Efficient numerical stability analysis of detonation waves in ZND
Authors:
Jeffrey Humpherys and Kevin Zumbrun
Journal:
Quart. Appl. Math. 70 (2012), 685-703
MSC (2000):
Primary 76E99; Secondary 76L05
DOI:
https://doi.org/10.1090/S0033-569X-2012-01276-X
Published electronically:
June 20, 2012
MathSciNet review:
3052085
Full-text PDF Free Access
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Abstract: As described in the classic works of Lee–Stewart and Short–Stewart, the numerical evaluation of linear stability of planar detonation waves is a computationally intensive problem of considerable interest in applications. Reexamining this problem from a modern numerical Evans function point of view, we derive a new algorithm for their stability analysis, related to a much older method of Erpenbeck, that, while equally simple and easy to implement as the standard method introduced by Lee–Stewart, appears to be potentially faster and more stable.
References
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- Benjamin Texier and Kevin Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, Comm. Math. Phys. 302 (2011), no. 1, 1–51. MR 2770009, DOI https://doi.org/10.1007/s00220-010-1175-8
- Kevin Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser Boston, Boston, MA, 2001, pp. 307–516. MR 1842778
- Kevin Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 311–533. With an appendix by Helge Kristian Jenssen and Gregory Lyng. MR 2099037
- Kevin Zumbrun, Stability of detonation profiles in the ZND limit, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 141–182. MR 2781588, DOI https://doi.org/10.1007/s00205-010-0342-6
- Kevin Zumbrun, A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces, Quart. Appl. Math. 68 (2010), no. 3, 557–561. MR 2676976, DOI https://doi.org/10.1090/S0033-569X-2010-01209-1
- K. Zumbrun. Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization. preprint, 2009.
- K. Zumbrun. High-frequency asymptotics and stability of ZND detonations in the high-overdrive and small-heat release limits. to appear, Arch. Ration. Mech. Anal.
References
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- G. Abouseif and T.Y. Toong. Theory of unstable one-dimensional detonations, Combust. Flame 45 (1982) 67–94.
- S. Alinhac. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Differential Equations, 14(2):173–230, 1989. MR 976971 (90h:35147b)
- B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun. Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231–249. MR 2403609 (2009c:35286)
- B. Barker, J. Humpherys, and K. Zumbrun. One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, J. Differential Equations, 249(9):2175–2213, 2010. MR 2718655 (2011j:76088)
- B. Barker, J. Humpherys, and K. Zumbrun. STABLAB: A MATLAB-based numerical library for Evans function computation. Available at: http://impact.byu.edu/stablab/.
- B. Barker, O. Lafitte, and K. Zumbrun. Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity. Acta Math. Sci. Ser. B Engl. Ed., 30(2):447–498, 2010. MR 2656550 (2011c:35391)
- B. Barker, M. Lewicka, and K. Zumbrun. Existence and stability of viscoelastic shock profiles, Arch. Ration. Mech. Anal. 200 (2011), 491–532. MR 2787588
- B. Barker, S. Shaw, S. Yarahmadian, and K. Zumbrun. Existence and stability of steady states of a reaction convection diffusion equation modeling microtubule formation, to appear, J. Math. Biology.
- B. Barker and K. Zumbrun. Numerical stability of ZND detonations for Majda’s model, preprint (2010).
- B. Barker and K. Zumbrun. Numerical stability of ZND detonations, in preparation.
- A. Bourlioux, A. Majda, and V. Roytburd. Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51 (1991) 303–343. MR 1095022 (91m:76090)
- T. J. Bridges, G. Derks, and G. Gottwald. Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D, 172(1-4):190–216, 2002. MR 1946769 (2004i:37148)
- L. Q. Brin. Numerical testing of the stability of viscous shock waves. Ph.D. thesis, Indiana University, Bloomington, 1998. MR 2697543
- L. Q. Brin. Numerical testing of the stability of viscous shock waves. Math. Comp., 70(235):1071–1088, 2001. MR 1710652 (2001j:65118)
- L. Q. Brin and K. Zumbrun. Analytically varying eigenvectors and the stability of viscous shock waves. Mat. Contemp., 22:19–32, 2002. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). MR 1965784 (2004c:15012)
- J. Buckmaster and J. Neves. One-dimensional detonation stability: the spectrum for infinite activation energy. Phys. Fluids 31 (1988) no. 12, 3572–3576.
- N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun. Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers. Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537–587. MR 2505363
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- J. J. Erpenbeck. Stability of steady-state equilibrium detonations. Physics of Fluids, 5(5):604–614, 1962.
- J. J. Erpenbeck. Stability of idealized one-reaction detonations, Phys. Fluids, 7 (1964).
- W. Fickett and W. C. Davis. Detonation: Theory and Experiment. Dover Publications, 2000.
- W. Fickett and W. Wood. Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9 (1966) 903–916.
