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

Abstract

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.

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Correspondence to David C. Del Rey Fernández.

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Portions of this paper were previously published in “Numerical Investigation of Tensor-Product Summation-by-Parts Discretization Strategies and Operators” AIAA paper 2017-0530.

Periodic Boundary Conditions for the Global SBP-Operator Approach

Periodic Boundary Conditions for the Global SBP-Operator Approach

The computation of the metrics and Jacobian involve derivatives of the physical nodal locations. The global SBP-operator approach includes the use of interface SATs to couple the derivative values in adjacent elements. Unfortunately, for problems with periodic boundary conditions, the nodal locations are not necessarily periodic, even if the solution is periodic. As a result, naively applying the SBP operator approach will not give the correct solution. In order to use the global SBP-operator approach, the interface SATs for periodic boundary conditions used in the computation of the metrics and Jacobian must be modified. This requires knowledge of the domain’s geometry.

For the test case used in Sect. 12, the size of the domain relative to any point is unity in each direction. In other words, traveling one unit along any of the coordinate directions in physical space will bring you back to the same point, even though the domain is not a unit cube. Therefore, the interface SATs involving nodal locations used at periodic faces only require the addition or subtraction of a unit value. A one-dimensional example is shown in Fig. 4.

Fig. 4
figure4

Interface SATs used for periodic faces in the computation of metric and Jacobian for the the global SBP-operator approach

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Del Rey Fernández, D.C., Boom, P.D., Carpenter, M.H. et al. Extension of Tensor-Product Generalized and Dense-Norm Summation-by-Parts Operators to Curvilinear Coordinates. J Sci Comput 80, 1957–1996 (2019) doi:10.1007/s10915-019-01011-3

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Keywords

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

Mathematics Subject Classification

  • 65M06
  • 65M60
  • 65M70