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Parallel Kinetic Schemes for Conservation Laws, with Large Time Steps

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Abstract

We propose a new parallel discontinuous Galerkin method for the approximation of hyperbolic systems of conservation laws. The method remains stable with large time steps, while keeping the complexity of an explicit scheme: it does not require the assembly and resolution of large linear systems for the time iterations. The approach is based on a kinetic representation of the system of conservation laws previously investigated by the authors. Coulette et al. (in: Springer proceedings in mathematics & statistics, Springer, Berlin, pp 171–178, 2017. https://doi.org/10.1007/978-3-319-57394-6_19), Badwaik et al. (ESAIM Proc Surv; 63:60–77, 2018. https://doi.org/10.1051/proc/201863060), Coulette et al. (Comput Fluids; 190:485–502, 2019. https://doi.org/10.1016/j.compfluid.2019.06.007), Drui et al. (CR Mécanique; 347(3):259–269, 2019. https://doi.org/10.1016/j.crme.2018.12.001) and Gerhard et al. (Comput Math Appl; 112:116–137, 2022. https://doi.org/10.1016/j.camwa.2022.02.015. https://hal.science/hal-03218086/document). In this paper, the approach is extended with a subdomain strategy that improves the parallel scaling of the method on computers with distributed memory.

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References

  1. Adams, M.P., Adams, M.L., Hawkins, W.D., Smith, T., Rauchwerger, L., Amato, N.M., Bailey, T.S., Falgout, R.D., Kunen, A., Brown, P.: Provably optimal parallel transport sweeps on semi-structured grids. J. Comput. Phys. 407, 109234 (2020). https://doi.org/10.1016/j.jcp.2020.109234

    Article  MathSciNet  Google Scholar 

  2. Alexander, R.: Diagonally implicit Runge–Kutta methods for Stiff O.D.E.’s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977). https://doi.org/10.1137/0714068

    Article  MathSciNet  Google Scholar 

  3. Altmann, C., Belat, T., Gutnic, M., Helluy, P., Mathis, H., Sonnendrücker, É., Angulo, W., Hérard, J.M.: A local time-stepping discontinuous Galerkin algorithm for the MHD system. ESAIM Proc. 28, 33–54 (2009). https://doi.org/10.1051/proc/2009038

    Article  MathSciNet  Google Scholar 

  4. Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37(6), 1973–2004 (2000). https://doi.org/10.1137/s0036142998343075

    Article  MathSciNet  Google Scholar 

  5. Ayachit, U.: The ParaView guide : updated for ParaView version 4.3. Kitware, Clifton Park (2015)

    Google Scholar 

  6. Badwaik, J., Boileau, M., Coulette, D., Franck, E., Helluy, P., Klingenberg, C., Mendoza, L., Oberlin, H.: Task-based parallelization of an implicit kinetic scheme. ESAIM: Proc. Surv. 63, 60–77 (2018). https://doi.org/10.1051/proc/201863060

    Article  MathSciNet  Google Scholar 

  7. Baker, R.S., Koch, K.R.: An \(S_n\) algorithm for the massively parallel CM-200 Computer. Nucl. Sci. Eng. 128(3), 312–320 (1998). https://doi.org/10.13182/nse98-1

  8. Baty, H., Drui, F., Helluy, P., Franck, E., Klingenberg, C., Thanhäuser, L.: A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations. Appl. Math. Comput. 440, 127667 (2023). https://doi.org/10.1016/j.amc.2022.127667

    Article  MathSciNet  Google Scholar 

  9. Boileau, M., Girard, C., Helluy, P., Houillon, M., Muot, N., Prin, G., Strub, T., Weber, B.: Simulation de l’interaction électromagnétique des objets connectés avec le corps humain . https://www.genci.fr/sites/default/files/grands-challenges-idris-2020_0.pdf (2020)

  10. Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95(1/2), 113–170 (1999). https://doi.org/10.1023/a:1004525427365.