- G. R. Fowles. On the evolutionary condition for stationary plane waves in inert and reactive substances. In Shock induced transitions and phase structures in general media, volume 52 of IMA Vol. Math. Appl., pages 93–110. Springer, New York, 1993. MR 1240334 (94g:76043)
- R. A. Gardner and K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math., 51(7):797–855, 1998. MR 1617251 (99c:35152)
- J. Humpherys, O. Lafitte, and K. Zumbrun. Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Comm. Math. Phys., 293(1):1–36, 2010. MR 2563797 (2010i:76085)
- J. Humpherys, G. Lyng, and K. Zumbrun. Spectral stability of ideal-gas shock layers. Arch. Ration. Mech. Anal., 194(3):1029–1079, 2009. MR 2563632 (2011b:35329)
- J. Humpherys and K. Zumbrun. An efficient shooting algorithm for Evans function calculations in large systems. Phys. D, 220(2):116–126, 2006. MR 2253406 (2007e:35006)
- H. K. Jenssen, G. Lyng, and M. Williams. Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion. Indiana Univ. Math. J., 54(1):1–64, 2005. MR 2126075 (2006a:35249)
- T. Kapitula and B. Sandstede. Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Phys. D, 124(1-3):58–103, 1998. MR 1662530 (99h:35199)
- T. Kato, Perturbation theory for linear operators. Springer–Verlag, Berlin, Heidelberg, 1976. MR 0407617 (53:11389)
- A.R. Kasimov and D.S. Stewart. Spinning instability of gaseous detonations. J. Fluid Mech. 466 (2002), 179–203. MR 1925152 (2003g:76093)
- O. Lafitte, M. Williams, and K. Zumbrun. High-frequency asymptotics and multi-d instability of ZND detonations. In preparation.
- H. I. Lee and D. S. Stewart. Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech., 216:103–132, 1990.
- G. Lyng and K. Zumbrun. One-dimensional stability of viscous strong detonation waves. Arch. Ration. Mech. Anal., 173(2):213–277, 2004. MR 2081031 (2005f:76061)
- G. Lyng and K. Zumbrun. A stability index for detonation waves in Majda’s model for reacting flow. Phys. D, 194(1-2):1–29, 2004. MR 2075662 (2005d:35134)
- A. Majda. The stability of multidimensional shock fronts. Mem. Amer. Math. Soc., 41(275):iv+95, 1983. MR 683422 (84e:35100)
- C. Mascia and K. Zumbrun. Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Ration. Mech. Anal., 172(1):93–131, 2004. MR 2048568 (2005d:35166)
- R. L. Pego and M. I. Weinstein. Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A, 340(1656):47–94, 1992. MR 1177566 (93g:35115)
- B. Sandstede. Stability of travelling waves. In Handbook of dynamical systems, Vol. 2, pages 983–1055. North-Holland, Amsterdam, 2002. MR 1901069 (2004e:37121)
- B. Sandstede. private communication, 1999.
- D. Serre. Systems of conservation laws. 1. Cambridge University Press, Cambridge, 1999. Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon. MR 1707279 (2000g:35142)
- D. Serre. Systems of conservation laws. 2. Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon. MR 1775057 (2001c:35146)
- M. Short and D. S. Stewart. The multi-dimensional stability of weak-heat-release detonations. J. Fluid Mech., 382:109–135, 1999. MR 1680620 (99m:80017)
- D. S. Stewart and A. R. Kasimov, On the State of Detonation Stability Theory and Its Application to Propulsion, Journal of Propulsion and Power, 22:6, 1230-1244, 2006.
- J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, second edition, 1994. MR 1301779 (95g:35002)
- B. Texier and K. Zumbrun. Galloping instability of viscous shock waves. Phys. D, 237(10-12):1553–1601, 2008. MR 2454606 (2009h:35271)
- B. Texier and K. Zumbrun. Hopf bifurcation of viscous shock waves in compressible gas dynamics and MHD. Arch. Ration. Mech. Anal., 190(1):107–140, 2008. MR 2434902 (2009g:35239)
- B. Texier and K. Zumbrun. Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, Comm. Math. Physics, 302:1–51, 2011. MR 2770009
- K. Zumbrun. Multidimensional stability of planar viscous shock waves. In Advances in the theory of shock waves, volume 47 of Progr. Nonlinear Differential Equations Appl., pages 307–516. Birkhäuser Boston, Boston, MA, 2001. MR 1842778 (2002k:35200)
- K. Zumbrun. Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In Handbook of mathematical fluid dynamics. Vol. III, pages 311–533. North-Holland, Amsterdam, 2004. With an appendix by Helge Kristian Jenssen and Gregory Lyng. MR 2099037 (2006f:35229)
- K. Zumbrun. Stability of detonation profiles in the ZND limit. Arch. Rational Mech. Anal., 200:141–182, 2011. MR 2781588
- K. Zumbrun. A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces. Quart. Appl. Math., 68:557–561, 2010. MR 2676976 (2011e:65124)
- K. Zumbrun. Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization. preprint, 2009.
- K. Zumbrun. High-frequency asymptotics and stability of ZND detonations in the high-overdrive and small-heat release limits. to appear, Arch. Ration. Mech. Anal.
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Additional Information
Jeffrey Humpherys
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
MR Author ID:
358503
Email:
jeffh@math.byu.edu
Kevin Zumbrun
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47402
MR Author ID:
330192
Email:
kzumbrun@indiana.edu
Received by editor(s):
November 6, 2010
Published electronically:
June 20, 2012
Additional Notes:
This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487, and National Science Fountation CAREER award DMS-0847074. Thanks to Mark Williams for stimulating discussions regarding the numerical literature on stability of ZND detonations.
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Brown University
The copyright for this article reverts to public domain 28 years after publication.