High-order multistep coupling: convergence, stability and PDE application

Antoine E. Simon; Laurent François; Marc Massot

doi:10.5802/crmeca.333

Article de recherche

High-order multistep coupling: convergence, stability and PDE application
[Couplage multipas d’ordre élevé : convergence, stabilité et application aux EDPs]

Antoine E. Simon ^1,² ; Laurent François ¹ ; Marc Massot ³

¹ ONERA, 6 Chemin de la Vauve aux Granges, 91123 Palaiseau, France
² CMAP, CNRS — École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau Cedex, France
³ CMAP, CNRS — École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau Cedex, France

Comptes Rendus. Mécanique, Volume 353 (2025), pp. 1159-1184

Cet article fait partie du numéro thématique Les environnements de simulation multiphysiques coordonné par Nicolas Bertier et al..

Abstract (VO)
Résumé

Designing coupling schemes for specialized advanced mono-physics solvers in order to conduct accurate and efficient multiphysics simulations is a key issue that has recently received a lot of attention. A novel high-order adaptive multistep coupling strategy has shown potential to improve the efficiency and accuracy of such simulations, but requires further analysis. The purpose of the present contribution is to conduct the numerical analysis of convergence of the explicit and implicit variants of the method and to provide a first analysis of its absolute stability. A simplified coupled problem is constructed to assess the stability of the method along the lines of the Dahlquist’s test equation for ODEs. We propose a connection with the stability analysis of other methods such as splitting and ImEx schemes. A stability analysis on a representative conjugate heat transfer case is also presented. This work constitutes a first building block to an a priori analysis of the stability of coupled PDEs.

Concevoir des schémas de couplage de solveurs mono-physiques spécialisés pour mener des simulations multi-physiques précises et efficaces est un problème clé qui a été l’objet de beaucoup de travaux récemment. Un nouveau schéma de couplage multipas d’ordre élevé et à pas de temps adaptatif a montré un potentiel certain pour l’amélioration de l’efficacité et la précision de telles simulations, mais requiert toutefois une analyse plus approfondie. L’objectif de cette contribution est de mener une analyse numérique de la convergence des variantes explicite et implicite de cette méthode et d’introduire une première analyse de sa stabilité absolue. Un problème de couplage simplifié est construit pour évaluer la stabilité de la méthode, à l’instar de l’équation test de Dahlquist pour l’analyse des équations différentielles ordinaires. Nous comparons aussi la stabilité d’autres méthodes, comme le splitting d’opérateurs et des schémas ImEx. Une analyse de stabilité sur un cas test représentatif de transfert de chaleur conjugué est aussi présentée.

Reçu le : 2025-01-20
Révisé le : 2025-09-17
Accepté le : 2025-10-20
Publié le : 2025-11-18

DOI : 10.5802/crmeca.333

Keywords: Multiphysics simulation, high-order, multistep coupling, stability, conjugate heat transfer
Mots-clés : Simulation multi-physique, ordre élevé, couplage multipas, stabilité, transfert de chaleur conjugué

Affiliations des auteurs :

Antoine E. Simon ^{1, 2} ; Laurent François ¹ ; Marc Massot ³

¹ ONERA, 6 Chemin de la Vauve aux Granges, 91123 Palaiseau, France
² CMAP, CNRS — École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau Cedex, France
³ CMAP, CNRS — École polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Antoine E. Simon and Laurent Fran\c{c}ois and Marc Massot},
     title = {High-order multistep coupling: convergence, stability and {PDE} application},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {1159--1184},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {353},
     doi = {10.5802/crmeca.333},
     language = {en},
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Antoine E. Simon; Laurent François; Marc Massot. High-order multistep coupling: convergence, stability and PDE application. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 1159-1184. doi: 10.5802/crmeca.333

Bibliographie
Cité par

[1] Charbel Farhat; M. Lesoinne Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems, Comput. Methods Appl. Mech. Eng., Volume 182 (2000) no. 3, pp. 499-515 | DOI

[2] Aymeric Bourlet; Fabien Tholin; Julien Labaune; François Pechereau; Axel Vincent-Randonnier; Christophe O. Laux Numerical model of restrikes in gliding arc discharges, Plasma Sources Sci. Technol., Volume 33 (2024) no. 1, 015010

