Abstract
We propose a novel framework for model-order reduction of hyperbolic differential equations. The approach combines a relaxation formulation of the hyperbolic equations with a discretization using shifted base functions. Model-order reduction techniques are then applied to the resulting system of coupled ordinary differential equations. On computational examples including in particular the case of shock waves we show the validity of the approach and the performance of the reduced system.
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: The authors thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the financial support through 20021702/GRK2326, 333849990/IRTG-2379, HE5386/19-2,22-1,23-1 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612. Supported also by the German Federal Ministry for Economic Affairs and Energy, in the joint project: “MathEnergy – Mathematical Key Technologies for Evolving Energy Grids”, sub-project: Model Order Reduction (Grant number: 0324019B).
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] A. C. Antoulas, C. A. Beattie, and S. Gugercin, Interpolatory Methods for Model Reduction, Philadelphia, PA, Computational Science & Engineering, Society for Industrial and Applied Mathematics, 2020.10.1137/1.9781611976083Search in Google Scholar
[2] A. C. Antoulas, D. C. Sorensen, and S. Gugercin, “A survey of model reduction methods for large-scale systems,” Contemp. Math., vol. 280, pp. 193–219, 2001. https://doi.org/10.1090/conm/280/04630.Search in Google Scholar
[3] M. Grepl, “Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations,” PhD thesis, Massachussetts Institute of Technology (MIT), Cambridge, USA, 2005.Search in Google Scholar
[4] B. Haasdonk and M. Ohlberger, “Reduced basis method for explicit finite volume approximations of nonlinear conservation laws,” in Proc. 12th International Conference on Hyperbolic Problems: Theory, Numerics, Application, College Park, MD, United States, Citeseer, 2008.10.1090/psapm/067.2/2605256Search in Google Scholar
[5] B. Haasdonk and M. Ohlberger, “Reduced basis method for finite volume approximations of parametrized linear evolution equations,” ESAIM: Math. Model. Numer. Anal., vol. 42, pp. 277–302, 2008.10.1051/m2an:2008001Search in Google Scholar
[6] A. T. Maday, Y. Patera and G. Turinici, “A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations,” J. Sci. Comput., vol. 17, pp. 437–446, 2002. https://doi.org/10.1023/a:1015145924517.10.1023/A:1015145924517Search in Google Scholar
[7] A. Quarteroni, G. Rozza, and A. Manzoni, “Certified reduced basis approximation for parametrized partial differential equations and applications,” J. Math. Ind., vol. 1, pp. 1–49, 2011. https://doi.org/10.1186/2190-5983-1-3.Search in Google Scholar
[8] X. Dai and Y. Maday, “Stable parareal in time method for first-and second-order hyperbolic systems,” SIAM J. Sci. Comput., vol. 35, pp. A52–A78, 2013. https://doi.org/10.1137/110861002.Search in Google Scholar
[9] C. Himpe and M. Ohlberger, “Model reduction for complex hyperbolic networks,” in 2014 European Control Conference (ECC), Strasbourg, France, IEEE, 2014, pp. 2739–2743.10.1109/ECC.2014.6862188Search in Google Scholar
[10] F. Laakmann and P. Petersen, “Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs,” Adv. Comput. Math., vol. 47, pp. 1–32, 2021. https://doi.org/10.1007/s10444-020-09834-7.Search in Google Scholar
[11] K.-S. Moon, A. Szepessy, R. Tempone, and G. Zouraris, Hyperbolic Differential Equations and Adaptive Numerics, Berlin, Heidelberg, Springer Berlin Heidelberg, 2001, pp. 231–280.10.1007/978-3-662-04354-7_5Search in Google Scholar
[12] N. Sarna and S. Grundel, Model Reduction of Time-dependent Hyperbolic Equations Using Collocated Residual Minimisation and Shifted Snapshots, e-prints 2003.06362, arXiv, 2020. cs.NA.Search in Google Scholar
[13] J. Reiss, P. Schulze, J. Sesterhenn, and V. Mehrmann, “The shifted proper orthogonal decomposition: a mode decomposition for multiple transport phenomena,” SIAM J. Sci. Comput., vol. 40, pp. A1322–A1344, 2018. https://doi.org/10.1137/17m1140571.Search in Google Scholar
[14] P. Benner, M. Ohlberger, A. Cohen, and K. Willcox, Model Reduction and Approximation, Philadelphia, PA, Society for Industrial and Applied Mathematics, 2017.10.1137/1.9781611974829Search in Google Scholar
