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Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Versbach, Lea Miko LU ; Linders, Viktor LU ; Klöfkorn, Robert LU orcid and Birken, Philipp LU (2023) In The SMAI Journal of computational mathematics 9. p.61-93
Abstract
We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed... (More)
We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems. (Less)
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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Space-time, Discontinuous Galerkin, DG-SEM
in
The SMAI Journal of computational mathematics
volume
9
pages
61 - 93
publisher
Centre Mersenne
external identifiers
  • scopus:85164375799
ISSN
2426-8399
DOI
10.5802/smai-jcm.95
language
English
LU publication?
yes
id
244d8e6b-9c16-4802-8c9a-1cac68e124cc
date added to LUP
2023-08-19 15:22:03
date last changed
2023-09-14 12:27:58
@article{244d8e6b-9c16-4802-8c9a-1cac68e124cc,
  abstract     = {{We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.}},
  author       = {{Versbach, Lea Miko and Linders, Viktor and Klöfkorn, Robert and Birken, Philipp}},
  issn         = {{2426-8399}},
  keywords     = {{Space-time; Discontinuous Galerkin; DG-SEM}},
  language     = {{eng}},
  pages        = {{61--93}},
  publisher    = {{Centre Mersenne}},
  series       = {{The SMAI Journal of computational mathematics}},
  title        = {{Theoretical and Practical Aspects of Space-Time DG-SEM Implementations}},
  url          = {{http://dx.doi.org/10.5802/smai-jcm.95}},
  doi          = {{10.5802/smai-jcm.95}},
  volume       = {{9}},
  year         = {{2023}},
}