High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

doi:10.1007/s42967-023-00355-w

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1954-1977.doi: 10.1007/s42967-023-00355-w

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High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

Michel Bergmann^1,2, Afaf Bouharguane^1,2,3, Angelo Iollo^1,2,3, Alexis Tardieu^1,2,3

1 Centre Inria de l'Université de Bordeaux, Memphis Team, Talence, France;
2 Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Talence, France;
3 Université de Bordeaux, Bordeaux, France

收稿日期:2023-02-25 修回日期:2023-09-19 接受日期:2023-11-28 发布日期:2024-12-20
通讯作者: Alexis Tardieu,alexis.tardieu@u-bordeaux.fr;Michel Bergmann,michel.bergmann@u-bordeaux.fr;Afaf Bouharguane,afaf.bouharguane@u-bordeaux.fr;Angelo Iollo,angelo.iollo@u-bordeaux.fr E-mail:alexis.tardieu@u-bordeaux.fr;michel.bergmann@u-bordeaux.fr;afaf.bouharguane@u-bordeaux.fr;angelo.iollo@u-bordeaux.fr

High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

Michel Bergmann^1,2, Afaf Bouharguane^1,2,3, Angelo Iollo^1,2,3, Alexis Tardieu^1,2,3

1 Centre Inria de l'Université de Bordeaux, Memphis Team, Talence, France;
2 Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Talence, France;
3 Université de Bordeaux, Bordeaux, France

Received:2023-02-25 Revised:2023-09-19 Accepted:2023-11-28 Published:2024-12-20
Contact: Alexis Tardieu,alexis.tardieu@u-bordeaux.fr;Michel Bergmann,michel.bergmann@u-bordeaux.fr;Afaf Bouharguane,afaf.bouharguane@u-bordeaux.fr;Angelo Iollo,angelo.iollo@u-bordeaux.fr E-mail:alexis.tardieu@u-bordeaux.fr;michel.bergmann@u-bordeaux.fr;afaf.bouharguane@u-bordeaux.fr;angelo.iollo@u-bordeaux.fr

摘要/Abstract

摘要： We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.

关键词: Advection-diffusion, Galerkin, Arbitrary high order DERivatives (ADER) approach, Interior Penalty Discontinuous Galerkin (IPDG), High-order schemes, Empirical convergence rates

Abstract: We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.

Key words: Advection-diffusion, Galerkin, Arbitrary high order DERivatives (ADER) approach, Interior Penalty Discontinuous Galerkin (IPDG), High-order schemes, Empirical convergence rates

中图分类号:

65M12
65M60

Michel Bergmann, Afaf Bouharguane, Angelo Iollo, Alexis Tardieu. High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1954-1977.

参考文献

1. Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35(7), 641–669 (2006). https:// doi. org/ 10. 1016/j. compfluid. 2005. 01. 007
2. Baldassari, C.: Modélisation et simulation numérique pour la migration terrestre par équation d’ondes. Theses, Université de Pau et des Pays de l’Adour (2009). https:// theses. hal. scien ce/ tel- 00472 810
3. Bergmann, M., Carlino, M.G., Iollo, A.: Second order ADER scheme for unsteady advection-diffusion on moving overset grids with a compact transmission condition. SIAM J. Sci. Comput. 44(1), 524–553 (2022). https:// doi. org/ 10. 1137/ 21M13 93911
4. Bergmann, M., Carlino, M.G., Iollo, A., Telib, H.: ADER scheme for incompressible Navier-Stokes equations on overset grids with a compact transmission condition. J. Comput. Phys. 467, 111414 (2022). https:// doi. org/ 10. 1016/j. jcp. 2022. 111414
5. Bergmann, M., Fondanèche, A., Iollo, A.: An Eulerian finite-volume approach of fluid-structure interaction problems on quadtree meshes. J. Comput. Phys. 471, 111647 (2022). https:// doi. org/ 10. 1016/j. jcp. 2022. 111647
6. Boscheri, W., Dumbser, M.: Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. J. Comput. Phys. 346, 449– 479 (2017). https:// doi. org/ 10. 1016/j. jcp. 2017. 06. 022
7. Boscheri, W., Loubère, R.: High order accurate direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for non-conservative hyperbolic systems with stiff source terms. Commun. Comput. Phys. 21(1), 271–312 (2017). https:// doi. org/ 10. 4208/ cicp. OA- 2015- 0024
8. Busto, S., Dumbser, M., Escalante, C., Favrie, N., Gavrilyuk, S.: On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. J. Sci. Comput. 87(2), 48 (2021). https:// doi. org/ 10. 1007/ s10915- 021- 01429-8
9. Busto, S., Toro, E.F., Vázquez-Cendón, M.E.: Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations. J. Comput. Phys. 327, 553–575 (2016). https:// doi. org/ 10. 1016/j. jcp. 2016. 09. 043
10. Cockburn, B.: Discontinuous Galerkin methods. ZAMM 83(11), 731–754 (2003). https:// doi. org/ 10. 1002/ zamm. 20031 0088
11. Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000). https:// doi. org/ 10. 1007/ 978-3- 642- 59721-3
12. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)
13. Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.-D.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209–8253 (2008). https:// doi. org/ 10. 1016/j. jcp. 2008. 05. 025
14. 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/ axiom s7030 063
15. Dumbser, M., Munz, C.-D.: Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comput. 27(1/2/3), 215–230 (2006). https:// doi. org/ 10. 1007/ s10915- 005- 9025-0
16. Fambri, F.: Discontinuous Galerkin methods for compressible and incompressible flows on space-time adaptive meshes: toward a novel family of efficient numerical methods for fluid dynamics. Arch. Comput. Methods Eng. 27(1), 199–283 (2020). https:// doi. org/ 10. 1007/ s11831- 018- 09308-6
17. Fambri, F., Dumbser, M.: Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids. Comput. Methods Appl. Mech. Eng. 324, 170–203 (2017). https:// doi. org/ 10. 1016/j. cma. 2017. 06. 003
18. Fambri, F., Dumbser, M., Zanotti, O.: Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations. Comput. Phys. Commun. 220, 297–318 (2017). https:// doi. org/ 10. 1016/j. cpc. 2017. 08. 001
19. Gassner, G., Dumbser, M., Hindenlang, F., Munz, C.-D.: Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors. J. Comput. Phys. 230(11), 4232–4247 (2011). https:// doi. org/ 10. 1016/j. jcp. 2010. 10. 024
20. Hartmann, R.: Numerical analysis of higher order discontinuous Galerkin finite element methods. In: VKI LS 2008-08: CFD-ADIGMA Course on very High Order Discretization Methods, Oct. 13–17 (2008). https:// elib. dlr. de/ 57074/1/ Har08b. pdf
21. Hidalgo, A., Dumbser, M.: ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J. Sci. Comput. 48(1/2/3), 173–189 (2011). https:// doi. org/ 10. 1007/ s10915- 010- 9426-6
22. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675–698 (2009). https:// doi. org/ 10. 1137/ 08072 0255
23. Loubère, R., Dumbser, M., Diot, S.: A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Commun. Comput. Phys. 16(3), 718–763 (2014). https:// doi. org/ 10. 4208/ cicp. 181113. 14031 4a
24. Ricchiuto, M., Abgrall, R.: Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case. J. Comput. Phys. 229(16), 5653–5691 (2010). https:// doi. org/ 10. 1016/j. jcp. 2010. 04. 002
25. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Applied Mathematics. SIAM, Philadelphia (2008) (OCLC: ocn226292048)
26. Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 609–618 (2002). https:// doi. org/ 10. 1023/A: 10151 26814 947
27. Toro, E.F., Montecinos, G.I.: Advection-diffusion-reaction equations: hyperbolization and high-order ADER discretizations. SIAM J. Sci. Comput. 36(5), 2423–2457 (2014). https:// doi. org/ 10. 1137/ 13093 7469
28. Zahran, Y.H.: Central ADER schemes for hyperbolic conservation laws. J. Math. Anal. Appl. 346(1), 120–140 (2008). https:// doi. org/ 10. 1016/j. jmaa. 2008. 05. 032

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High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

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