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  • 1.
    Compagnoni, Marco
    et al.
    Politecn Milan, Italy.
    Notari, Roberto
    Politecn Milan, Italy.
    Ruggiu, Andrea Alessandro
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Antonacci, Fabio
    Politecn Milan, Italy.
    Sarti, Augusto
    Politecn Milan, Italy.
    The Algebro-geometric Study of Range Maps2017In: Journal of nonlinear science, ISSN 0938-8974, E-ISSN 1432-1467, Vol. 27, no 1, p. 99-157Article in journal (Refereed)
    Abstract [en]

    Localizing a radiant source is a problem of great interest to many scientific and technological research areas. Localization based on range measurements is at the core of technologies such as radar, sonar and wireless sensor networks. In this manuscript, we offer an in-depth study of the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummers and Cayleys surfaces. Our work also gives new insights into the localization based on range differences.

  • 2.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ruggiu, Andrea Alessandro
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Dual Time-Stepping Using Second Derivatives2019Report (Other academic)
    Abstract [en]

    We present a modied formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive eects require that an approximation of the square root of the spatial operator is available and inexpensive.

  • 3.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ruggiu, Andrea Alessandro
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Dual Time-Stepping Using Second Derivatives2019In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 81, no 2, p. 1050-1071Article in journal (Refereed)
    Abstract [en]

    We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.

  • 4.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ruggiu, Andrea Alessandro
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    On conservation and stability properties for summation-by-parts schemes2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 344, p. 14p. 451-464Article in journal (Refereed)
    Abstract [en]

    We discuss conservative and stable numerical approximations in summation-by-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.

  • 5.
    Ruggiu, Andrea Alessandro
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Many physical phenomena can be described mathematically by means of partial differential equations. These mathematical formulations are said to be well-posed if a unique solution, bounded by the given data, exists. The boundedness of the solution can be established through the so-called energy-method, which leads to an estimate of the solution by means of integration-by-parts. Numerical approximations mimicking integration-by-parts discretely are said to fulfill the Summation-By-Parts (SBP) property. These formulations naturally yield bounded approximate solutions if the boundary conditions are weakly imposed through Simultaneous-Approximation-Terms (SAT). Discrete problems with bounded solutions are said to be energy-stable.

    Energy-stable and high-order accurate SBP-SAT discretizations for well-posed linear problems were first introduced for centered finite-difference methods. These mathematical formulations, based on boundary conforming grids, allow for an exact mimicking of integration-by-parts. However, other discretizations techniques that do not include one or both boundary nodes, such as pseudo-spectral collocation methods, only fulfill a generalized SBP (GSBP) property but still lead to energy-stable solutions.

    This thesis consists of two main topics. The first part, which is mostly devoted to theoretical investigations, treats discretizations based on SBP and GSBP operators. A numerical approximation of a conservation law is said to be conservative if the approximate solution mimics the physical conservation property. It is shown that conservative and energy-stable spatial discretizations of variable coefficient problems require an exact numerical mimicking of integration-by-parts. We also discuss the invertibility of the algebraic problems arising from (G)SBP-SAT discretizations in time of energy-stable spatial approximations. We prove that pseudo-spectral collocation methods for the time derivative lead to invertible fully-discrete problems. The same result is proved for second-, fourth- and sixth-order accurate finite-difference based time integration methods.

    Once the invertibility of (G)SBP-SAT discrete formulations is established, we are interested in efficient algorithms for the unique solution of such problems. To this end, the second part of the thesis has a stronger experimental flavour and deals with convergence acceleration techniques for SBP-SAT approximations. First, we consider a modified Dual Time-Stepping (DTS) technique which makes use of two derivatives in pseudo-time. The new DTS formulation, compared to the classical one, accelerates the convergence to steady-state and reduces the stiffness of the problem. Next, we investigate multi-grid methods. For parabolic problems, highly oscillating error modes are optimally damped by iterative methods, while smooth residuals are transferred to coarser grids. In this case, we show that the Galerkin condition in combination with the SBP-preserving interpolation operators leads to fast convergence. For hyperbolic problems, low frequency error modes are rapidly expelled by grid coarsening, since coarser grids have milder stability restrictions on time steps. For such problems, Total Variation Dimishing Multi-Grid (TVD-MG) allows for faster wave propagation of first order upwind discretizations. In this thesis, we extend low order TVD-MG schemes to high-order SBP-SAT upwind discretizations.

    List of papers
    1. On conservation and stability properties for summation-by-parts schemes
    Open this publication in new window or tab >>On conservation and stability properties for summation-by-parts schemes
    2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 344, p. 14p. 451-464Article in journal (Refereed) Published
    Abstract [en]

    We discuss conservative and stable numerical approximations in summation-by-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.

    Publisher
    p. 14
    Keywords
    Hyperbolic problems Summation-by-parts Boundary conditions Interface conditions Stability Conservation
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-137544 (URN)10.1016/j.jcp.2017.05.002 (DOI)000402481300023 ()
    Note

    Funding agencies: VINNOVA [2013-01209]

    Available from: 2017-05-21 Created: 2017-05-21 Last updated: 2019-09-03
    2. On pseudo-spectral time discretizations in summation-by-parts form
    Open this publication in new window or tab >>On pseudo-spectral time discretizations in summation-by-parts form
    2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 192-201Article in journal (Refereed) Published
    Abstract [en]

    Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used.