    Article  MathSciNet  Google Scholar 

  11. Bourdel, F., Mazet, P.A., Helluy, P.: Resolution of the non-stationary or harmonic Maxwell equations by a discontinuous finite element method. Application to an EMI (electromagnetic impulse) case. In: 10th International Conference on Computing Methods in Applied Sciences and Engineering on Computing Methods in Applied Sciences and Engineering, pp. 405–422. Nova Science Publishers, Inc. Commack, NY, USA (1992)

  12. Brenier, Y.: Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21(6), 1013–1037 (1984). https://doi.org/10.1137/0721063

    Article  MathSciNet  Google Scholar 

  13. Breuer, A., Heinecke, A., Rettenberger, S., Bader, M., Gabriel, A.A., Pelties, C.: Sustained Petascale Performance of Seismic Simulations with SeisSol on SuperMUC. In: Lecture Notes in Computer Science, pp. 1–18. Springer International Publishing (2014). https://doi.org/10.1007/978-3-319-07518-1_1

  14. Catella, A., Dolean, V., Lanteri, S.: An implicit discontinuous Galerkin time-domain method for two-dimensional electromagnetic wave propagation. COMPEL Int. J. Comput. Math. Electric. Electron. Eng. 29(3), 602–625 (2010). https://doi.org/10.1108/03321641011028215

    Article  MathSciNet  Google Scholar 

  15. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-59721-3

  16. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Theory, computation and applications, Papers from the 1st International Symposium held in Newport, RI, May 24–26 (1999)

  17. Costa, R., Clain, S., Loubère, R., Machado, G.J.: Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition. Appl. Math. Model. 54, 752–767 (2018). https://doi.org/10.1016/j.apm.2017.10.016

    Article  MathSciNet  Google Scholar 

  18. Costa, R., Nóbrega, J.M., Clain, S., Machado, G.J., Loubère, R.: Very high-order accurate finite volume scheme for the convection-diffusion equation with general boundary conditions on arbitrary curved boundaries. Int. J. Numer. Methods Eng. 117(2), 188–220 (2018). https://doi.org/10.1002/nme.5953

    Article  MathSciNet  Google Scholar 

  19. Coulette, D., Franck, E., Helluy, P., Mehrenberger, M., Navoret, L.: Palindromic Discontinuous Galerkin Method. In: Springer Proceedings in Mathematics & Statistics, pp. 171–178. Springer International Publishing (2017). https://doi.org/10.1007/978-3-319-57394-6_19

  20. Coulette, D., Franck, E., Helluy, P., Mehrenberger, M., Navoret, L.: High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation. Comput. Fluids 190, 485–502 (2019). https://doi.org/10.1016/j.compfluid.2019.06.007

    Article  MathSciNet  Google Scholar 

  21. Crestetto, A., Helluy, P.: Resolution of the Vlasov–Maxwell system by PIC discontinuous Galerkin method on GPU with OpenCL. ESAIM Proc. 38, 257–274 (2012). https://doi.org/10.1051/proc/201238014

    Article  MathSciNet  Google Scholar 

  22. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques et Applications, vol. 69. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-22980-0

    Book  Google Scholar 

  23. Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31(3), 1985–2014 (2009). https://doi.org/10.1137/070709414

    Article  MathSciNet  Google Scholar 

  24. Dolean, V., Fahs, H., Fezoui, L., Lanteri, S.: Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. 229(2), 512–526 (2010). https://doi.org/10.1016/j.jcp.2009.09.038

    Article  MathSciNet  Google Scholar 

  25. Drui, F., Franck, E., Helluy, P., Navoret, L.: An analysis of over-relaxation in a kinetic approximation of systems of conservation laws. CR Mécanique 347(3), 259–269 (2019). https://doi.org/10.1016/j.crme.2018.12.001

    Article  Google Scholar 

  26. Dubois, F.: Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes. Int. J. Modern Phys. C 25(12), 1441014 (2014). https://doi.org/10.1142/s0129183114410149

