Abstract
We present a conservative Arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems. The key characteristics of the method is that it preserves all the convex invariants of the hyperbolic system in question. The method is explicit in time, uses continuous finite elements and is first-order accurate in space and high-order in time. The stability of the method is obtained by introducing an artificial viscosity that is unambiguously defined irrespective of the mesh geometry/anisotropy and does not depend on any ad hoc parameter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ainsworth M (2014) Pyramid algorithms for Bernstein-Bézier finite elements of high, nonuniform order in any dimension. SIAM J Sci Comput 36(2):A543–A569
Bangerth W, Hartmann R, Kanschat G (2007) deal.II – a general purpose object oriented finite element library. ACM Trans Math Softw 33(4):Art. 24, 27 (27 pp)
Caramana E, Shashkov M, Whalen P (1998) Formulations of artificial viscosity for multi-dimensional shock wave computations. J Comput Phys 144(1):70–97
Chen G-Q (2005) Euler equations and related hyperbolic conservation laws. In: Evolutionary equations, Vol. II, Handbook of Differential Equations. Elsevier/North-Holland, Amsterdam, pp 1–104
Chueh KN, Conley CC, Smoller JA (1977) Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ Math J 26(2):373–392
Dafermos CM (2000) Hyperbolic conservation laws in continuum physics, vol 325. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin
Farhat C, Geuzaine P, Grandmont C (2001) The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. J Comput Phys 174(2):669–694
Ferracina L, Spijker MN (2005) An extension and analysis of the Shu-Osher representation of Runge-Kutta methods. Math Comp 74(249):201–219
Frid H (2001) Maps of convex sets and invariant regions for finite-difference systems of conservation laws. Arch Ration Mech Anal 160(3):245–269
Gottlieb S, Ketcheson DI, Shu C-W (2009) High order strong stability preserving time discretizations. J Sci Comput 38(3):251–289
Guermond J-L, Popov B (2016) Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations. J Comput Phys 321:908–926
Guermond J-L, Popov B (2016) Invariant domains and first-order continuous finite element approximation for hyperbolic systems. SIAM J Numer Anal 54(4):2466–2489
Guermond J-L, Popov B, Saavedra L, Yang Y (2017) Invariant domains preserving ALE approximation of hyperbolic systems with continuous finite elements. SIAM J Sci Comput 39(2):A385–A414. arXiv:1603.01184 [math.NA].
Higueras I (2005) Representations of Runge-Kutta methods and strong stability preserving methods. SIAM J Numer Anal 43(3):924–948
Hoff D (1985) Invariant regions for systems of conservation laws. Trans Am Math Soc 289(2):591–610
Lai M-J, Schumaker LL (2007) Spline functions on triangulations, vol 110. Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge
Loubère R, Maire P-H, Shashkov M, Breil J, Galera S (2010) ReALE: a reconnection-based arbitrary-Lagrangian-Eulerian method. J Comput Phys 229(12):4724–4761
Pironneau O (1981/82) On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer Math 38(3):309–332
Pironneau O (1989) Finite element methods for fluids. Wiley, Chichester (translated from the French)
Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics: a practical introduction, 3rd edn. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Guermond, JL., Popov, B., Saavedra, L., Yang, Y. (2019). Arbitrary Lagrangian-Eulerian Finite Element Method Preserving Convex Invariants of Hyperbolic Systems. In: Chetverushkin, B., Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Periaux, J., Pironneau, O. (eds) Contributions to Partial Differential Equations and Applications. Computational Methods in Applied Sciences, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-78325-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-78325-3_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78324-6
Online ISBN: 978-3-319-78325-3
eBook Packages: EngineeringEngineering (R0)