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Conservative and stable degree preserving SBP operators for non-conforming meshes
Mathematical Institute, University of Cologne, Cologne, Germany.
Institute for Aerospace Studies, University of Toronto, Toronto, Canada.
Mathematical Institute, University of Cologne, Cologne, Germany.ORCID iD: 0000-0002-5902-1522
Mathematical Institute, University of Cologne, Cologne, Germany.
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2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 75, no 2, p. 657-686Article in journal (Refereed) Published
Abstract [en]

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree ≥ 2p, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is 2p − 1 accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2018. Vol. 75, no 2, p. 657-686
Keywords [en]
First derivative, Summation-by-parts, Simultaneous-approximation-term, Conservation, Energy stability, Finite difference methods, Non-conforming methods, Intermediate grids
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-156858DOI: 10.1007/s10915-017-0563-zISI: 000428565100004Scopus ID: 2-s2.0-85030107035OAI: oai:DiVA.org:liu-156858DiVA, id: diva2:1315785
Funder
German Research Foundation (DFG), TA 2160/1-1Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-24Bibliographically approved

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