    Article  Google Scholar 

  27. Dumbser, M., Fambri, F., Tavelli, M., Bader, M., Weinzierl, T.: Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Axioms 7(3), 63 (2018). https://doi.org/10.3390/axioms7030063

    Article  Google Scholar 

  28. Dumbser, M., Käser, M., Toro, E.F.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—V. Local time stepping and \(p\)-adaptivity. Geophys. J. Int. 171(2), 695–717 (2007). https://doi.org/10.1111/j.1365-246x.2007.03427.x

    Article  Google Scholar 

  29. Ecer, A., Gopalaswamy, N., Akay, H.U., Chien, Y.P.: Digital filtering techniques for parallel computation of explicit schemes. Int. J. Comput. Fluid Dyn. 13(3), 211–222 (2000). https://doi.org/10.1080/10618560008940899

    Article  Google Scholar 

  30. Fernández-Fidalgo, J., Clain, S., Ramírez, L., Colominas, I., Nogueira, X.: Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids. Comput. Methods Appl. Mech. Engrg. 360, 112782 (2020). https://doi.org/10.1016/j.cma.2019.112782

    Article  MathSciNet  Google Scholar 

  31. Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM Math. Model. Numer. Anal. 39(6), 1149–1176 (2005). https://doi.org/10.1051/m2an:2005049

    Article  MathSciNet  Google Scholar 

  32. Gaffar, M., Jiao, D.: An explicit and unconditionally stable FDTD method for electromagnetic analysis. IEEE Trans. Microw. Theory Technol. 62(11), 2538–2550 (2014). https://doi.org/10.1109/tmtt.2014.2358557

    Article  Google Scholar 

  33. Gaffar, M., Jiao, D.: Alternative method for making explicit FDTD unconditionally stable. IEEE Trans. Microw. Theory Techn. 63(12), 4215–4224 (2015). https://doi.org/10.1109/tmtt.2015.2496255

    Article  Google Scholar 

  34. Gerhard, P., Helluy, P., Michel-Dansac, V.: Unconditionally stable and parallel Discontinuous Galerkin solver. Comput. Math. Appl. 112, 116–137 (2022). https://doi.org/10.1016/j.camwa.2022.02.015.https://hal.science/hal-03218086/document

  35. Geuzaine, C., Remacle, J.F.: Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009). https://doi.org/10.1002/nme.2579.

    Article  MathSciNet  Google Scholar 

  36. Girard, C., Weber, B., Cirou, B., Ponz, V.C.: SHAPE project AxesSim: CINES Partnership: HPC for connected Objects. https://prace-ri.eu/wp-content/uploads/AXESSIM-%E2%80%93-CINES-Partnership-HPC-for-connected-Objects.pdf (2018)

  37. Guiffaut, C., Reineix, A., Pecqueux, B.: New oblique thin wire formalism in the FDTD method with multiwire junctions. IEEE T. Antenn. Propag. 60(3), 1458–1466 (2012). https://doi.org/10.1109/tap.2011.2180304

    Article  MathSciNet  Google Scholar 

  38. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, New York (2008). https://doi.org/10.1007/978-0-387-72067-8

    Book  Google Scholar 

  39. Higueras, I., Happenhofer, N., Koch, O., Kupka, F.: Optimized strong stability preserving IMEX Runge–Kutta methods. J. Comput. Appl. Math. 272, 116–140 (2014). https://doi.org/10.1016/j.cam.2014.05.011

    Article  MathSciNet  Google Scholar 

  40. Hochbruck, M., Pažur, T.: Implicit Runge–Kutta Methods and Discontinuous Galerkin discretizations for Linear Maxwell’s equations. SIAM J. Numer. Anal. 53(1), 485–507 (2015). https://doi.org/10.1137/130944114

    Article  MathSciNet  Google Scholar 

  41. Houillon, M.: Schémas Galerkin Discontinu optimisés pour les problèmes d’électromagnétisme avec des géométries complexes. Ph.D. Thesis, Université de Strasbourg (2020). https://hal.archives-ouvertes.fr/tel-03023095

