Top

Journal of Scientific Computing

Published in:

01-06-2014

A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids and Fluids

Authors: Lulu Tian, Yan Xu, J. G. M. Kuerten, J. J. W. Van der Vegt

Published in: Journal of Scientific Computing | Issue 3/2014

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

A local discontinuous Galerkin (LDG) finite element method for the solution of a hyperbolic–elliptic system modeling the propagation of phase transition in solids and fluids is presented. Viscosity and capillarity terms are added to select the physically relevant solution. The \(L^2-\)stability of the LDG method is proven for basis functions of arbitrary polynomial order. In addition, using a priori error analysis, we provide an error estimate for the LDG discretization of the phase transition model when the stress–strain relation is linear, assuming that the solution is sufficiently smooth and the system is hyperbolic. Also, results of a linear stability analysis to determine the time step are presented. To obtain a reference exact solution we solved a Riemann problem for a trilinear strain–stress relation using a kinetic relation to select the unique admissible solution. This exact solution contains both shocks and phase transitions. The LDG method is demonstrated by computing several model problems representing phase transition in solids and in fluids with a Van der Waals equation of state. The results show the convergence properties of the LDG method.

previous article Superconvergence of Jacobi–Gauss-Type Spectral Interpolation

next article Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Abeyaratne, R., Knowles, J.K.: Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids. SIAM J. Appl. Math. 51, 1205–1221 (1991)CrossRefMATHMathSciNet

Abeyaratne, R., Knowles, J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119–154 (1991)CrossRefMATHMathSciNet

Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions: A Continuum Theory. Cambridge University Press, Cambridge (2006)CrossRef

Berezovski, A., Maugin, G.A.: Numerical simulation of phase-transition front propagation in thermoelastic solids. In: Quintela, P., Salgado, P., Bermudez de Castro, A., Gomez, D. (eds.) Numerical Mathematics and Advanced Applications (Proceedings of ENUMATH 2005), pp. 703–711. Springer, Berlin (2006)

Boutin, B., Chalons, C., Lagoutiere, F., LeFloch, P.G.: Convergent and conservative schemes for nonclassical solutions based on kinetic relations. Interfaces Free Boundaries 10, 399–421 (2008)CrossRefMATHMathSciNet

Chalons, C.: Transport-equilibrium schemes for computing nonclassical shocks. Comptes Rendus Mathematique 342, 623–626 (2006)CrossRefMATHMathSciNet

Chalons, C., Coulombel, J.F.: Relaxation approximation of the Euler equations. J. Math. Anal. Appl. 348, 872–893 (2008)CrossRefMATHMathSciNet

Chalons, C., Goatin, P.: Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces Free Boundaries 10, 197–221 (2008)CrossRefMATHMathSciNet

Chalons, C., LeFloch, P.G.: High-order entropy-conservative schemes and kinetic relations for van der Waals fluids. J. Comput. Phys. 168, 184–206 (2001)CrossRefMATHMathSciNet

10.

Chalons, C., LeFloch, P.G.: Computing under-compressive waves with the random choice scheme. Nonclassical shock waves. Interfaces Free Boundaries 5, 129–158 (2003)CrossRefMATHMathSciNet

11.

Chalons, C., Coquel, F., Engel, P., Rohde, C.: Fast relaxation solvers for hyperbolic-elliptic phase transition problems. SIAM J. Sci. Comput. 34, 1753–1776 (2012)CrossRefMathSciNet

12.

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)MATH

13.

Cockburn, B., Gau, H.: A model numerical scheme for the propagation of phase transitions in solids. SIAM J. Sci. Comput. 17, 1092–1121 (1996)CrossRefMATHMathSciNet

14.

Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 141, 2440–2463 (1998)CrossRefMathSciNet

15.

Colombo, R.M., Corli, A.: Sonic and kinetic phase transitions with applications to Chapman–Jouguet deflagrations. Math. Methods Appl. Sci. 27, 843–864 (2004)CrossRefMATHMathSciNet

16.

Coquel, F., Perthame, B.: Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35, 2223–2249 (1998)CrossRefMATHMathSciNet

17.

Haink, J., Rohde, C.: Local discontinuous Galerkin schemes for model problems in phase transition theory. Commun. Comput. Phys. 4, 860–893 (2008)MathSciNet

18.

LeFloch, P.: Propagating phase boundaries: formulation of the problem and existence via the Glimm method. Arch. Ration. Mech. Anal. 123, 153–197 (1993)CrossRefMathSciNet

19.

LeFloch, P.G.: Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, vol. 35. Birkhäuser, Basel (2002)CrossRef

20.

Merkle, C., Rohde, C.: The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques. Math. Model. Numer. Anal. 41, 1089–1123 (2007)CrossRefMATHMathSciNet

21.

Pecenko, A., Van Deurzen, L.G.M., Kuerten, J.G.M., Van der Geld, C.W.M.: Non-isothermal two-phase flow with a diffuse-interface model. Int. J. Multiph. Flow 37, 149–165 (2011)CrossRef

22.

Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)CrossRefMATHMathSciNet

23.

Slemrod, M.: Dynamic phase transitions in a van der Waals fluid. J. Differ. Equ. 52, 1–23 (1984)CrossRefMATHMathSciNet

24.

Truskinovsky, L.: Kinks versus shocks, Shock induced transitions and phase structures in general media. IMA Vol. Math. Appl. 52, 185–229 (1993)CrossRefMathSciNet

25.

Xu, Y., Shu, C.W.: A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Numer. Anal. 46, 1998–2021 (2008)CrossRefMATHMathSciNet

26.

Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)MathSciNet

27.

Xu, Y., Shu, C.W.: Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50, 79–104 (2012)CrossRefMATHMathSciNet

28.

Yan, J., Shu, C.W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)CrossRefMATHMathSciNet

29.

Zhong, X.G., Hou, T.Y., LeFloch, P.G.: Computational methods for propagating phase boundaries. J. Comput. Phys. 124, 192–216 (1996)

Title: A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids and Fluids
Authors: Lulu Tian
Yan Xu
J. G. M. Kuerten
J. J. W. Van der Vegt
Publication date: 01-06-2014
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2014
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-013-9778-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2014

Erratum to: Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

Simulation of an $$n^{+}\hbox {-}n\hbox {-}n^{+}$$ n + - n - n + Diode by Using Globally-Hyperbolically-Closed High-Order Moment Models

Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

Convergence Analysis of a Symmetric Dual-Wind Discontinuous Galerkin Method

High Order Spatial Generalization of 2D and 3D Isotropic Discrete Gradient Operators with Fast Evaluation on GPUs

Premium Partner