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  • 1.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindbo, Dag
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant2014In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, 50-63 p.Article in journal (Refereed)
    Abstract [en]

    The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

  • 2.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Saffar Shamshirgar, Davoud
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for free-space Stokes potentials2017In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed)
    Abstract [en]

    We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.

  • 3.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

  • 4.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansion2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 326, 420-445 p.Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

    The full text will be freely available from 2018-09-09 10:45
  • 5.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Error estimation for quadrature by expansion in layer potential evaluation2017In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 43, no 1, 195-234 p.Article in journal (Refereed)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

  • 6.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Estimation of quadrature errors in layer potential evaluation using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

  • 7.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for Stokesian particle suspensions2014In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 76, no 10, 669-698 p.Article in journal (Refereed)
    Abstract [en]

    We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to O(N log N), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. 

  • 8. Engblom, S.
    et al.
    Do-Quang, Minh
    KTH, School of Engineering Sciences (SCI), Mechanics, Physicochemical Fluid Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Amberg, Gustav
    KTH, School of Engineering Sciences (SCI), Mechanics, Physicochemical Fluid Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    On diffuse interface modeling and simulation of surfactants in two-phase fluid flow2013In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 14, no 4, 879-915 p.Article in journal (Refereed)
    Abstract [en]

    An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.

  • 9.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    On Consistent Boundary Conditions for the Yee Scheme in 3DManuscript (preprint) (Other academic)
    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.

  • 10.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tornberg, Anna Karin
    High-frequency wave propagation by the segment projection method2002In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 178, no 2, 373-390 p.Article in journal (Refereed)
    Abstract [en]

    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.

  • 11.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna Karin
    A finite element based level-set method for multi-phase flow, Proceedings of Conference on Progress in Numerical Solutions of Partial Differential Equations2002In: Scientific World Journal, ISSN 1537-744X, E-ISSN 1537-744X, 86-110 p.Article in journal (Refereed)
  • 12. Engquist, Björn
    et al.
    Tornberg, Anna-Karin
    Tsai, R.
    Discretization of Dirac delta functions in level set methods2005In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 207, no 1, 28-51 p.Article in journal (Refereed)
    Abstract [en]

    Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions. It is independent of the irregularity relative to the grid. In the second method, the regularization is constructed from a one-dimensional regularization that is extended to multi-dimensions with a variable support depending on the orientation of the singularity relative to the computational grid. Convergence analysis and numerical results are given. © 2005 Published by Elsevier Inc.

  • 13.
    Gustavsson, Katarina
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Numerical Simulations of Rigid Fiber Suspensions2008Conference paper (Refereed)
    Abstract [en]

    In this paper, we present a numerical method designed to simulate the

    challenging problem of the dynamics of slender fibers immersed in an incompressible

    fluid. Specifically, we consider microscopic, rigid fibers, that

    sediment due to gravity. Such fibers make up the micro-structure of many

    suspensions for which the macroscopic dynamics are not well understood.

    Our numerical algorithm is based on a non-local slender body approximation

    that yields a system of coupled integral equations, relating the forces

    exerted on the fibers to their velocities, which takes into account the hydrodynamic

    interactions of the fluid and the fibers. The system is closed by

    imposing the constraints of rigid body motions.

    The fact that the fibers are straight have been further exploited in the

    design of the numerical method, expanding the force on Legendre polynomials

    to take advantage of the specific mathematical structure of a finite-part

    integral operator, as well as introducing analytical quadrature in a manner

    possible only for straight fibers.

    We have carefully treated issues of accuracy, and present convergence

    results for all numerical parameters before we finally discuss the results from

    simulations including a larger number of fibers.

  • 14.
    Gustavsson, Katarina
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Gravity induced sedimentation of slender fibers2009In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 21, no 12Article in journal (Refereed)
    Abstract [en]

