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

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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.

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References

  1. 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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  4. 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)

  5. 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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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Acknowledgments

L.Tian acknowledges a fellowship from the China Scholarship Council (CSC) giving the opportunity and financial support to study at the University of Twente.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE , Enschede, The Netherlands

    Lulu Tian, J. G. M. Kuerten & J. J. W. Van der Vegt

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026 , Anhui, People’s Republic of China

    Yan Xu

Authors
  1. Lulu Tian
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  2. Yan Xu
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  3. J. G. M. Kuerten
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  4. J. J. W. Van der Vegt
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Corresponding author

Correspondence to Yan Xu.

Additional information

The research of L. Tian was supported by the China Scholarship Council (CSC). Research of J.J.W. van der Vegt was partially supported by the High-end Foreign Experts Recruitment Program (GDW20137100168), while the author was in residence at the University of Science and Technology of China.

Yan Xu received research supported from NSFC Grant Nos. 11371342, 11031007, FANEDD No. 200916, NCET No. 09-0922, Fok Ying Tung Education Foundation No. 131003.

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Tian, L., Xu, Y., Kuerten, J.G.M. et al. A Local Discontinuous Galerkin Method for the Propagation of Phase Transition in Solids and Fluids. J Sci Comput 59, 688–720 (2014). https://doi.org/10.1007/s10915-013-9778-9

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  • DOI: https://doi.org/10.1007/s10915-013-9778-9

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