nr 3, s. 615-637Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Yee scheme is one of the most popular methods for electromagnetic wave propagation. A main advantage is the structured staggered grid, making it simple and efficient on modern computer architectures. A downside to this is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In this paper we present a method to improve the boundary treatment in two dimensions by, starting from a staircase approximation, modifying the coefficients of the update stencil so that we can obtain a consistent approximation while preserving the energy conservation, structure and the optimal CFL-condition of the original Yee scheme. We prove this in L_2 and verify it by numerical experiments.

  • 77.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Tornberg, Anna-Karin
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    On Consistent Boundary Conditions for the Yee Scheme in 3DManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The standard staircase approximation of curved boundaries in the Yee scheme is inconsistent. Consistency can however be achieved by modifying the algorithm close to the boundary.  We consider a technique to consistently model curved boundaries where the coefficients of the update stencil is modified, thus preserving the Yee structure.  The method has previously been successfully applied to acoustics in two and three dimension, as well as electromagnetics in two dimensions.  In this paper we generalize to electromagnetics in three dimensions.  Unlike in previous cases there is a non-zero divergence growth along the boundary that needs to be projected away.  We study the convergence and provide numerical examples that demonstrates the improved accuracy.

  • 78.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Jiang, A
    Osher, S
    Zhong, S
    Wavelet based algorithms for linear initial value problems1994Konferensbidrag (Refereegranskat)
  • 79.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Johnsson, Lennart
    KTH, Skolan för datavetenskap och kommunikation (CSC), Centra, Parallelldatorcentrum, PDC.
    Hammil, M
    Short, F
    Proceedings PDC Seventh Annual Conference: Simulation and Visualization on the Grid2000 (uppl. 13)Bok (Refereegranskat)
    Abstract [en]

    The Grid is an emerging computational infrastructure, similar to the pervasive energy infrastructure provided by national power grids. Simulation and Visualization on the Grid focuses on applications and technologies on this emerging computational Grid. Readers will find interesting discussions of such Grid technologies as distributed file I/O, clustering, CORBA software infrastructure, tele-immersion, interaction environments, visualization steering and virtual reality as well as applications in biology, chemistry and physics. A lively panel discussion addresses current successes and pitfalls of the Grid. This book provides an understanding of the Grid that offers a persistent, wide-scale infrastructure for solving problems.

  • 80.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Kreiss, H.O.
    Difference and finite element methods for hyperbolic differential equations1979Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 17-18, nr 3, s. 581-596Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 81.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Kriegsmann, G.A.
    IMA Volumes in Mathematics and its Applications: Computational wave propagation1997 (uppl. 86)Bok (Refereegranskat)
  • 82.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Liu, J.-G.
    Numerical methods for oscillatory solutions to hyperbolic problems1993Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 46, nr 10, s. 1327-1361Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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

  • 83.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Lotstedt, P.
    Runborg, O.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Multiscale Modeling and Simulation in Science2009Bok (Refereegranskat)
  • 84.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Lotstedt, P.
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Multiscale methods in science and engineering2005Bok (Refereegranskat)
  • 85.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Luo, E
    Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients1997Ingår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 34, nr 6, s. 2254-2273Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 86.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Luo, E
    Multigrid methods for differential equations with highly oscillatory coefficients1993Konferensbidrag (Refereegranskat)
    Abstract [en]

    New coarse grid multigrid operators for problems with highly oscillatory coefficients aredeveloped. These types of operators are necessary when the characters of the differentialequations on coarser grids or longer wavelengths are different from that on the fine grid.Elliptic problems for composite materials and different classes of hyperbolic problems arepractical examples.The new coarse grid operators can be constructed directly based on the homogenizeddifferential operators or hierarchally computed from the finest grid. Convergence analysisbased on the homogenization theory is given for elliptic problems with periodic coefficientsand some hyperbolic problems. These are classes of equations for which there exists afairly complete theory for the interaction between shorter and longer wavelengths in theproblems. Numerical examples are presented.

  • 87.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Luo, Erding
    Erding New coarse grid operators for highly oscillatory coefficient elliptic problems1996Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 2, s. 296-306Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 88.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Luskin, M
    Majda, A
    Volumes in Mathematics and its Applications: Computational fluid dynamics and reacting gas flows1988 (uppl. 12)Bok (Refereegranskat)
    Abstract [en]

    This volume contains papers presented at the workshop on Computational Fluid Dynamics and Reacting Gas Flows held at the Institute for Mathematics and Its Applications during September, 1986. Computational fluid dynamics has become a research area of central importance to mathematics, science, and technology. It is a subject which brings together applied mathematics and numerical analysis to solve problems in fluid dynamics. Included in this volume is the description of new algorithms which can make possible the discovery of important new scientific phenomena and the development of new technological processes. This volume will be of interest to mathematicians, scientists, and engineers who are interested in the current research of international leaders in numerical analysis and scientific computing

  • 89.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Lötstedt, P
    Sjögreen, B
    Nonlinear filters for efficient shock computation1989Ingår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 52, nr 186, s. 509-537Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

  • 90.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Majda, A
    Absorbing boundary conditions for the numerical simulation of waves1997Ingår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 1, s. 629-651Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artifical boundaries. These boundary conditions not only guarantee stable difference approximations, but also minimize the (unphysical) artificial reflections that occur at the boundaries.

