Abstract
We consider solutions to nonlinear hyperbolic systems of balance laws with stiff relaxation and formally derive a parabolic-type effective system describing the late-time asymptotics of these solutions. We show that many examples from continuous physics fall into our framework, including the Euler equations with (possibly nonlinear) friction. We then turn our attention to the discretization of these stiff problems and introduce a new finite volume scheme which preserves the late-time asymptotic regime. Importantly, our scheme requires only the classical CFL (Courant–Friedrichs–Lewy) condition associated with the hyperbolic system under consideration, rather than the more restrictive, parabolic-type stability condition.
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References
Berthon, C., Turpault, R.: Asymptotic preserving HLL schemes. Numer. Methods Partial Differ. Equ. doi:10.1002/num.20586
Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions. J. Sci. Comput. 31, 347–389 (2007)
Berthon, C., LeFloch, P.G., Turpault, R.: Late-time relaxation limits of nonlinear hyperbolic systems. A general framework. Math. Comput. (2012)
Bouchut, F., Ounaissa, H., Perthame, B.: Upwinding of the source term at interfaces for Euler equations with high friction. J. Comput. Math. Appl. 53, 361–375 (2007)
Buet, C., Cordier, S.: An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math. 108, 199–221 (2007)
Buet, C., Després, B.: Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215, 717–740 (2006)
Chen, G.Q., Levermore, C.D., Liu, T.P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1995)
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)
Donatelli, D., Marcati, P.: Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems. Trans. Am. Math. Soc. 356, 2093–2121 (2004)
Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983)
Jin, S., Xin, Z.: The relaxation scheme for systems of conservation laws in arbitrary space dimension. Commun. Pure Appl. Math. 45, 235–276 (1995)
Marcati, P.: Approximate solutions to conservation laws via convective parabolic equations. Commun. Partial Differ. Equ. 13, 321–344 (1988)
Marcati, P., Milani, A.: The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84, 129–146 (1990)
Marcati, P., Rubino, B.: Hyperbolic to parabolic relaxation theory for quasilinear first order systems. J. Differ. Equ. 162, 359–399 (2000)
Acknowledgements
The author was partially supported by the Agence Nationale de la Recherche (ANR) through the grant 06-2-134423, and by the Centre National de la Recherche Scientifique (CNRS).
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LeFloch, P.G. (2013). A Framework for Late-Time/Stiff Relaxation Asymptotics. In: de Moura, C., Kubrusly, C. (eds) The Courant–Friedrichs–Lewy (CFL) Condition. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8394-8_8
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DOI: https://doi.org/10.1007/978-0-8176-8394-8_8
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