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  • 1. Almquist, Martin
    et al.
    Wang, Siyang
    Werpers, Jonatan
    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.
    Order-preserving interpolation for summation-by-parts operators at nonconforming grid interfaces2019In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 41, p. A1201-A1227Article in journal (Refereed)
  • 2. Appelö, Daniel
    et al.
    Kreiss, Gunilla
    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.
    Wang, Siyang
    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.
    An explicit Hermite–Taylor method for the Schrödinger equation2017In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 21, p. 1207-1230Article in journal (Refereed)
  • 3. Appelö, Daniel
    et al.
    Wang, Siyang
    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.
    An energy based discontinuous Galerkin method for acoustic–elastic waves2017In: Proc. 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Minneapolis, MN: University of Minnesota Press, 2017, p. 389-390Conference paper (Other academic)
  • 4.
    Ludvigsson, Gustav
    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.
    Steffen, Kyle R.
    Sticko, Simon
    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.
    Wang, Siyang
    Xia, Qing
    Epshteyn, Yekaterina
    Kreiss, Gunilla
    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.
    High-order numerical methods for 2D parabolic problems in single and composite domains2018In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 76, p. 812-847Article in journal (Refereed)
  • 5.
    Wang, Siyang
    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.
    An improved high order finite difference method for non-conforming grid interfaces for the wave equation2018In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, p. 775-792Article in journal (Refereed)
  • 6.
    Wang, Siyang
    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.
    Analysis of boundary and interface closures for finite difference methods for the wave equation2015Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    We consider high order finite difference methods for the wave equations in the second order form, where the finite difference operators satisfy the summation-by-parts principle. Boundary conditions and interface conditions are imposed weakly by the simultaneous-approximation-term method, and non-conforming grid interfaces are handled by an interface operator that is based on either interpolating directly between the grids or on projecting to piecewise continuous polynomials on an intermediate grid.

    Stability and accuracy are two important aspects of a numerical method. For accuracy, we prove the convergence rate of the summation-by-parts finite difference schemes for the wave equation. Our approach is based on Laplace transforming the error equation in time, and analyzing the solution to the boundary system in the Laplace space. In contrast to first order equations, we have found that the determinant condition for the second order equation is less often satisfied for a stable numerical scheme. If the determinant condition is satisfied uniformly in the right half plane, two orders are recovered from the boundary truncation error; otherwise we perform a detailed analysis of the solution to the boundary system in the Laplace space to obtain an error estimate. Numerical experiments demonstrate that our analysis gives a sharp error estimate.

    For stability, we study the numerical treatment of non-conforming grid interfaces. In particular, we have explored two interface operators: the interpolation operators and projection operators applied to the wave equation. A norm-compatible condition involving the interface operator and the norm related to the SBP operator is essential to prove stability by the energy method for first order equations. In the analysis, we have found that in contrast to first order equations, besides the norm-compatibility condition an extra condition must be imposed on the interface operators to prove stability by the energy method. Furthermore, accuracy and efficiency studies are carried out for the numerical schemes.

    List of papers
    1. Convergence of summation-by-parts finite difference methods for the wave equation
    Open this publication in new window or tab >>Convergence of summation-by-parts finite difference methods for the wave equation
    2017 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 71, p. 219-245Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-264752 (URN)10.1007/s10915-016-0297-3 (DOI)000398062500009 ()
    Available from: 2016-09-27 Created: 2015-10-16 Last updated: 2017-05-17Bibliographically approved
    2. High order finite difference methods for the wave equation with non-conforming grid interfaces
    Open this publication in new window or tab >>High order finite difference methods for the wave equation with non-conforming grid interfaces
    2016 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 68, p. 1002-1028Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-264754 (URN)10.1007/s10915-016-0165-1 (DOI)000380693700006 ()
    External cooperation:
    Available from: 2016-01-27 Created: 2015-10-16 Last updated: 2017-12-01Bibliographically approved
  • 7.
    Wang, Siyang
    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.
    Finite Difference and Discontinuous Galerkin Methods for Wave Equations2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost.

