High order simulations are necessary in order to capture fine details in resolving supersonic reactive flows. However, high Mach number compressible flows feature sharp gradients and discontinuities, which present a challenge to successful simulations using high order methods. