Journal of Scientific Computing

, Volume 80, Issue 3, pp 1957–1996 | Cite as

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

  • David C. Del Rey FernándezEmail author
  • Pieter D. Boom
  • Mark H. Carpenter
  • David W. Zingg


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.


Summation by parts Simultaneous approximation terms Curvilinear coordinates Linear stability 

Mathematics Subject Classification

65M06 65M60 65M70 


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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  • David C. Del Rey Fernández
    • 1
    • 2
    Email author
  • Pieter D. Boom
    • 3
  • Mark H. Carpenter
    • 1
  • David W. Zingg
    • 3
  1. 1.NASA LaRCHamptonUSA
  2. 2.NIAHamptonUSA
  3. 3.UTIASNorth YorkCanada

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