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Analysis of boundary and interface closures for finite difference methods for the wave equation
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.
2015 (English)Licentiate 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.

Place, publisher, year, edition, pages
Uppsala University, 2015.
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2015-005
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-264761OAI: oai:DiVA.org:uu-264761DiVA: diva2:861458
Supervisors
Available from: 2015-10-14 Created: 2015-10-16 Last updated: 2017-08-31Bibliographically approved
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, 219-245 p.Article 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, 1002-1028 p.Article 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

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