[15] N. Cagniart, Y. Maday, and B. Stamm, “Model order reduction for problems with large convection effects,” in Contributions to Partial Differential Equations and Applications, Cham, Springer International Publishing, 2019, pp. 131–150.10.1007/978-3-319-78325-3_10Search in Google Scholar
[16] V. Ehrlacher, D. Lombardi, O. Mula, and F.-X. Vialard, “Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces,” ESAIM Math. Model. Numer. Anal., vol. 54, 2019.10.1051/m2an/2020013Search in Google Scholar
[17] N. J. Nair and M. Balajewicz, “Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks,” Int. J. Numer. Methods Eng., vol. 117, pp. 1234–1262, 2019. https://doi.org/10.1002/nme.5998.Search in Google Scholar
[18] B. Peherstorfer, “Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling,” SIAM J. Sci. Comput., vol. 42, pp. A2803–A2836, 2020. https://doi.org/10.1137/19m1257275.Search in Google Scholar
[19] T. Taddei, S. Perotto, and A. Quarteroni, “Reduced basis techniques for nonlinear conservation laws,” ESAIM: M2AN, vol. 49, pp. 787–814, 2015. https://doi.org/10.1051/m2an/2014054.Search in Google Scholar
[20] G. Welper, “Interpolation of functions with parameter dependent jumps by transformed snapshots,” SIAM J. Sci. Comput., vol. 39, pp. A1225–A1250, 2017. https://doi.org/10.1137/16m1059904.Search in Google Scholar
[21] S. Bianchini, “Hyperbolic limit of the Jin-Xin relaxation model,” Commun. Pure Appl. Math., vol. 59, pp. 688–753, 2006. https://doi.org/10.1002/cpa.20114.Search in Google Scholar
[22] S. Jin and Z. Xin, “The relaxation schemes for systems of conservation laws in arbitrary space dimensions,” Commun. Pure Appl. Math., vol. 48, pp. 235–276, 1995. https://doi.org/10.1002/cpa.3160480303.Search in Google Scholar
[23] R. Natalini, “Recent mathematical results on hyperbolic relaxation problems,” Quaderno IAC, vol. 7, pp. 128–198, 1999.Search in Google Scholar
[24] C. M. Dafermos, “Hyperbolic conservation laws in continuum physics,” in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 2nd ed., Berlin, Springer-Verlag, 2005.10.1007/3-540-29089-3Search in Google Scholar
[25] U. Pallaske, “Ein verfahren zur ordnungsreduktion mathematischer prozessmodelle,” Chem. Ing. Tech., vol. 59, pp. 604–605, 1987. https://doi.org/10.1002/cite.330590720.Search in Google Scholar
[26] D. Aregba-Driollet and R. Natalini, “Convergence of relaxation schemes for conservation laws,” Int. J. Phytoremediation, vol. 61, pp. 163–193, 1996. https://doi.org/10.1080/00036819608840453.Search in Google Scholar
[27] A. Chalabi, “Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms,” Math. Comput., vol. 68, pp. 955–970, 1999. https://doi.org/10.1090/s0025-5718-99-01089-3.Search in Google Scholar
[28] A. Klar, “Relaxation scheme for a lattice–Boltzmann-type discrete velocity model and numerical Navier–Stokes limit,” J. Comput. Phys., vol. 148, pp. 416–432, 1999. https://doi.org/10.1006/jcph.1998.6123.Search in Google Scholar
[29] H. Liu, J. Wang, and G. Warnecke, “The lip + -stability and error estimates for a relaxation scheme,” SIAM J. Numer. Anal., vol. 38, pp. 1154–1170, 2001.10.1137/S0036142999358949Search in Google Scholar
[30] L. Pareschi and G. Russo, “High order asymptotically strong-stability-preserving methods for hyperbolic systems with stiff relaxation,” in Hyperbolic Problems: Theory, Numerics, Applications, Berlin, Heidelberg, Springer, 2003, pp. 241–251.10.1007/978-3-642-55711-8_21Search in Google Scholar
[31] L. Pareschi and G. Russo, “Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation,” J. Sci. Comput., vol. 25, pp. 129–155, 2005. https://doi.org/10.1007/s10915-004-4636-4.Search in Google Scholar
[32] W.-A. Yong, “An interesting class of partial differential equations,” J. Math. Phys., vol. 49, p. 03350321, 2008. https://doi.org/10.1063/1.2884710.Search in Google Scholar
[33] Z. Chen, “On nonsingularity of circulant matrices,” Lin. Algebra Appl., vol. 612, pp. 162–176, 2021. https://doi.org/10.1016/j.laa.2020.12.010.Search in Google Scholar
[34] S. Boscarino, L. Pareschi, and G. Russo, “Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit,” SIAM J. Sci. Comput., vol. 35, pp. A22–A51, 2013. https://doi.org/10.1137/110842855.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
