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  • 1.
    Berg, Jens
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software.

    We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models.

    The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

    List of papers
    1. A stable and high-order accurate conjugate heat transfer problem
    Open this publication in new window or tab >>A stable and high-order accurate conjugate heat transfer problem
    2010 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, p. 5440-5456Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-122740 (URN)10.1016/j.jcp.2010.04.010 (DOI)000279139200010 ()
    Available from: 2010-04-13 Created: 2010-04-17 Last updated: 2018-01-12Bibliographically approved
    2. Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains
    Open this publication in new window or tab >>Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains
    2012 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, p. 1620-1638Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-177315 (URN)10.1016/j.apnum.2012.05.002 (DOI)000309027700002 ()
    Available from: 2012-05-15 Created: 2012-07-08 Last updated: 2017-12-07Bibliographically approved
    3. Stable Robin solid wall boundary conditions for the Navier-Stokes equations
    Open this publication in new window or tab >>Stable Robin solid wall boundary conditions for the Navier-Stokes equations
    2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, p. 7519-7532Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-156456 (URN)10.1016/j.jcp.2011.06.027 (DOI)000294979400027 ()
    Available from: 2011-07-05 Created: 2011-07-23 Last updated: 2017-12-08Bibliographically approved
    4. Conjugate heat transfer for the unsteady compressible Navier–Stokes equations using a multi-block coupling
    Open this publication in new window or tab >>Conjugate heat transfer for the unsteady compressible Navier–Stokes equations using a multi-block coupling
    2013 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 72, p. 20-29Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-187192 (URN)10.1016/j.compfluid.2012.11.018 (DOI)000314442900002 ()
    Available from: 2012-12-19 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved
    5. Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form
    Open this publication in new window or tab >>Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form
    2012 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, p. 6846-6860Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-177320 (URN)10.1016/j.jcp.2012.06.032 (DOI)000307299400014 ()
    Available from: 2012-07-05 Created: 2012-07-08 Last updated: 2017-12-07Bibliographically approved
    6. On the impact of boundary conditions on dual consistent finite difference discretizations
    Open this publication in new window or tab >>On the impact of boundary conditions on dual consistent finite difference discretizations
    2013 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 236, p. 41-55Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-187194 (URN)10.1016/j.jcp.2012.11.019 (DOI)000314801500005 ()
    Available from: 2012-12-12 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved
  • 2.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    A stable and dual consistent boundary treatment using finite differences on summation-by-parts form2012In: Proc. ECCOMAS Congress 2012, Austria: Tech. Univ. Wien , 2012, p. 7557-7570Conference paper (Refereed)
  • 3.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    A stable and dual consistent boundary treatment using finite differences on summation-by-parts form2012Report (Other academic)
  • 4.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations2013Report (Other academic)
  • 5.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 259, p. 135-153Article in journal (Refereed)
  • 6.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Duality based boundary treatment for the Euler and Navier-Stokes equations2013In: Proc. 21st AIAA CFD Conference, AIAA , 2013Conference paper (Refereed)
  • 7.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    On the impact of boundary conditions on dual consistent finite difference discretizations2012Report (Other academic)
  • 8.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    On the impact of boundary conditions on dual consistent finite difference discretizations2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 236, p. 41-55Article in journal (Refereed)
  • 9.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains2012In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, p. 1620-1638Article in journal (Refereed)
  • 10.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Stable Robin solid wall boundary conditions for the Navier-Stokes equations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, p. 7519-7532Article in journal (Refereed)
  • 11.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form2012Report (Other academic)
  • 12.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form2012In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, p. 6846-6860Article in journal (Refereed)
  • 13.
    Berg, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A unified deep artificial neural network approach to partial differential equations in complex geometries2018In: Neurocomputing, ISSN 0925-2312, E-ISSN 1872-8286, Vol. 317, p. 28-41Article in journal (Refereed)
    Abstract [en]

    In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

  • 14.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Bejhed, Johan
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Measurements and numerical modelling of orifice flow in microchannels2009In: Proc. 41st AIAA Thermophysics Conference, AIAA , 2009Conference paper (Refereed)
  • 15.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A stable and high order accurate conjugate heat transfer problem2009Report (Other academic)
  • 16.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A stable and high order interface procedure for conjugate heat transfer problems2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 599-607Conference paper (Refereed)
  • 17.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A stable and high-order accurate conjugate heat transfer problem2010In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, p. 5440-5456Article in journal (Refereed)
  • 18.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains2010Report (Other academic)
  • 19.
    Lindström, Jens
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Stable Robin boundary conditions for the Navier-Stokes equations2011Report (Other academic)
  • 20. Nordström, Jan
    et al.
    Berg, Jens
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Conjugate heat transfer for the unsteady compressible Navier–Stokes equations using a multi-block coupling2013In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 72, p. 20-29Article in journal (Refereed)
1 - 20 of 20
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