    There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

    List of papers
    1. High order finite difference methods for the wave equation with non-conforming grid interfaces
    Open this publication in new window or tab >>High order finite difference methods for the wave equation with non-conforming grid interfaces
    2016 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 68, p. 1002-1028Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-264754 (URN)10.1007/s10915-016-0165-1 (DOI)000380693700006 ()
    External cooperation:
    Available from: 2016-01-27 Created: 2015-10-16 Last updated: 2017-12-01Bibliographically approved
    2. An improved high order finite difference method for non-conforming grid interfaces for the wave equation
    Open this publication in new window or tab >>An improved high order finite difference method for non-conforming grid interfaces for the wave equation
    2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, p. 775-792Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-320600 (URN)10.1007/s10915-018-0723-9 (DOI)000446594600004 ()
    Available from: 2018-05-09 Created: 2017-04-23 Last updated: 2018-11-29Bibliographically approved
    3. Convergence of summation-by-parts finite difference methods for the wave equation
    Open this publication in new window or tab >>Convergence of summation-by-parts finite difference methods for the wave equation
    2017 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 71, p. 219-245Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-264752 (URN)10.1007/s10915-016-0297-3 (DOI)000398062500009 ()
    Available from: 2016-09-27 Created: 2015-10-16 Last updated: 2017-05-17Bibliographically approved
    4. Convergence of finite difference methods for the wave equation in two space dimensions
    Open this publication in new window or tab >>Convergence of finite difference methods for the wave equation in two space dimensions
    2018 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 314, p. 2737-2763Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-320603 (URN)10.1090/mcom/3319 (DOI)
    Available from: 2018-02-02 Created: 2017-04-23 Last updated: 2018-08-23Bibliographically approved
    5. An energy based discontinuous Galerkin method for acoustic–elastic waves
    Open this publication in new window or tab >>An energy based discontinuous Galerkin method for acoustic–elastic waves
    2017 (English)In: Proc. 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Minneapolis, MN: University of Minnesota Press, 2017, p. 389-390Conference paper, Oral presentation with published abstract (Other academic)
    Place, publisher, year, edition, pages
    Minneapolis, MN: University of Minnesota Press, 2017
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-320601 (URN)
    Conference
    WAVES 2017
    Available from: 2017-05-19 Created: 2017-04-23 Last updated: 2017-06-30Bibliographically approved
  • 8.
    Wang, Siyang
    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.
    Elf, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational and Systems Biology. Uppsala University, Science for Life Laboratory, SciLifeLab.
    Hellander, Stefan
    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.
    Lötstedt, Per
    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.
    Stochastic reaction–diffusion processes with embedded lower dimensional structures2012Report (Other academic)
  • 9.
    Wang, Siyang
    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.
    Elf, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational and Systems Biology. Uppsala University, Science for Life Laboratory, SciLifeLab.
    Hellander, Stefan
    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.
    Lötstedt, Per
    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.
    Stochastic reaction–diffusion processes with embedded lower-dimensional structures2014In: Bulletin of Mathematical Biology, ISSN 0092-8240, E-ISSN 1522-9602, Vol. 76, p. 819-853Article in journal (Refereed)
  • 10.
    Wang, Siyang
    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.
    Kreiss, Gunilla
    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.
    Convergence of summation-by-parts finite difference methods for the wave equation2017In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 71, p. 219-245Article in journal (Refereed)
  • 11.
    Wang, Siyang
    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.
    Nissen, Anna
    Kreiss, Gunilla
    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.
    Convergence of finite difference methods for the wave equation in two space dimensions2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 314, p. 2737-2763Article in journal (Refereed)
  • 12.
    Wang, Siyang
    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.
    Virta, Kristoffer
    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.
    Kreiss, Gunilla
    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.
    High order finite difference methods for the wave equation with non-conforming grid interfaces2016In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 68, p. 1002-1028Article in journal (Refereed)
1 - 12 of 12
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