A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

doi:10.1007/s42967-023-00307-4

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1742-1778.doi: 10.1007/s42967-023-00307-4

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A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

Saray Busto¹, Michael Dumbser²

1 Departamento de Matemática Aplicada I, Universidade de Vigo, Campus As Lagoas Marcosende s/n, E-36310 Vigo, Spain;
2 Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, I-38123 Trento, Italy

收稿日期:2023-01-30 修回日期:2023-05-30 接受日期:2023-08-15 发布日期:2024-12-20
通讯作者: Michael Dumbser,michael.dumbser@unitn.it;Saray Busto,saray.busto@unitn.it E-mail:michael.dumbser@unitn.it;saray.busto@unitn.it
基金资助:
The research presented in this paper was funded by the Spanish Ministry of Science and Innovation under project number PID2021-122625OB-I00 and by the Italian Ministry of Education, University and Research (MIUR) in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications, the PRIN 2022 project High order structure-preserving semiimplicit schemes for hyperbolic equations and under the Department of Excellence Initiative 2023-2027. M.D. is member of INdAM GNCS and was also co-funded by the European Union NextGenerationEU (PNRR, Spoke 7 CN HPC). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors would like to thank the Leibniz Rechenzentrum (LRZ) in Garching, Germany, for the access to SuperMUC-NG under project pr83no as well as the Spanish Supercomputing Network (RES) under project RES-IM-2022-3-0017 and the CESGA supercomputing center in Santiago de Compostela, Spain, for granting access to FinisTerrae 3. The authors are very grateful to the two anonymous referees for their constructive and insightful comments, which helped to improve the clarity and quality of this paper.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

Saray Busto¹, Michael Dumbser²

1 Departamento de Matemática Aplicada I, Universidade de Vigo, Campus As Lagoas Marcosende s/n, E-36310 Vigo, Spain;
2 Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, I-38123 Trento, Italy

Received:2023-01-30 Revised:2023-05-30 Accepted:2023-08-15 Published:2024-12-20
Contact: Michael Dumbser,michael.dumbser@unitn.it;Saray Busto,saray.busto@unitn.it E-mail:michael.dumbser@unitn.it;saray.busto@unitn.it
Supported by:
The research presented in this paper was funded by the Spanish Ministry of Science and Innovation under project number PID2021-122625OB-I00 and by the Italian Ministry of Education, University and Research (MIUR) in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications, the PRIN 2022 project High order structure-preserving semiimplicit schemes for hyperbolic equations and under the Department of Excellence Initiative 2023-2027. M.D. is member of INdAM GNCS and was also co-funded by the European Union NextGenerationEU (PNRR, Spoke 7 CN HPC). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors would like to thank the Leibniz Rechenzentrum (LRZ) in Garching, Germany, for the access to SuperMUC-NG under project pr83no as well as the Spanish Supercomputing Network (RES) under project RES-IM-2022-3-0017 and the CESGA supercomputing center in Santiago de Compostela, Spain, for granting access to FinisTerrae 3. The authors are very grateful to the two anonymous referees for their constructive and insightful comments, which helped to improve the clarity and quality of this paper.

摘要/Abstract

摘要： In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

关键词: Overdetermined thermodynamically compatible hyperbolic systems, Hyperbolic and thermodynamically compatible (HTC) finite volume schemes, Abgrall framework, Discrete entropy inequality, Nonlinear stability in the energy norm, Applications to ideal magnetohydrodynamics (MHD), Godounov-Peshkov-Romenski (GPR) model of continuum mechanics, Turbulent shallow water (TSW) flows

Abstract: In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Key words: Overdetermined thermodynamically compatible hyperbolic systems, Hyperbolic and thermodynamically compatible (HTC) finite volume schemes, Abgrall framework, Discrete entropy inequality, Nonlinear stability in the energy norm, Applications to ideal magnetohydrodynamics (MHD), Godounov-Peshkov-Romenski (GPR) model of continuum mechanics, Turbulent shallow water (TSW) flows

中图分类号:

35L40
65M08

Saray Busto, Michael Dumbser. A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1742-1778.

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A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

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