    Gravity induced sedimentation of slender, rigid fibers in a highly viscous fluid is investigated by large scale numerical simulations. The fiber suspension is considered on a microscopic level and the flow is described by the Stokes equations in a three dimensional periodic domain. Numerical simulations are performed to study in great detail the complex dynamics of a cluster of fibers. A repeating cycle is identified. It consists of two main phases: a densification phase, where the cluster densifies and grows, and a coarsening phase, during which the cluster becomes smaller and less dense. The dynamics of these phases and their relation to fluctuations in the sedimentation velocity are analyzed. Data from the simulations are also used to investigate how average fiber orientations and sedimentation velocities are influenced by the microstructure in the suspension. The dynamical behavior of the fiber suspension is very sensitive to small random differences in the initial configuration and a number of realizations of each numerical experiment are performed. Ensemble averages of the sedimentation velocity and fiber orientation are presented for different values of the effective concentration of fibers and the results are compared to experimental data. The numerical code is parallelized using the Message Passing Instructions (MPI) library and numerical simulations with up 800 fibers can be run for very long times which is crucial to reach steady levels of the averaged quantities. The influence of the periodic boundary conditions on the process is also carefully investigated.

  • 15. Jung, Sunghwan
    et al.
    Spagnolie, S. E.
    Parikh, K.
    Shelley, M.
    Tornberg, Anna-Karin
    Periodic sedimentation in a Stokesian fluid2006In: Physical Review E, ISSN 1539-3755, Vol. 74, no 3Article in journal (Refereed)
    Abstract [en]

    We study the sedimentation of two identical but nonspherical particles sedimenting in a Stokesian fluid. Experiments and numerical simulations reveal periodic orbits wherein the bodies mutually induce an in-phase rotational motion accompanied by periodic modulations of sedimentation speed and separation distance. We term these tumbling orbits and find that they appear over a broad range of body shapes.

  • 16. Kanevsky, Alex
    et al.
    Shelley, Michael J.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Modeling simple locomotors in Stokes flow2010In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 4, 958-977 p.Article in journal (Refereed)
    Abstract [en]

    Motivated by the locomotion of flagellated micro-organisms and by recent experiments of chemically driven nanomachines, we study the dynamics of bodies of simple geometric shape that are propelled by specified tangential surface stresses. We develop a mathematical description of the body dynamics based on a mixed-type boundary integral formulation. We also derive analytic axisymmetric solutions for the case of a single locomoting sphere and ellipsoid based on spherical and ellipsoidal harmonics, and compare our numerical results to these. The hydrodynamic interactions between two spherical and ellipsoidal swimmers in an infinite fluid are then simulated using second-order accurate spatial and temporal discretizations. We find that the near-field interactions result in complex and interesting changes in the locomotors' orientations and trajectories. Stable as well as unstable pairwise swimming motions are observed, similar to the recent findings of Pooley et al. [C.M. Pooley, G.P. Alexander, J.M. Yeomans, Hydrodynamic interaction between two swimmers at low Reynolds number, Phys. Rev. Lett. 99 (2007) 228103].

  • 17. Khatri, Shilpa
    et al.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    A numerical method for two phase flows with insoluble surfactants2011In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 49, no 1, 150-165 p.Article in journal (Refereed)
    Abstract [en]

    In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier-Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier-Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach. Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.

  • 18. Khatri, Shilpa
    et al.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An embedded boundary method for soluble surfactants with interface tracking for two-phase flows2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 256, 768-790 p.Article in journal (Refereed)
    Abstract [en]

    Surfactants, surface reacting agents, lower the surface tension of the interface between fluids in multiphase flow. This capability of surfactants makes them ideal for many applications, including wetting, foaming, and dispersing. Due to their molecular composition, surfactants are adsorbed from the bulk fluid to the interface between the fluids, leading to different concentrations on the interface and in the fluid. In a previous paper [21], we introduced a new second order method using uniform grids to simulate insoluble surfactants in multiphase flow. This method used Strang splitting allowing for a fully second order treatment in time. Here, we use the same numerical methods to explicitly represent the singular interface, treat the interfacial surfactant concentration, and couple with the Navier-Stokes equations. Now, we introduce a second order method for the surfactants in the bulk that continues to allow the use of regular grids for the full problem. Difficulties arise since the boundary condition for the bulk concentration, which handles the flux of surfactant between the interface and bulk fluid, is applied at the interface which cuts arbitrarily through the regular grid. We extend the embedded boundary method, introduced in [22], to handle this challenge. Through our results, we present the effect of the solubility of the surfactants. We show results of drop dynamics due to resulting Marangoni stresses and of drop deformations in shear flow in the presence of soluble surfactants. There is a large nondimensional parameter space over which we try to understand the drop dynamics.

  • 19.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems2012In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 136, no 16, 164111-1-164111-16 p.Article in journal (Refereed)
    Abstract [en]

    A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.

