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  • 1. Abdulle, Assyr
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Localized orthogonal decomposition method for the wave equation with a continuum of scales2017In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 304, p. 549-587Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

  • 2.
    Ariel, Gil
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Engquist, Björn
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Tsai, Richard
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    A multiscale method for stiff ordinary differential equations with resonance2009In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 78, no 266, p. 929-956Article in journal (Refereed)
    Abstract [en]

    A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

  • 3. Björn, Anders
    et al.
    Riesel, Hans
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Factors of generalized fermat numbers1998In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 67, no 221, p. 441-446Article in journal (Refereed)
  • 4. Björn, Anders
    et al.
    Riesel, Hans
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Factors of generalized fermat numbers (vol 67, pg 441, 1998)2005In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 74, no 252, p. 2099-2099Article in journal (Refereed)
    Abstract [en]

    We note that three factors are missing from Table 1 in Factors of generalized Fermat numbers by A. Bjorn and H. Riesel published in Math. Comp. 67 (1998), 441-446.

  • 5. Björn, Anders
    et al.
    Riesel, Hans
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    FACTORS OF GENERALIZED FERMAT NUMBERS (vol 67, pg 441, 1998): Table errata 22011In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 80, no 275, p. 1865-1866Article in journal (Refereed)
    Abstract [en]

    We note that one more factor is missing from Table 1 in Bjorn-Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441 446, in addition to the three already reported upon in Bjorn-Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), p. 2099.

  • 6. Brenan, K.E.
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Backward differentiation approximations of nonlinear differential/algebraic systems1988In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 51, no 84, p. 659-676Article in journal (Refereed)
    Abstract [en]

    Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

  • 7.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustafsson, B
    Steady state computations for wave propagation problems1987In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 49, p. 39-64Article in journal (Refereed)
    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

  • 8.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Lötstedt, P
    Sjögreen, B
    Nonlinear filters for efficient shock computation1989In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 52, no 186, p. 509-537Article in journal (Refereed)
    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.

  • 9.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Majda, A
    Absorbing boundary conditions for the numerical simulation of waves1997In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 1, p. 629-651Article in journal (Refereed)
    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.

  • 10.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Osher, S
    Stable and entropy satisfying approximations for transonic flow calculations1980In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 34, no 149, p. 45-75Article in journal (Refereed)
    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

  • 11.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tsai, Y. H.
    Heterogeneous multiscale methods for stiff ordinary differential equations2005In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 74, no 252, p. 1707-1742Article in journal (Refereed)
    Abstract [en]

    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 different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

  • 12.
    Gallouët, Thierry
    et al.
    Université de Provence.
    Larcher, Aurélien
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Latché, Jean-Claude
    Institut de Sûreté et de Radioprotection Nucléaire.
    Convergence of a finite volume scheme for the convection-diffusion equation with L1 data2012In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 81, no 279, p. 1429-1454Article in journal (Refereed)
    Abstract [en]

    In this paper, we prove the convergence of a finite-volume schemefor the time-dependent convection–diffusion equation with an L1 right-handside. To this purpose, we first prove estimates for the discrete solution andfor its discrete time and space derivatives. Then we show the convergence of asequence of discrete solutions obtained with more and more refined discretiza-tions, possibly up to the extraction of a subsequence, to a function which metsthe regularity requirements of the weak formulation of the problem; to thispurpose, we prove a compactness result, which may be seen as a discrete ana-logue to Aubin-Simon’s lemma. Finally, such a limit is shown to be indeed aweak solution.

  • 13. Johnson, Claes
    et al.
    Szepessy, Anders
    Hansbo, Peter
    On the convergence of shock-capturing  streamline diffusion finite element methods for hyperbolic conservation laws1990In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 54, no 189, p. 107-129Article in journal (Refereed)
  • 14. Liu, Hailiang
    et al.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tanushev, Nicolay M.
    Error estimates for Gaussian beam superpositions2013In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 82, no 282, p. 919-952Article in journal (Refereed)
    Abstract [en]

    Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrodinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength epsilon. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrodinger equations subject to highly oscillatory initial data of the form Ae(i Phi/) (epsilon). Through a careful estimate of an oscillatory integral operator, we prove that the k-th order Gaussian beam superposition converges to the original wave field at a rate proportional to epsilon(k/2) in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, epsilon-scaled, energy norm and for the Schrodinger equation in the L-2 norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in R-2 to analyze the sharpness of the theoretical results.

  • 15.
    Möller, Niels
    KTH, School of Electrical Engineering (EES), Automatic Control.
    On Schonhage's algorithm and subquadratic integer GCD computation2008In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 77, no 261, p. 589-607Article in journal (Refereed)
    Abstract [en]

    We describe a new subquadratic left-to-right GCD algorithm, inspired by Schonhage's algorithm for reduction of binary quadratic forms, and compare it to the first subquadratic GCD algorithm discovered by Knuth and Schonhage, and to the binary recursive GCD algorithm of Stehle and Zimmer-mann. The new GCD algorithm runs slightly faster than earlier algorithms, and it is much simpler to implement. The key idea is to use a stop condition for HGCD that is based not on the size of the remainders, but on the size of the next difference. This subtle change is sufficient to eliminate the back-up steps that are necessary in all previous subquadratic left-to-right GCD algorithms. The subquadratic GCD algorithms all have the same asymptotic running time, O(n(log n)(2) log log n).

  • 16.
    Sjögreen, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Gustavsson, Katarina
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gudmundsson, Reynir Levi
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    A model for peak formation in the two-phase equations2007In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 76, no 260, p. 1925-1940Article in journal (Refereed)
    Abstract [en]

    We present a hyperbolic-elliptic model problem related to the equations of two-phase fluid flow. The model problem is solved numerically, and properties of its solution are presented. The model equation is well-posed when linearized around a constant state, but there is a strong focusing effect, and very large solutions exist at certain times. We prove that the model problem has a smooth solution for bounded times.

  • 17. Szepessy, Anders
    Convergence of a shock-capturing streamline diffusion finite element method for a conservation law in two space dimensions1989In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, p. 527-545Article in journal (Refereed)
  • 18. Szepessy, Anders
    et al.
    Johnson, Claes
    On the convergence of a finite element method for a nonlinear hyperbolic conservation law1987In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 49, no 180, p. 427-444Article in journal (Refereed)
1 - 18 of 18
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