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  • 1.
    Bozorgnia, Farid
    Faculty of Sciences, Persian Gulf University, Boushehr 75168, Iran.
    Numerical solutions of the two-phase membrane problem2011Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 61, nr 1, s. 92-107Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.

  • 2.
    Efraimsson, Gunilla
    et al.
    KTH, Tidigare Institutioner (före 2005), Numerisk analys och datalogi, NADA.
    Kreiss, Gunilla
    KTH, Tidigare Institutioner (före 2005), Numerisk analys och datalogi, NADA.
    A note on the effect of artificial viscosity on solutions of conservation laws1996Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 21, s. 155-173Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider central difference schemes with artificial viscosity terms for nonlinear hyperbolic systems. We analyze the influence of artificial viscosity on solutions with shocks of nonlinear hyperbolic problems in one dimension. Both stationary and moving shocks are considered. The analysis shows that for the Euler equations one can obtain well behaved (sharp) shock layers when a scalar viscosity coefficient is used. This is not true for a general system. Numerical computations of Burgers' equation and the Euler equations are presented. They support the results from the linear theory.

  • 3.
    Engblom, Stefan
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    On well-separated sets and fast multipole methods2011Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 61, nr 10, s. 1096-1102Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The notion of well-separated sets is crucial in fast multipole methods as the main idea is to approximate the interaction between such sets via cluster expansions. We revisit the one-parameter multipole acceptance criterion in a general setting and derive a relative error estimate. This analysis benefits asymmetric versions of the method, where the division of the multipole boxes is more liberal than in conventional codes. Such variants offer a particularly elegant implementation with a balanced multipole tree, a feature which might be very favorable on modern computer architectures.

  • 4.
    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.

  • 5.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Zhao, H-K
    Absorbing boundary conditions for domain decomposition1998Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 27, nr 4, s. 315-324Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we would like to point out some similarities between two artificial boundary conditions. One is the far field or absorbing boundary conditions for computations over unbounded domain. The other is the boundary conditions used at the boundary between subdomains in domain decomposition. We show some convergence result for the generalized Schwarz alternating method (GSAM), in which a convex combination of Dirichlet data and Neumann data is exchanged at the artificial boundary. We can see clearly how the mixed boundary condition and the relative size of overlap will affect the convergence rate. These results can be extended to more general coercive elliptic partial differential equations using the equivalence of elliptic operators. Numerically first- and second-order approximations of the Dirichlet-to-Neumann operator are constructed using local operators, where information tangential to the boundary is included. Some other possible extensions and applications are pointed out. Finally numerical results are presented.

  • 6.
    Engquist, Björn
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Zhao, H-K
    Absorbing boundary conditions for domain decomposition1998Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 27, nr 4, s. 341-365Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we would like to point out some similarities between two artificial boundary conditions. One is the far field or absorbing boundary conditions for computations over unbounded domain. The other is the boundary conditions used at the boundary between subdomains in domain decomposition. We show some convergence result for the generalized Schwarz alternating method (GSAM), in which a convex combination of Dirichlet data and Neumann data is exchanged at the artificial boundary. We can see clearly how the mixed boundary condition and the relative size of overlap will affect the convergence rate. These results can be extended to more general coercive elliptic partial differential equations using the equivalence of elliptic operators. Numerically first- and second-order approximations of the Dirichlet-to-Neumann operator are constructed using local operators, where information tangential to the boundary is included. Some other possible extensions and applications are pointed out. Finally numerical results are presented.

  • 7.
    Forsgren, A
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Inertia-controlling factorizations for optimization algorithms2002Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 43, nr 1-2, s. 91-107, artikel-id PII S0168-9274(02)00119-8Artikel i tidskrift (Refereegranskat)
  • 8. Hansbo, Peter
    et al.
    Larson, Mats G.
    Zahedi, Sara
    Uppsala Univ, Dept Informat Technol, SE-75105 Uppsala, Sweden.
    A cut finite element method for a Stokes interface problem2014Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 85, nr 0, s. 90-114Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.

  • 9. Harten, A
    et al.
    Osher, S
    Engquist, Björn
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
    Chakravarthy, Sukumar R
    Some results on uniformly high-order accurate essentially nonoscillatory schemes1986Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 2, nr 3-5, s. 347-377Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We continue the construction and the analysis of essentially nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy in the sense of truncation error, at extrema of the solution. In this paper we construct an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory. This means that, for piecewise smooth solutions, the variation of the numerical approximation is bounded by that of the true solution up to O(hR - 1), for 0 <R and h sufficiently small. The design involves an essentially nonoscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. To solve this reconstruction problem we use a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities.

  • 10.
    Jarlebring, Elias
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Poloni, F.
    Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning2019Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 135, s. 173-185Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems x˙(t)=A0x(t)+A1x(t−τ)+B0u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value U(τ/2)∈Rn×n, i.e., the delay Lyapunov matrix at time τ/2. This linear matrix equation with n2 unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n2×n2 matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation MX+XTN=C, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n≈1000, i.e., a linear system with n2≈106 unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.