[3] V. Kazemi-Kamyab; A. H. Van Zuijlen; H. Bijl A high order time-accurate loosely-coupled solution algorithm for unsteady conjugate heat transfer problems, Comput. Methods Appl. Mech. Eng., Volume 264 (2013), pp. 205-217 | DOI

[4] Lancelot Boulet; Pierre Bénard; Ghislain Lartigue; Vincent Moureau; Sheddia Didorally; N Chauvet; F. Duchaine Modeling of conjugate heat transfer in a kerosene/air spray flame used for aeronautical fire resistance tests, Flow Turbul. Combust., Volume 101 (2018), pp. 579-602 | DOI

[5] Adrien Langenais; François Vuillot; Julien Troyes; Christophe Bailly Accurate simulation of the noise generated by a hot supersonic jet including turbulence tripping and nonlinear acoustic propagation, Phys. Fluids, Volume 31 (2019), 016105, 31 pages | DOI

[6] A. Refloch; B. Courbet; A. Murrone; P. Villedieu; C. Laurent; P. Gilbank; Julien Troyes; L. Tessé; G. Chaineray; J. B. Dargaud; E. Quémerais; François Vuillot CEDRE software, AerospaceLab J. (2011) no. 2, pp. 1-10

[7] ONERA onera/cwipi: Library for coupling parallel scientific codes via MPI communications to perform multi-physics simulations https://github.com/onera/cwipi (Accessed 2025-10-29)

[8] A. Grenouilloux; R. Letournel; N. Dellinger; K. Bioche; Vincent Moureau Large-eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite material, Proceedings of the 14th Direct and Large Eddy Simulation (DLES) Workshop, ERCOFTAC (2024)

[9] David E Keyes; Lois C. McInnes; Carol S. Woodward; William Gropp; Eric Myra; Michael Pernice; John Bell; Jed Brown; Alain Clo; Jeffrey Connors; Emil M. Constantinescu; Don Estep; Kate Evans; Charbel Farhat; Ammar Hakim; Glenn Hammond; Glen Hansen; Judith Hill; Tobin Isaac; Xiangmin Jiao; Kirk Jordan; Dinesh Kaushik; Efthimios Kaxiras; Alice Koniges; Kihwan Lee; Aaron Lott; Qiming Lu; John Magerlein; Reed Maxwell; Michael McCourt; Miriam Mehl; Roger Pawlowski; Amanda P Randles; Daniel R. Reynolds; Beatrice Rivière; Ulrich Rüde; Tim Scheibe; John Shadid; Brendan Sheehan; Mark Shephard; Andrew Siegel; Barry Smith; Xianzhu Tang; Cian Wilson; Barbara Wohlmuth Multiphysics simulations: challenges and opportunities, Int. J. High Perform. Comput. Appl., Volume 27 (2013) no. 1, pp. 4-83 | DOI

[10] Christopher A. Kennedy; Mark H. Carpenter Additive Runge–Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., Volume 44 (2003) no. 1, pp. 139-181 | DOI

[11] Michael Günther; Anne Kværnø; P. Rentrop Multirate Partitioned Runge–Kutta Methods, BIT Numer. Math., Volume 41 (2001) no. 3, pp. 504-514 | DOI

[12] C. William Gear Automatic multirate methods for ordinary differential equations, Information processing 80 (Simon H. Lavington, ed.) (IFIP Congress Series), North-Holland, 1980 no. 8, pp. 717-722

[13] C. William Gear; D. R. Wells Multirate linear multistep methods, BIT, Volume 24 (1984) no. 4, pp. 484-502 | DOI

[14] Shinhoo Kang; Alp Dener; Aidan Hamilton; Hong Zhang; Emil M. Constantinescu; Robert L. Jacob Multirate partitioned Runge–Kutta methods for coupled Navier–Stokes equations, Comput. Fluids, Volume 264 (2023), 105964 | DOI | MR

[15] Jörg Wensch; Oswald Knoth; Alexander Galant Multirate infinitesimal step methods for atmospheric flow simulation, BIT Numer. Math., Volume 49 (2009), pp. 449-473 | DOI

[16] John J. Loffeld; Andy Nonaka; Daniel R. Reynolds; David J. Gardner; Carol S. Woodward Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks, Int. J. High Perform. Comput. Appl., Volume 38 (2024) no. 4, pp. 263-281 | DOI

