A Framework for Late-Time/Stiff Relaxation Asymptotics
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 f
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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 latetime 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. Keywords Hyperbolic system · Late-time · Stiff relaxation · Finite volume method · Asymptotic preserving
1 Introduction This short presentation is based on the joint work [3] in collaboration with C. Berthon and R. Turpault. We are interested in hyperbolic models arising in continuum physics and, especially, describing complex multi-fluid flows involving several time-scales. The partial differential equations under consideration are nonlinear hyperbolic systems of balance laws with stiff relaxation sources. We investigate here the late-time behavior of entropy solutions. Precisely, our objective is to derive the relevant effective system—which turns out to be of parabolic type—and to investigate the role of a convex entropy associated with the given system of balance laws. As we show it, many examples from continuous physics fall into our framework. In addition, we investigate here the discretization of such problems, and we propose a new finite volume scheme which preserves the late-time asymptotic regime identified in the first part of this paper. P.G. LeFloch () Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France e-mail: [email protected] url: philippelefloch.org C.A. de Moura, C.S. Kubrusly (eds.), The Courant–Friedrichs–Lewy (CFL) Condition, DOI 10.1007/978-0-8176-8394-8_8, © Springer Science+Business Media New York 2013
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An outline of this paper follows. In Sect. 2, we present a formal derivation of the effective equations associated with our problem. In Sect. 3, we demonstrate that many examples of continuous physics are covered by our theory. Finally, in Sect. 4, we are in a position to present the new discretization method and state several properties of interest.
2 Late-Time/Stiff Relaxation Framework 2.1 Hyperbolic Systems of Balance Laws We consider systems of partial differential equations of the form ε ∂t U + ∂x F (U ) = −
R(U ) , ε
U = U (t, x) ∈ Ω ⊂ RN ,
(1)
where t > 0, x ∈ R denote the (time and space) independent variables. We make the following standard assumptions. The flux F : Ω → RN is defined on a convex and open subset Ω. The first-order part of (1) is a hyperbolic system, that is, the matrixvalued field A(U ) := DU F (U
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