  • 11.
    Kreiss, G.
    et al.
    Uppsala University, Department of Information Technology, Sweden.
    Krank, B.
    Technical University of Munich, Institute for Computational Mechanics, Germany.
    Efraimsson, Gunilla
    KTH, Skolan för teknikvetenskap (SCI), Farkost och flyg. KTH, Skolan för teknikvetenskap (SCI), Centra, VinnExcellence Center for ECO2 Vehicle design.
    Analysis of stretched grids as buffer zones in simulations of wave propagation2016Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 107, s. 1-17Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A zone of increasingly stretched grid is a robust and easy-to-use way to avoid unwanted reflections at artificial boundaries in wave propagating simulations. In such a buffer zone there are two main damping mechanisms, dissipation and under-resolution that turns a traveling wave into an evanescent wave. We present analysis in one and two space dimensions showing that evanescent decay through under-resolution is a very efficient way to damp waves. The analysis is supported by numerical computations.

  • 12.
    Loubenets, Alexei
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Ali, Tanweer
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Hanke, Michael
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Highly accurate finite element method for one-dimensional elliptic interface problems2009Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, nr 1, s. 119-134Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A high order finite element method for one-dimensional elliptic interface problems is presented. Due to presence of these interfaces the problem will contain discontinuities in the coefficients and singularities in the right hand side that are represented by delta functional with the support on the interfaces. As a result, the solution to the interface problem and its derivatives may have jump discontinuities. The proposed method is specifically designed to handle this features of the solution using non-body fitted grids, i.e. the grids are not aligned with the interfaces.The finite element method will be based on third order Hermitian interpolation. The main idea is to modify the basis functions in the vicinity of the interface such that the jump conditions are well approximated. A rigorous error analysis shows that the presented finite element method is fourth order accurate in L-2 norm. The numerical results agree well with the theoretical analysis. The basic idea can easily be generalized to other finite element ansatz functions.

  • 13. Saadvandi, M.
    et al.
    Meerbergen, K.
    Jarlebring, Elias
    On dominant poles and model reduction of second order time-delay systems2012Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, nr 1, s. 21-34Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the systemʼs input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.

  • 14.
    Scheffel, Jan
    et al.
    KTH, Skolan för elektro- och systemteknik (EES), Fusionsplasmafysik.
    Håkansson, Cristian
    KTH, Skolan för elektro- och systemteknik (EES), Fusionsplasmafysik.
    Solution of systems of nonlinear equations - a semi-implicit approach2009Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, nr 10, s. 2430-2443Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    An iterative method for globally convergent solution of nonlinear equations and systems of nonlinear equations is presented. Convergence is quasi-monotonous and approaches second order in the proximity of the real roots. The algorithm is related to semi-implicit methods, earlier being applied to partial differential equations. It is shown that the Newton-Raphson and Newton methods are special cases of the method. The degrees of freedom introduced by the semi-implicit parameters are used to control convergence. When applied to a single equation, efficient global convergence and convergence to a nearby root makes the method attractive in comparison with methods as those of Newton-Raphson and van Wijngaarden-Dekker-Brent. An extensive standard set of systems of equations is solved and convergence diagrams are introduced, showing the robustness, efficiency and simplicity of the method as compared to Newton's method using linesearch.

  • 15. Svärd, Magnus
    et al.
    Gong, Jing
    Uppsala Univ., IT dept..
    Nordstroem, Jan
    An accuracy evaluation of unstructured node-centred finite volume methods2008Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 58, nr 8, s. 1142-1158Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Node-centred edge-based finite volume approximations are very common in computational fluid dynamics since they are assumed to run on structured, unstructured and even on mixed grids. We analyse the accuracy properties of both first and second derivative approximations and conclude that these schemes cannot be used on arbitrary grids as is often assumed. For the Euler equations first-order accuracy can be obtained if care is taken when constructing the grid. For the Navier-Stokes equations, the grid restrictions are so severe that these finite volume schemes have little advantage over structured finite difference schemes. Our theoretical results are verified through extensive computations.

  • 16. Svärd, Magnus
    et al.
    Gong, Jing
    Uppsala Univ., IT dept..
    Nordstrom, Jan
    Stable artificial dissipation operators for finite volume schemes on unstructured grids2006Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 56, nr 12, s. 1481-1490Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Our objective is to derive stable first-, second- and fourth-order artificial dissipation operators for node based finite volume schemes. Of particular interest are general unstructured grids where the strength of the finite volume method is fully utilised. A commonly used finite volume approximation of the Laplacian will be the basis in the construction of the artificial dissipation. Both a homogeneous dissipation acting in all directions with equal strength and a modification that allows different amount of dissipation in different directions are derived. Stability and accuracy of the new operators are proved and the theoretical results are supported by numerical computations.

  • 17. Svärd, Magnus
    et al.
    Nordstrom, J.
    Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids2004Ingår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 51, nr 1, s. 101-125Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Our objective is to analyse a commonly used edge based finite volume approximation of the Laplacian and construct an accurate and stable way to implement boundary conditions for time dependent problems. Of particular interest are unstructured grids where the strength of the finite volume method is fully utilised. As a model problem we consider the heat equation. We analyse the Cauchy problem in one and several space dimensions and prove stability on unstructured grids. Next, the initial-boundary value problem is considered and a scheme is constructed in a summation-by-parts framework. The boundary conditions are imposed in a stable and accurate manner, using a penalty formulation. Numerical computations of the wave equation in two-dimensions are performed, verifying stability and order of accuracy for structured grids. However, the results are not satisfying for unstructured grids. Further investigation reveals that the approximation is not consistent for general unstructured grids. However, grids consisting of equilateral polygons recover the convergence.

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