  • 20.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fast and spectrally accurate summation of 2-periodic Stokes potentialsManuscript (preprint) (Other academic)
    Abstract [en]

    We derive a Ewald decomposition for the Stokeslet in planar periodicity and a novel PME-type O(N log N) method for the fast evaluation of the resulting sums. The decomposition is the natural 2P counterpart to the classical 3P decomposition by Hasimoto, and is given in an explicit form not found in the literature. Truncation error estimates are provided to aid in selecting parameters. The fast, PME-type, method appears to be the first fast method for computing Stokeslet Ewald sums in planar periodicity, and has three attractive properties: it is spectrally accurate; it uses the minimal amount of memory that a gridded Ewald method can use; and provides clarity regarding numerical errors and how to choose parameters. Analytical and numerical results are give to support this. We explore the practicalities of the proposed method, and survey the computational issues involved in applying it to 2-periodic boundary integral Stokes problems.

  • 21.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Interface tracking using patches2011Manuscript (preprint) (Other academic)
  • 22.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Spectral accuracy in fast Ewald-based methods for particle simulations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 24, 8744-8761 p.Article in journal (Refereed)
    Abstract [en]

    A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.

  • 23.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Spectrally accurate fast summation for periodic Stokes potentials2010In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 23, 8994-9010 p.Article in journal (Refereed)
    Abstract [en]

    A spectrally accurate method for the fast evaluation of N-particle sums of the periodic Stokeslet is presented. Two different decomposition methods, leading to one sum in real space and one in reciprocal space, are considered. An FFT based method is applied to the reciprocal part of the sum, invoking the equivalence of multiplications in reciprocal space to convolutions in real space, thus using convolutions with a Gaussian function to place the point sources on a grid. Due to the spectral accuracy of the method, the grid size needed is low and also in practice, for a fixed domain size, independent of N. The leading cost, which is linear in N, arises from the to-grid and from-grid operations. Combining this FFT based method for the reciprocal sum with the direct evaluation of the real space sum, a spectrally accurate algorithm with a total complexity of 0(N log N) is obtained. This has been shown numerically as the system is scaled up at constant density. (C) 2010 Elsevier Inc. All rights reserved.

  • 24.
    Marin, Oana
    et al.
    KTH, School of Engineering Sciences (SCI).
    Gustavsson, Katarina
    KTH, School of Engineering Sciences (SCI).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI).
    A fast summation method for fiber simulationsManuscript (preprint) (Other academic)
  • 25.
    Marin, Oana
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Gustavsson, Katarina
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    A highly accurate boundary treatment for confined Stokes flow2012In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 66, 215-230 p.Article in journal (Refereed)
    Abstract [en]

    Fluid flow phenomena in the Stokesian regime abounds in nature as well as in microfluidic applications. Discretizations based on boundary integral formulations for such flow problems allow for a reduction in dimensionality but have to deal with dense matrices and the numerical evaluation of integrals with singular kernels. The focus of this paper is the discretization of wall confinements, and specifically the numerical treatment of flat solid boundaries (walls), for which a set of high-order quadrature rules that accurately integrate the singular kernel of the Stokes equations are developed. Discretizing by Nystrom's method, the accuracy of the numerical integration determines the accuracy of the solution of the boundary integral equations, and a higher order quadrature method yields a large gain in accuracy at negligible cost. The structure of the resulting submatrix associated with each wall is exploited in order to substantially reduce the memory usage. The expected convergence of the quadrature rules is validated through numerical tests, and this boundary treatment is further applied to the classical problem of a sedimenting sphere in the vicinity of solid walls.

  • 26.
    Marin, Oana
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustavsson, Katarina
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    A wall treatment for confined Stokes flowArticle in journal (Other academic)
    Abstract [en]

     

    The study of bodies immersed in Stokes flow is crucial in various microfluidic applications. Recasting the governing equations in a boundary integral formulation reduces the three-dimensional problem to two-dimensional integral equations to be discretized over the surface of the submerged objects. The present work focuses on the development and validation of a wall treatment where the wall is discretized in the same fashion as the immersed bodies. For this purpose, a set of high-order quadrature rules for the numerical integration of integrals containing the singular Green’s function-the so-called Stokeslet - has been developed. By coupling the wall discretization to the discretization of immersed objects, we exploit the structure of the block matrix corresponding to the wall discretization in order to substantially reduce the memory usage. For validation, the classical problem of a sedimenting sphere in the vicinity of solid walls is studied.