[17] Benjamin Rüth; Benjamin Uekermann; Miriam Mehl; Philipp Birken; Azahar Monge; Hans-Joachim Bungartz Quasi-Newton waveform iteration for partitioned surface-coupled multiphysics applications, Int. J. Numer. Methods Eng., Volume 122 (2021) no. 19, pp. 5236-5257 | DOI

[18] Laurent François; Marc Massot Multistep interface coupling for high-order adaptive black-box multiphysics simulations, X International Conference on Computational Methods for Coupled Problems in Science and Engineering (M. Papadrakakis; B. Schrefler; E. Oñate, eds.), 2023 | DOI

[19] Laurent François Multiphysics modelling and simulation of solid rocket motor ignition, Ph. D. Thesis, Institut Polytechnique de Paris (France) (2022)

[20] Ali Asad; Romain de Loubens; Laurent François; Marc Massot High-order adaptive multi-domain time integration scheme for microscale lithium-ion batteries simulations, SMAI J. Comput. Math., Volume 11 (2025), pp. 369-404 | DOI | MR

[21] Ralf Kübler; Werner Schiehlen Two methods of simulator coupling, Math. Comput. Model. Dyn. Syst., Volume 6 (2000) no. 2, pp. 93-113 | DOI

[22] M. Busch Zur effizienten Kopplung von Simulationsprogrammen, Ph. D. Thesis, Universität Kassel (Germany) (2012)

[23] A. Toselli; O. Widlund Domain decomposition methods — Algorithms and theory, Springer Series in Computational Mathematics, Springer, 2004 no. 34 | DOI | MR

[24] M. Gander; P. Henry; G. Wanner Landmarks in the history of iterative methods https://www.unige.ch/...

[25] E. Hairer; G. Wanner Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, 14, Springer, 1996 | MR

[26] Gilbert Strang On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968) no. 3, pp. 506-517 | DOI

[27] Christopher A. Kennedy; Mark H. Carpenter Higher-order additive Runge–Kutta schemes for ordinary differential equations, Appl. Numer. Math., Volume 136 (2019), pp. 183-205 | DOI

[28] Joris Degroote Partitioned simulation of fluid-structure interaction: coupling black-box solvers with quasi-Newton techniques, Arch. Comput. Methods Eng., Volume 20 (2013) no. 3, pp. 185-238 | DOI

[29] Uri M. Ascher; Linda R. Petzold Computer methods for ordinary differential equations and differential-algebraic equations, Society for Industrial and Applied Mathematics, 1998 | DOI

[30] Michelle Schatzman Numerical analysis: a mathematical introduction, Oxford University Press, 2002 | Zbl

[31] V. Kazemi-Kamyab; A. H. Van Zuijlen; H. Bijl Analysis and application of high order implicit Runge–Kutta schemes for unsteady conjugate heat transfer: a strongly-coupled approach, J. Comput. Phys., Volume 272 (2014), pp. 471-486 | DOI

[32] Philipp Birken; Tobias Gleim; Detlef Kuhl; Andreas Meister Fast solvers for unsteady thermal fluid structure interaction, Int. J. Numer. Methods Fluids, Volume 79 (2015) no. 1, pp. 16-29 | DOI

[33] Anne Kværnø Stability of multirate Runge–Kutta schemes, Int. J. Differ. Equ. Appl., Volume 1A (2000) no. 1, pp. 97-105 | Zbl

[34] Gilbert Strang On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968) no. 3, pp. 506-517 | DOI

[35] Stéphane Descombes; Max Duarte; Thierry Dumont; Violaine Louvet; Marc Massot Adaptive time splitting method for multi-scale evolutionary partial differential equations, Confluentes Math., Volume 3 (2011) no. 3, pp. 413-443 | DOI

[36] Adrian Sandu; Michael Günther A generalized-structure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., Volume 53 (2015) no. 1, pp. 17-42 | DOI

[37] M. B. Giles Stability analysis of numerical interface conditions in fluid-structure thermal analysis, Int. J. Numer. Methods Fluids, Volume 25 (1997) no. 4, pp. 421-436 | DOI

[38] Laurent François hpc-maths/Rhapsopy: Demonstrator of high-order adaptive code coupling, implicit or explicit https://github.com/hpc-maths/rhapsopy (Accessed 2025-10-29)

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