    Place, publisher, year, edition, pages
    Springer Publishing Company, 2018
    Keywords
    Time integration; Initial boundary value problem; Summation-by-parts operators; Pseudo-spectral methods; Eigenvalue problem
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-145083 (URN)10.1016/j.jcp.2018.01.043 (DOI)000428966300011 ()2-s2.0-85041575964 (Scopus ID)
    Available from: 2018-02-09 Created: 2018-02-09 Last updated: 2019-09-03Bibliographically approved
    3. Eigenvalue analysis for summation-by-parts finite difference time discretizations
    Open this publication in new window or tab >>Eigenvalue analysis for summation-by-parts finite difference time discretizations
    2019 (English)Report (Other academic)
    Abstract [en]

    Diagonal norm finite-difference based time integration methods in summation-by-parts form are investigated. The second, fourth and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully-discrete approximations of initial boundary value problems.

    Our findings also allow us to conclude that the second, fourth and sixth order time discretizations are stiffly accurate, strongly S-stable and dissipatively stable Runge-Kutta methods. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

    Place, publisher, year, edition, pages
    Linköping: Linköping University Electronic Press, 2019. p. 35
    Series
    LiTH-MAT-R, ISSN 0348-2960 ; 2019:9
    Keywords
    Time integration, Initial value problem, Summation-by-parts operators, Finite difference methods, Eigenvalue problem
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-160009 (URN)LiTH-MAT-R-2019/09-SE (ISRN)
    Available from: 2019-09-02 Created: 2019-09-02 Last updated: 2019-09-03Bibliographically approved
    4. Dual Time-Stepping Using Second Derivatives
    Open this publication in new window or tab >>Dual Time-Stepping Using Second Derivatives
    2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 81, no 2, p. 1050-1071Article in journal (Refereed) Published
    Abstract [en]

    We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.

    Place, publisher, year, edition, pages
    Springer, 2019
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-160245 (URN)10.1007/s10915-019-01047-5 (DOI)000491440200017 ()
    Note

    Funding agencies:  Linkoping University; Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]; VINNOVAVinnova

    Available from: 2019-09-13 Created: 2019-09-13 Last updated: 2019-11-05
    5. A new multigrid formulation for high order finite difference methods on summation-by-parts form
    Open this publication in new window or tab >>A new multigrid formulation for high order finite difference methods on summation-by-parts form
    2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 359, p. 216-238Article in journal (Refereed) Published
    Abstract [en]

    Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

    Keywords
    High order finite difference methodsSummation-by-partsMultigridRestriction and prolongation operatorsConvergence acceleration
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-145086 (URN)10.1016/j.jcp.2018.01.011 (DOI)000427396200011 ()
    Note

    Funding agencies:  VINNOVA, the Swedish Governmental Agency for Innovation Systems [2013-01209]

    Available from: 2018-02-09 Created: 2018-02-09 Last updated: 2019-09-03
    6. Multigrid schemes for high order discretizations of hyperbolic problems
    Open this publication in new window or tab >>Multigrid schemes for high order discretizations of hyperbolic problems
    2019 (English)In: 2019 AIAA Aerospace Sciences Meeting, AIAA Scitech Forum, American Institute of Aeronautics and Astronautics, 2019, p. 1-25, article id AIAA 2019-0103Conference paper, Published paper (Refereed)
    Abstract [en]

    Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed.

    Place, publisher, year, edition, pages
    American Institute of Aeronautics and Astronautics, 2019
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:liu:diva-154393 (URN)10.2514/6.2019-0103 (DOI)978-1-62410-578-4 (ISBN)
    Conference
    2019 AIAA Aerospace Sciences Meeting, AIAA Scitech Forum, San Diego, California, 7-11 January 2019
    Available from: 2019-02-11 Created: 2019-02-11 Last updated: 2019-09-03
  • 6.
    Ruggiu, Andrea Alessandro
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Eigenvalue analysis for summation-by-parts finite difference time discretizations2019Report (Other academic)
    Abstract [en]

    Diagonal norm finite-difference based time integration methods in summation-by-parts form are investigated. The second, fourth and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully-discrete approximations of initial boundary value problems.

    Our findings also allow us to conclude that the second, fourth and sixth order time discretizations are stiffly accurate, strongly S-stable and dissipatively stable Runge-Kutta methods. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

  • 7.
    Ruggiu, Andrea Alessandro
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Multigrid schemes for high order discretizations of hyperbolic problems2019In: 2019 AIAA Aerospace Sciences Meeting, AIAA Scitech Forum, American Institute of Aeronautics and Astronautics, 2019, p. 1-25, article id AIAA 2019-0103Conference paper (Refereed)
    Abstract [en]

    Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed.

  • 8.
    Ruggiu, Andrea Alessandro
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    On pseudo-spectral time discretizations in summation-by-parts form2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 192-201Article in journal (Refereed)
    Abstract [en]

    Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used.

    The full text will be freely available from 2020-01-31 14:56
  • 9.
    Ruggiu, Andrea Alessandro
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Weinerfelt, Per
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Saab Aerospace.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A new multigrid formulation for high order finite difference methods on summation-by-parts form2017Report (Other academic)
    Abstract [en]

    Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

  • 10.
    Ruggiu, Andrea Alessandro
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Weinerfelt, Per
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Saab Aerospace, SE-581 88 Linköping, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A new multigrid formulation for high order finite difference methods on summation-by-parts form2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 359, p. 216-238Article in journal (Refereed)
    Abstract [en]

    Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

    The full text will be freely available from 2020-01-12 15:01
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