  • 27.
    Marin, Oana
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Corrected trapezoidal rules for a class of singular functions2014In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 34, no 4, 1509-1540 p.Article in journal (Refereed)
    Abstract [en]

    A set of accurate quadrature rules applicable to a class of integrable functions with isolated singularities is designed and analysed theoretically in one and two dimensions. These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. To compute the correction weights, small-size ill-conditioned systems have to be solved. The convergence of the correction weights is accelerated by the use of compactly supported functions that annihilate boundary errors. Convergence proofs with error estimates for the resulting quadrature rules are given in both one and two dimensions. The tabulated weights are specific for the singularities under consideration, but the methodology extends to a large class of functions with integrable isolated singularities. Furthermore, in one dimension we have obtained a closed form expression based on which the modified weights can be computed directly.

  • 28.
    Ojala, Rikard
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An accurate integral equation method for simulating multi-phase Stokes flow2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 298, 145-160 p.Article in journal (Refereed)
    Abstract [en]

    We introduce a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane. It builds upon the method introduced by Kropinski in 2001 [17], but improves on it by adding an interpolatory quadrature approach for handling near-singular integrals. Such integrals typically arise when drop boundaries come close to one another, and are difficult to compute accurately using standard quadrature rules. Adapting the interpolatory quadrature method introduced by Helsing and Ojala in 2008 [11] to the current application, very general drop configurations can be handled while still maintaining stability and high accuracy. The performance of the new method is demonstrated by some challenging numerical examples.

  • 29.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Hess, Berk
    KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACSManuscript (preprint) (Other academic)
    Abstract [en]

    The smooth particle mesh Ewald (SPME) method is an FFT based methodfor the fast evaluation of electrostatic interactions under periodic boundaryconditions. A highly optimized implementation of this method is availablein GROMACS, a widely used software for molecular dynamics simulations.In this article, we compare a more recent method from the same family ofmethods, the spectral Ewald (SE) method, to the SPME method in termsof performance and efficiency. We consider serial and parallel implementa-tions of both methods for single and multiple core computations on a desktopmachine as well as the Beskow supercomputer at KTH Royal Institute ofTechnology. The implementation of the SE method has been well optimized,however not yet comparable to the level of the SPME implementation thathas been improved upon for many years. We show that the SE method isvery efficient whenever used to achieve high accuracy and that it already atthis level of optimization can be competitive for low accuracy demands.

  • 30.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Hess, Berk
    KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Yokota, Rio
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Regularized FMM for MD simulationsManuscript (preprint) (Other academic)
    Abstract [en]

    A regularized fast multipole method (FMM) which approximately conserves the total energy in Molecular dynamics (MD) simulations is presented. The new algorithm introduces a regularization which eliminates the discontinuity inherent in the FMM. This allows us to use FMM in simulations as a substitute for widely used FFT based methods. For a system of N particles, the computational complexity of the resulting method is still of order N though with a larger constant compared to the plain FMM. Numerical examples are provided to confirm that the new algorithm improves the accuracy and approximately conserves the long term total energy.

  • 31.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A fast multipole method for evaluating exponential integral type kernelsManuscript (preprint) (Other academic)
    Abstract [en]

    We present a fast multipole method for evaluation of sums with exponential  integral type kernels. These sums appear while solving free space Poisson problems in two dimensions and in the derivation of 1d-periodic Ewald sums. The presented method uses recurrence relations to derive multipole expansions for computing interactions between particles and far clusters.

  • 32.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for electrostatic potentials with arbitrary periodicityManuscript (preprint) (Other academic)
    Abstract [en]

    A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only marginally increase the total run time. Our comparisons also show that the computational cost of the SE method for the free-space case is typically about four times more expensive as compared to the triply periodic case.

    The Gaussian window function previously used in the SE method, is here compared to an approximation of the Kaiser-Bessel window function, recently introduced. With a carefully tuned shape parameter that is selected based on an error estimate for this new window function, runtimes for the SE method can be further reduced.

