Asymptotic and Numerical Analysis of Dynamics in a Generalized Free-Interfacial Combustion Model

  • Jun YuEmail author
  • Kewang Chen
  • Laura K. Gross
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


Dynamics of temperature distribution and interfacial front propagation in a generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. Finally, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters.


  1. 1.
    Merzhanov, A.G.: SHS processes: combustion theory and practice. Arch. Combust. 1, 2–48 (1981)Google Scholar
  2. 2.
    Munir, Z.A., Anselmi-Tamburini, U.: Self-propagating exothermic reactions: the synthesis of high-temperature materials by combustion. Mat. Sci. Rep. 3, 277–365 (1989)CrossRefGoogle Scholar
  3. 3.
    Varma, A., Rogachev, A.S., Mukasyan, A.S., Huang, S.: Combustion synthesis of advanced materials: principles and applications. Adv. Chem. Eng. 24, 7–226 (1998)Google Scholar
  4. 4.
    Yang, Y., Gross, L.K., Yu, J.: Comparison study of dynamics in one-sided and two-sided solid-combustion models. SIAM J. Appl. Math. 70(8), 3022–3038 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Matkowsky, B.J., Sivashinsky, G.I.: Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math. 35(93), 465–478 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, K., Gross, L.K., Yu J., Yang, Y.: On a generalized free-interface model of solid combustion. J. Eng. Math. (submitted)Google Scholar
  7. 7.
    Bayliss, A., Matkowsky, B.J.: Two routes to chaos in condensed phase combustion. SIAM J. Appl. Math. 50(2), 437–459 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brailovsky, I., Sivashinsky, G.: Chaotic dynamics in solid fuel combustion. Physica D 65, 191–198 (1993)CrossRefGoogle Scholar
  9. 9.
    Frankel, M.L., Roytburd, V., Sivashinsky, G.: Complex dynamics generated by a sharp interface model of self-propagating high-temperature synthesis. Combust. Theory Model. 2, 1–18 (1998)CrossRefGoogle Scholar
  10. 10.
    Frankel, M.L., Roytburd, V.: Dynamical portrait of a model of thermal instability: cascades, chaos, reversed cascades and infinite period bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Eng. 4(3), 579–593 (1994)CrossRefGoogle Scholar
  11. 11.
    Chen, K.: Mathematical analysis of some partial differential equations with applications. Ph.D. thesis, The University of Vermont (in progress)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA
  2. 2.Departments of Mathematics and Computer ScienceBridgewater State UniversityBridgewaterUSA

Personalised recommendations