  42. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998). https://doi.org/10.1137/s1064827595287997

    Article  MathSciNet  Google Scholar 

  43. Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge–Kutta methods for stiff ODEs. Appl. Numer. Math. 146, 221–244 (2019). https://doi.org/10.1016/j.apnum.2019.07.008

    Article  MathSciNet  Google Scholar 

  44. Klöckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863–7882 (2009). https://doi.org/10.1016/j.jcp.2009.06.041

    Article  MathSciNet  Google Scholar 

  45. Matsakis, N., Stone, J.: Rayon: A data parallelism library for Rust. https://github.com/rayon-rs/rayon (2022)

  46. Michel-Dansac, V., Thomann, A.: TVD-MOOD schemes based on implicit-explicit time integration. Appl. Math. Comput. 433, 127397 (2022). https://doi.org/10.1016/j.amc.2022.127397

    Article  MathSciNet  Google Scholar 

  47. Munz, C.D., Omnes, P., Schneider, R., Sonnendrücker, E., Voß, U.: Divergence correction techniques for maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161(2), 484–511 (2000). https://doi.org/10.1006/jcph.2000.6507

    Article  MathSciNet  Google Scholar 

  48. Müller, S., Stiriba, Y.: Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comput. 30(3), 493–531 (2006). https://doi.org/10.1007/s10915-006-9102-z

    Article  MathSciNet  Google Scholar 

  49. Pautz, S.D.: An algorithm for parallel \(S_n\) sweeps on unstructured meshes. Nucl. Sci. Eng. 140(2), 111–136 (2002). https://doi.org/10.13182/nse02-1

    Article  Google Scholar 

  50. Perthame, B.: Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27(6), 1405–1421 (1990)

    Article  MathSciNet  Google Scholar 

  51. Shi, X., Lin, J., Yu, Z.: Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Internat. J. Numer. Methods Fluids 42(11), 1249–1261 (2003). https://doi.org/10.1002/fld.594

    Article  Google Scholar 

  52. Weber, B.: Optimisation de code Galerkin Discontinu sur ordinateur hybride. Application à la simulation numérique en électromagnétisme. Ph.D. Thesis, Université de Strasbourg (2018). https://theses.hal.science/tel-01911261

  53. Yan, J., Jiao, D.: Explicit and unconditionally stable FDTD method without eigenvalue solutions. In: 2016 IEEE MTT-S International Microwave Symposium (IMS). IEEE (2016). https://doi.org/10.1109/mwsym.2016.7540416

  54. Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966). https://doi.org/10.1109/tap.1966.1138693

    Article  Google Scholar 

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Funding

This work of the Interdisciplinary Thematic Institute IRMIA++, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10- IDEX-0002), and by SFRI-STRAT’US project (ANR-20-SFRI-0012) under the framework of the French Investments for the Future Program. This work was also supported by France Relance.

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Authors and Affiliations

  1. IRMA, CNRS UMR 7501, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg, France

    Pierre Gerhard & Philippe Helluy

  2. CNRS, Inria, IRMA, Université de Strasbourg, 67000, Strasbourg, France

    Philippe Helluy & Victor Michel-Dansac

  3. AxesSim, Bâtiment Le Pythagore, 1 Rue Jean Sapidus, 67400, Illkirch-Graffenstaden, France

    Bruno Weber

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  1. Pierre Gerhard
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Contributions

Conceptualization: PH, VMD, formal analysis: PH, VMD, funding acquisition: PH, software: PG, BW, VMD, PH, validation: PG, visualization: PG, VMD, writing—original draft: PH, VMD, writing—review and editing: PG, BW, VMD, PH

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Gerhard, P., Helluy, P., Michel-Dansac, V. et al. Parallel Kinetic Schemes for Conservation Laws, with Large Time Steps. J Sci Comput 99, 5 (2024). https://doi.org/10.1007/s10915-024-02468-7

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