  • 33.
    Saffar Shamshirgar, Davoud
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    The Spectral Ewald method for singly periodic domains2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 347, 341-366 p.Article in journal (Refereed)
    Abstract [en]

    We present a fast and spectrally accurate method for efficient computation of the three dimensional Coulomb potential with periodicity in one direction. The algorithm is FFT-based and uses the so-called Ewald decomposition, which is naturally most efficient for the triply periodic case. In this paper, we show how to extend the triply periodic Spectral Ewald method to the singly periodic case, such that the cost of computing the singly periodic potential is only marginally larger than the cost of computing the potential for the corresponding triply periodic system. In the Fourier space contribution of the Ewald decomposition, a Fourier series is obtained in the periodic direction with a Fourier integral over the non-periodic directions for each discrete wave number. We show that upsampling to resolve the integral is only needed for modes with small wave numbers. For the zero wave number, this Fourier integral has a singularity. For this mode, we effectively need to solve a free-space Poisson equation in two dimensions. A very recent idea by Vico et al. makes it possible to use FFTs to solve this problem, allowing us to unify the treatment of all modes. An adaptive 3D FFT can be established to apply different upsampling rates locally. The computational cost for other parts of the algorithm is essentially unchanged as compared to the triply periodic case, in total yielding only a small increase in both computational cost and memory usage for this singly periodic case.

  • 34. Shelley, M. J.
    et al.
    Tornberg, Anna-Karin
    NYU.
    Microstructural Dynamics in Complex Fluids2005In: Handbook of Materials Modeling / [ed] M. Bazant, Kluwer Academic Publishers, 2005Chapter in book (Refereed)
  • 35. Tornberg, Anna Karin
    Finite element based methods for interface flow simulations2000Conference paper (Refereed)
  • 36. Tornberg, Anna Karin
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    A finite element based level-set method for multi-phase flow applications2000In: Computing and Visualization in Science, ISSN 1432-9360, E-ISSN 1433-0369, Vol. 3, no 1-2, 93-101 p.Article in journal (Refereed)
    Abstract [en]

    A numerical method for simulating incompressibletwo-dimensional multiphase flow is presented. The methodis based on a level-set formulation discretized by a finiteelementtechnique. The treatment of the specific features ofthis problem, such as surface tension forces acting at the interfacesseparating two immiscible fluids, as well as the densityand 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, thesingular surface tension forces are included through line integralsalong the interfaces, which are easily approximatedquantities. In addition, differentiation of the discontinuousviscosity is avoided. The discontinuous density and viscosityare included in the finite element integrals. A strategy forthe evaluation of integrals with discontinuous integrands hasbeen developed based on a rigorous analysis of the errors associatedwith the evaluation of such integrals. Numerical testshave been performed. For the case of a rising buoyant bubblethe results are in good agreement with results from a fronttrackingmethod. The run presented here is a run includingtopology changes, where initially separated areas of one fluidmerge in different stages due to buoyancy effects.

  • 37.
    Tornberg, Anna Karin
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    A Finite Element Based Level-Set Method for Multiphase Flows2002Conference paper (Refereed)
    Abstract [en]

    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.

  • 38.
    Tornberg, Anna Karin
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Interface Tracking in Two-Phase Flows2000In: State of the Art: Multifield Problems, Springer Berlin/Heidelberg, 2000, 58-66 p.Chapter in book (Refereed)
  • 39. Tornberg, Anna Karin
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Interface tracking in two-phase flows2000Conference paper (Refereed)
  • 40. Tornberg, Anna Karin
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Regularization techniques for numerical approximations of PDEs with singularities2003In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 19, no 1-3, 527-552 p.Article in journal (Refereed)
    Abstract [en]

    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.

  • 41. Tornberg, Anna Karin
    et al.
    Metcalfe, R.W.
    Scott, R
    Bagheri, B
    A Fluid Particle Simulation Method1997Conference paper (Refereed)
  • 42. Tornberg, Anna Karin
    et al.
    Metcalfe, R.W.
    Scott, R
    Bagheri, B
    A Front-Tracking Method for Simulating Fluid Particle Motion using High-Order Finite Element Methods1997Conference paper (Refereed)
  • 43.
    Tornberg, Anna-Karin
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Multi-dimensional quadrature of singular and discontinuous functions2002In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 42, no 3, 644-669 p.Article in journal (Refereed)
    Abstract [en]

    In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.

