Volume 1, Issue 1
Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Tianheng Chen & Chi-Wang Shu

CSIAM Trans. Appl. Math., 1 (2020), pp. 1-52.

Published online: 2020-03

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  • Abstract

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

  • Keywords

System of conservation laws, entropy stability, discontinuous Galerkin method, summation-by-parts.

  • AMS Subject Headings

65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

tianheng chen@alumni.brown.edu (Tianheng Chen)

chi-wang shu@brown.edu (Chi-Wang Shu)

  • BibTex
  • RIS
  • TXT
@Article{CSIAM-AM-1-1, author = {Chen , Tianheng and Shu , Chi-Wang }, title = {Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {1}, pages = {1--52}, abstract = {

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0003}, url = {http://global-sci.org/intro/article_detail/csiam-am/16793.html} }
TY - JOUR T1 - Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes AU - Chen , Tianheng AU - Shu , Chi-Wang JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 1 EP - 52 PY - 2020 DA - 2020/03 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0003 UR - https://global-sci.org/intro/article_detail/csiam-am/16793.html KW - System of conservation laws, entropy stability, discontinuous Galerkin method, summation-by-parts. AB -

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dissipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

Tianheng Chen & Chi-Wang Shu. (2020). Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes. CSIAM Transactions on Applied Mathematics. 1 (1). 1-52. doi:10.4208/csiam-am.2020-0003
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