Abstract
We present here an explicit finite volume scheme on unstructured meshes adapted to first-order hyperbolic systems under constraints in bounded domains. This scheme is based on the work (Coudière, Vila, Villedieu in C R Acad Sci Paris Sér I Math 331:95–100, 2000, [3]) in the unconstrained case and the splitting strategy of Després, Lagoutière, Seguin (Nonlinearity 24:3055–3081, 2011, [4]). We show that this scheme is stable under a Courant–Friedrichs–Lewy condition (and convergent for problems posed in the whole space), and we illustrate the solution constructed by this scheme on the example of the simplified model of perfect plasticity. From the theoretical point of view, the interaction between the constraint and the boundary of the domain in the model of perfect plasticity is encoded by a nonlinear boundary condition. With this numerical approach, we will show that, even if this scheme uses the underlying linear boundary condition, the results are consistent with the nonlinear model (and in particular with the nonlinear boundary condition).
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References
J.-F. Babadjian, C. Mifsud, Hyperbolic structure for a simplified model of dynamical perfect plasticity. Arch. Ration. Mech. Anal. 223(2), 761–815 (2017)
B. Cockburn, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52(186), 411–435 (1989)
Y. Coudière, J.-P. Vila, P. Villedieu, Convergence d’un schéma volumes finis explicite en temps pour les systèmes hyperboliques linéaires symétriques en domaines bornés. C. R. Acad. Sci. Paris Sér. I Math. 331(1), 95–100 (2000)
B. Després, F. Lagoutière, N. Seguin, Weak solutions to Friedrichs systems with convex constraints. Nonlinearity 24(11), 3055–3081 (2011)
R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, vol. VII (North-Holland, Amsterdam, 2000), pp. 713–1020
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002)
C. Mifsud, Variational and hyperbolic methods applied to constrained mechanical systems. Ph.D. thesis, Université Pierre et Marie Curie (2016)
C. Mifsud, B. Després, N. Seguin, Dissipative formulation of initial boundary value problems for Friedrichs’ systems. Commun. Partial Differ. Equ. 41(1), 51–78 (2016)
J.-P. Vila, P. Villedieu, Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math. 94(3), 573–602 (2003)
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Mifsud, C., Després, B. (2018). A Numerical Approach of Friedrichs’ Systems Under Constraints in Bounded Domains. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_25
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DOI: https://doi.org/10.1007/978-3-319-91548-7_25
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