Extension of Tensor-Product Generalized and Dense-Norm Summation-by-Parts Operators to Curvilinear Coordinates

David C. Del Rey Fernández, Pieter D. Boom, Mark H. Carpenter, David W. Zingg

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Methodologies are presented that enable the construction of provably linearly stable and conservative high-order discretizations of partial differential equations in curvilinear coordinates based on generalized summation-by-parts operators, including operators with dense-norm matrices. Specifically, three approaches are presented for the construction of stable and conservative schemes in curvilinear coordinates using summation-by-parts (SBP) operators that have a diagonal norm but may or may not include boundary nodes: (1) the mortar-element approach, (2) the global SBP-operator approach, and (3) the staggered-grid approach. Moreover, the staggered-grid approach is extended to enable the development of stable dense-norm operators in curvilinear coordinates. In addition, collocated upwind simultaneous approximation terms for the weak imposition of boundary conditions or inter-element coupling are extended to curvilinear coordinates with the new approaches. While the emphasis in the paper is on tensor-product SBP operators, the approaches that are covered are directly applicable to multidimensional SBP operators.

Original languageEnglish
Pages (from-to)1957-1996
Number of pages40
JournalJournal of Scientific Computing
Issue number3
Early online date22 Jul 2019
Publication statusPublished - 15 Sep 2019


  • Curvilinear coordinates
  • Linear stability
  • Simultaneous approximation terms
  • Summation by parts


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