  • 91.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Majda, A
    Numerical radiation boundary conditions for unsteady transonic flow1981Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 40, nr 1, s. 91-103Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A family of numerical boundary conditions for far-field-computational boundaries in calculations involving unsteady transonic flow is devised. These boundary conditions are developed in a systematic fashion from general principles. Both numerical and analytic comparisons with other currently used methods are given.

  • 92.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Majda, A
    Radiation boundary conditions for acoustic and elastic wave calculations1979Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 32, nr 3, s. 314-358Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A technique for developing radiating boundary conditions for artificial computational boundaries is described and applied to a class of problems typical in exploration seismology involving acoustic and elastic wave equations. First, one considers a constant coefficient scalar wave equation where the artificial boundary is one edge of a rectangular domain. By using continued fraction expansions, a systematic sequence of stable highly absorbing boundary conditions with successively better absorbing properties as the order of the boundary conditions increases is obtained. There follows a systematic derivation of a hierarchy of local radiating boundary conditions for the elastic wave equation. A theoretical procedure to guarantee stability at corners of the rectangular domain is worked out. A technique for fitting the discrete radiating boundary conditions directly to the difference scheme itself is proposed.

  • 93.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Osher, S
    One-sided difference approximations for nonlinear conservation laws1981Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 36, s. 321-351Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.

  • 94.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Osher, S
    One-sided difference schemes and transonic flow1980Ingår i: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 77, nr 6, s. 3071-3074Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two one-sided conservation form difference approximations to a scalar one-dimensional convex conservation law are introduced. These are respectively of first- and second-order accuracy and each has the minimum possible band-width. They are nonlinearly stable, they converge only to solutions satisfying the entropy condition, and they have sharp monotone profiles. No such stable approximation of order higher than two is possible. Dimensional splitting algorithms are constructed and used to approximate the small-disturbance equation of transonic flow. These approximations are also nonlinearly stable and without nonphysical limit solutions.

  • 95.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Osher, S
    Stable and entropy satisfying approximations for transonic flow calculations1980Ingår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 34, nr 149, s. 45-75Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Finite difference approximations for the small disturbance equation of transonic flow are developed and analyzed. New schemes of the Cole-Murman type are presented fpr which nonlinear stability is proved. The Cole-Murman scheme may have entropy violating expansion shocks as solutions. In the new schemes the switch between the subsonic and supersonic domains is designed such that these nonphysical shocks are guaranteed not to occur. Results from numercial calculations are given which illustrate these conclusions

  • 96.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Osher, S
    Sommerville, R.C.J.
    Large-scale computations in fluid mechanics: Proceedings of the fifteenth AMS-SIAM summer seminar on applied mathematics held at Scripps Institution of Oceanography1985 (uppl. 1/2)Bok (Refereegranskat)
    Abstract [en]

    This work covers the proceedings of an AMS-SIAM Summer Seminar on Applied Mathematics held at Scripps Institution of Oceanography in 1983, whose purpose was to bring scientists interested in computational fluid mechanics together with numerical analysts and mathematicians working in large-scale computations. The complexity of many contemporary problems of fluid mechanics is so great as to tax the capabilities of present-day computers. There is a real need and opportunity for numerical analysis to aid research on the physical problems of achieving optimal utilization of current computers. Fifty lectures were given on subjects equally divided between mathematics and applications. The numerical modeling included geophysical problems of the atmosphere, ocean, and interior of the earth, and planetary, solar, and stellar atmospheres. Applications ranged from idealized turbulence in laboratory convection models to operational weather prediction.Engineering applications included aerodynamics, combustion, and flow in porous media. Recent advances in numerical analysis which have applications to these problems were stressed.

    These include shock capturing algorithms, spectral methods, boundary treatments, vortex methods, and parallel computing. In addition to specialized research lectures, several speakers gave talks surveying important areas of numerical analysis and computational fluid dynamics.

  • 97.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Osher, S
    Zhong, S
    Fast wavelet based algorithms for linear evolution equations1994Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, ISSN 1064-8275, Vol. 15, nr 4, s. 755-775Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The authors devise a class 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 [Comm. Pure Appl. Math., 44 (1991), pp. 141-1841, which they applied to general Calderon-Zygmund type integral operators. The authors apply a modification of their idea to linear hyperbolic and parabolic equations, with spatially varying coefficients. The complexity for hyperbolic equations in one dimension is reduced from O(N2) to O(N log3 N). There are somewhat better gains for parabolic equations in multidimensions

  • 98.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, O
    Computational high frequency wave propagation2003Ingår i: Acta Numerica, ISSN 0962-4929, E-ISSN 1474-0508, Vol. 12, s. 181-266Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 99.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, O
    Multi-phase computations in geometrical optics1996Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 74, nr 1-2, s. 175-192Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    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.

  • 100.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Runborg, O
    Projection generated homogenisation2002Konferensbidrag (Refereegranskat)
123456 51 - 100 av 293
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