A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

doi:10.1007/s42967-021-00120-x

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (2): 679-721.doi: 10.1007/s42967-021-00120-x

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A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Johannes Markert¹, Gregor Gassner², Stefanie Walch³

1 Department for Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
2 Department for Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
3 I. Physikalisches Institut, Center for Data and Simulation Science, University of Cologne, Zülpicher Str. 77, 50937 Cologne, Germany

收稿日期:2019-12-18 修回日期:2021-01-08 出版日期:2023-06-20 发布日期:2023-05-26
通讯作者: Johannes Markert, jmarkert@math.uni-koeln.de E-mail:jmarkert@math.uni-koeln.de
基金资助:
JM acknowledgesfunding through theKlaus-Tschira Stiftung via the project "DG2RAV". GG thanks the Klaus-Tschira Stiftung and the European Research Council for funding through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws" (EXTREME, project no. 71448). SW thanks the Klaus-Tschira Stiftung, and acknowledges the Deutsche Forschungsgemeinschaft (DFG) for funding the sub-project C5 in the SFB956 and the European Research Council for funding the ERC Starting Grant "The radiative interstellar medium" (RADFEEDBACK, project no. 679852). This work was performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK). Furthermore, we thank Dr. Andrés Rueda-Ramírez for his help with the generation of the 3D supernova plot in Fig. 17.

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Johannes Markert¹, Gregor Gassner², Stefanie Walch³

1 Department for Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
2 Department for Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
3 I. Physikalisches Institut, Center for Data and Simulation Science, University of Cologne, Zülpicher Str. 77, 50937 Cologne, Germany

Received:2019-12-18 Revised:2021-01-08 Online:2023-06-20 Published:2023-05-26
Contact: Johannes Markert, jmarkert@math.uni-koeln.de E-mail:jmarkert@math.uni-koeln.de
Supported by:
JM acknowledgesfunding through theKlaus-Tschira Stiftung via the project "DG2RAV". GG thanks the Klaus-Tschira Stiftung and the European Research Council for funding through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws" (EXTREME, project no. 71448). SW thanks the Klaus-Tschira Stiftung, and acknowledges the Deutsche Forschungsgemeinschaft (DFG) for funding the sub-project C5 in the SFB956 and the European Research Council for funding the ERC Starting Grant "The radiative interstellar medium" (RADFEEDBACK, project no. 679852). This work was performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK). Furthermore, we thank Dr. Andrés Rueda-Ramírez for his help with the generation of the 3D supernova plot in Fig. 17.

摘要/Abstract

摘要： In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

关键词: High-order methods, Discontinuous Galerkin spectral element method, Finite volume method, Shock capturing, Astrophysics, Stellar physics

Abstract: In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Key words: High-order methods, Discontinuous Galerkin spectral element method, Finite volume method, Shock capturing, Astrophysics, Stellar physics

中图分类号:

76M99

Johannes Markert, Gregor Gassner, Stefanie Walch. A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 679-721.

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A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

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