Japan Journal of Industrial and Applied Mathematics

, Volume 35, Issue 3, pp 1213–1244

Structure-preserving finite difference schemes for a semilinear thermoelastic system with second order time derivative

Original Paper Area 2

Abstract

In this article we study a semilinear thermoelastic system which consists of heat equation and beam equation with second-order time derivative. Although in Yoshikawa (ZAMM 95(12):1393–1410, 2015; IMA J Numer Anal 37:477–504, 2017) the second author proposed the structure-preserving finite difference scheme for the system transformed to first order system with respect to time variable, here we propose structure-preserving finite difference schemes for the original system with second-order time derivative. The main purpose of this article is to give mathematical analysis for the scheme such as existence of solution and error estimate.

Keywords

Finite difference method Discrete variational derivative method Semilinear thermoelastic system Error estimate

65M06 74S20

References

1. 1.
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)
2. 2.
Falk, F., Laedke, E.W., Spatschek, K.H.: Stability of solitary-wave pulses in shape-memory alloys. Phys. Rev. B 36, 3031–3041 (1987)
3. 3.
Friedman, A., Sprekels, J.: Steady states of austenitic–martensitic domains in the Ginzburg–Landau theory of shape memory alloys. Contin. Mech. Thermodyn. 2, 199–213 (1990)
4. 4.
Furihata, D.: Finite-difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134, 37–57 (2001)
5. 5.
Furihata, D., Matsuo, M.: Discrete Variational Derivative Method. CRC Press/Taylor & Francis, Numerical Analysis and Scientific Computing series (2010)
6. 6.
Garcke, H.: Travelling wave solutions as dynamic phase transitions in shape memory alloys. J. Differ. Equ. 121, 203–231 (1995)
7. 7.
Hoffmann, K.-H., Zochowski, A.: Analysis of the thermoelastic model of a plate with nonlinear shape memory alloy reinforcements. Math. Methods Appl. Sci. 15, 631–645 (1992)
8. 8.
Hoffmann, K.-H., Zou, Jun: Finite element approximations of Landau-Ginzburg’s equation model for structural phase transitions in shape memory alloys. RAIRO Model. Math. Anal. Numer. 29, 629–655 (1995)
9. 9.
Jiang, S., Racke, R.: Evolution Equations in Thermoelasticity, Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 112. Chapman & Hall/CRC, Boca Raton, FL (2000)Google Scholar
10. 10.
Niezgodka, M., Sprekels, J.: Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys. Numer. Math. 58, 759–778 (1991)
11. 11.
Shimura, K.: Master thesis, Oita University (in preparation) Google Scholar
12. 12.
Sprekels, J., Zheng, Songmu: Global solutions to the equations of a Ginzburg–Landau theory for structural phase transitions in shape memory alloys. Phys. D 39, 59–76 (1989)
13. 13.
Sprekels, J., Zheng, Songmu, Zhu, Peicheng: Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys. SIAM J. Math. Anal. 29, 69–84 (1998)
14. 14.
Suzuki, T., Yoshikawa, S.: Stability of the steady state for the Falk model system of shape memory alloys. Math. Methods Appl. Sci. 30, 2233–2245 (2007)
15. 15.
Suzuki, T., Tasaki, S.: Stationary solutions to the Falk system on shape memory alloys. Math. Methods Appl. Sci. 33, 994–1011 (2010)
16. 16.
Yoshikawa, S.: Weak solutions for the Falk model system of shape memory alloys in energy class. Math. Methods Appl. Sci. 28, 1423–1443 (2005)
17. 17.
Yoshikawa, S.: A conservative finite difference scheme for the Falk model system of shape memory alloys. ZAMM Z. Angew. Math. Mech. 95(12), 1393–1410 (2015)
18. 18.
Yoshikawa, S.: Energy method for structure-preserving finite difference schemes and some properties of difference quotient. J. Comput. Appl. Math. 311, 394–413 (2017)
19. 19.
Yoshikawa, S.: An error estimate for structure-preserving finite difference scheme for the Falk model system of shape memory alloys. IMA J. Numer. Anal. 37, 477–504 (2017) 