  • 44.
    Tornberg, Anna-Karin
    NYU.
    Numerical Simulations of the Dynamics of Fiber Suspensions2005In: Multiscale Methods in Science and Engineering, Springer, 2005, 275-289 p.Chapter in book (Refereed)
    Abstract [en]

    The dynamics of flexible fibers or filaments immersed in a fluid are important to understanding many interesting problems arising in biology, engineering, and physics. For most applications, the flows are at very low Reynolds numbers, and the fibers can have aspect ratios of length to radius from a few tens to several thousands. This class of problems is difficult to solve accurately to a reasonable cost with grid based methods, partly due to the different scales in length and radius of the fibers and the fact that elastic equations must be solved within the fibers. Making explicit use both of the fact that we are considering Stokes flow, as well as of the slenderness of the fibers, we have designed a cost-effective method to simulate multiple interacting elastic fibers in a three dimensional Stokes flow. The key points are that for Stokes flow, boundary integral methods can be employed to reduce the three-dimensional dynamics to the dynamics of the two-dimensional fiber surfaces, and that using slender body asymptotics, this can be further reduced to the dynamics of the one-dimensional fiber center-lines. The resulting integral equations include both the effect of the fibers on the flow field, as well as the interactions of fibers, as mediated by the flow. We have developed a numerical method based on this theory that allows for simulating multiple interacting highly flexible fibers. Considering the efficiency of the method, another important fact is that the framework is suitable for introducing a semi-implicit time-stepping scheme, eliminating the severe constraint on the time-step size arising from the elasticity. Our numerical approach is based on second-order divided differences for spatial derivatives, combined with special product integration methods that reflect the nearly singular nature of the integral operators

  • 45.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Regularization techniques for singular source terms in differential equations2005In: European Congress of Mathematics: Stockholm, June 27-July 2, 2004 / [ed] Ari Laptev, Zurich: European Mathematical Society Publishing House, 2005, 477-499 p.Chapter in book (Refereed)
  • 46.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    The Ewald sums for singly, doubly and triply periodic electrostatic systems2015In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 42, 227-248 p.Article in journal (Refereed)
    Abstract [en]

    When evaluating the electrostatic potential, periodic boundary conditions in one, two or three of the spatial dimensions are often required for different applications. The triply periodic Ewald summation formula is classical, and Ewald summation formulas for the other two cases have also been derived. In this paper, derivations of the Ewald sums in the doubly and singly periodic cases are presented in a uniform framework based on Fourier analysis, which also yields a natural starting point for FFT-based fast summation methods.

  • 47.
    Tornberg, Anna-Karin
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC).
    Consistent boundary conditions for the Yee scheme2008In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 227, no 14, 6922-6943 p.Article in journal (Refereed)
    Abstract [en]

    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 0(l) 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.

  • 48.
    Tornberg, Anna-Karin
    et al.
    Courant Institute of Mathematical Sciences, New York University.
    Engquist, Björn
    Department of Mathematics and PACM, Princeton University.
    High order difference methods for wave propagation in discontinuous media2006In: Methods and Applications of Analysis, ISSN 1073-2772, E-ISSN 1945-0001, no 13, 217-273 p.Article in journal (Refereed)
  • 49. Tornberg, Anna-Karin
    et al.
    Engquist, Björn
    Numerical approximations of singular source terms in differential equations2004In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 200, no 2, 462-488 p.Article in journal (Refereed)
    Abstract [en]

    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.

  • 50.
    Tornberg, Anna-Karin
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Regularization for accurate numerical wave propagation in discontinuous media2006In: Methods and Applications of Analysis, ISSN 1073-2772, E-ISSN 1945-0001, Vol. 13, no 3, 247-274 p.Article in journal (Refereed)
    Abstract [en]

    Structured computational grids are the basis for highly efficient numerical approximations of wave propagation. When there are discontinuous material coefficients the accuracy is typically reduced and there may also be stability problems. In a sequence of recent papers Gustafsson et al. proved stability of the Yee scheme and a higher order difference approximation based on a similar staggered structure, for the wave equation with general coefficients. In this paper, the Yee discretization is improved from first to second order by modifying the material coefficients close to the material interface. This is proven in the $L^2$ norm. The modified higher order discretization yields a second order error component originating from the discontinuities, and a fourth order error from the smooth regions. The efficiency of each original method is retained since there is no special structure in the difference stencil at the interface. The main focus of this paper is on one spatial dimension, with the derivation of a second order algorithm for a two dimensional example given in the last section

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