Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems

Ma, Zhiting; Huang, Juntao; Yong, Wen-An

doi:10.1090/mcom/4111

Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems
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by Zhiting Ma, Juntao Huang and Wen-An Yong;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4111

Published electronically: June 23, 2025

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Abstract:

This work is concerned with the uniform accuracy of implicit-explicit backward differentiation formulas for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed previously by the third author. We prove the uniform stability and accuracy of a class of implicit-explicit backward differentiation formulas (IMEX-BDF) schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory.

References

Georgios Akrivis and Emmanuil Katsoprinakis, Backward difference formulae: new multipliers and stability properties for parabolic equations, Math. Comp. 85 (2016), no. 301, 2195–2216. MR 3511279, DOI 10.1090/mcom3055
Giacomo Albi, Giacomo Dimarco, and Lorenzo Pareschi, Implicit-explicit multistep methods for hyperbolic systems with multiscale relaxation, SIAM J. Sci. Comput. 42 (2020), no. 4, A2402–A2435. MR 4137043, DOI 10.1137/19M1303290
Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), no. 2-3, 151–167. Special issue on time integration (Amsterdam, 1996). MR 1485812, DOI 10.1016/S0168-9274(97)00056-1
Uri M. Ascher, Steven J. Ruuth, and Brian T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32 (1995), no. 3, 797–823. MR 1335656, DOI 10.1137/0732037
A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000), no. 3, 916–938. MR 1750085, DOI 10.1137/S0036139997332099
P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954), no. 3, 511–525.
Sebastiano Boscarino, Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems, SIAM J. Numer. Anal. 45 (2007), no. 4, 1600–1621. MR 2338401, DOI 10.1137/060656929
J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. Fluids 7 (1964), no. 8, 1243–1247.
Zhenning Cai, Yuwei Fan, and Ruo Li, Globally hyperbolic regularization of Grad’s moment system, Comm. Pure Appl. Math. 67 (2014), no. 3, 464–518. MR 3158573, DOI 10.1002/cpa.21472
Germund Dahlquist, $G$-stability is equivalent to $A$-stability, BIT 18 (1978), no. 4, 384–401. MR 520750, DOI 10.1007/BF01932018
Yana Di, Yuwei Fan, Ruo Li, and Lingchao Zheng, Linear stability of hyperbolic moment models for Boltzmann equation, Numer. Math. Theory Methods Appl. 10 (2017), no. 2, 255–277. MR 3647977, DOI 10.4208/nmtma.2017.s04
Giacomo Dimarco and Lorenzo Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal. 51 (2013), no. 2, 1064–1087. MR 3036998, DOI 10.1137/12087606X
Giacomo Dimarco and Lorenzo Pareschi, Implicit-explicit linear multistep methods for stiff kinetic equations, SIAM J. Numer. Anal. 55 (2017), no. 2, 664–690. MR 3623210, DOI 10.1137/16M1063824
Robert Geroch and Lee Lindblom, Dissipative relativistic fluid theories of divergence type, Phys. Rev. D (3) 41 (1990), no. 6, 1855–1861. MR 1048881, DOI 10.1103/PhysRevD.41.1855
Vincent Giovangigli, Multicomponent flow modeling, Sci. China Math. 55 (2012), no. 2, 285–308. MR 2886540, DOI 10.1007/s11425-011-4346-y
Harold Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331–407. MR 33674, DOI 10.1002/cpa.3160020403
Bruce D Greenshields, JR Bibbins, WS Channing, and HH Miller, A study of traffic capacity, Highway research board proceedings, vol. 14, Washington, DC, 1935, pp. 448–477.
E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR 1111480, DOI 10.1007/978-3-662-09947-6
Bernard Hanouzet and Philippe Huynh, Approximation par relaxation d’un système de Maxwell non linéaire, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 3, 193–198 (French, with English and French summaries). MR 1748307, DOI 10.1016/S0764-4442(00)00129-4
Jan S. Hesthaven, Sigal Gottlieb, and David Gottlieb, Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, 2007. MR 2333926, DOI 10.1017/CBO9780511618352
Jingwei Hu and Ruiwen Shu, On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations, Math. Comp. 90 (2021), no. 328, 641–670. MR 4194158, DOI 10.1090/mcom/3602
Jingwei Hu and Ruiwen Shu, Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation, Math. Comp. 94 (2025), no. 351, 209–240. MR 4807808, DOI 10.1090/mcom/3967
Willem Hundsdorfer and Steven J. Ruuth, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys. 225 (2007), no. 2, 2016–2042. MR 2349693, DOI 10.1016/j.jcp.2007.03.003
Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, DOI 10.1002/cpa.3160480303
Christopher A. Kennedy and Mark H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44 (2003), no. 1-2, 139–181. MR 1951292, DOI 10.1016/S0168-9274(02)00138-1
C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021–1065. MR 1392419, DOI 10.1007/BF02179552
Zhiting Ma and Wen-An Yong, Nonrelativistic limit of the Euler-$\mathrm {HMP}_N$ approximation models arising in radiation hydrodynamics, Math. Methods Appl. Sci. 46 (2023), no. 13, 13741–13780. MR 4631204, DOI 10.1002/mma.9288
Dimitri Mihalas and Barbara Weibel Mihalas, Foundations of radiation hydrodynamics, Oxford University Press, New York, 1984. MR 781346
Olavi Nevanlinna and F. Odeh, Multiplier techniques for linear multistep methods, Numer. Funct. Anal. Optim. 3 (1981), no. 4, 377–423. MR 636736, DOI 10.1080/01630568108816097
Lorenzo Pareschi and Giovanni Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25 (2005), no. 1-2, 129–155. MR 2231946, DOI 10.1007/s10915-004-4636-4
Tadeusz Płatkowski and Reinhard Illner, Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory, SIAM Rev. 30 (1988), no. 2, 213–255. MR 941111, DOI 10.1137/1030045
G. C. Pomraning, The Equations of Radiation Hydrodynamics, Courier Corporation, 2005.
Wen-An Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations 155 (1999), no. 1, 89–132. MR 1693210, DOI 10.1006/jdeq.1998.3584
W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in the Theory of Shock Waves, 2001, pp. 259–305.
W.-A. Yong, An interesting class of partial differential equations, J. Math. Phys. 49 (2008), no. 3, 033503.
Yanni Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225–279. MR 1738119, DOI 10.1007/s002050050188
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B: Methodological 36 (2002), no. 3, 275–290.
Weifeng Zhao, Wen-An Yong, and Li-Shi Luo, Stability analysis of a class of globally hyperbolic moment system, Commun. Math. Sci. 15 (2017), no. 3, 609–633. MR 3620597, DOI 10.4310/CMS.2017.v15.n3.a3

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Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems
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Abstract: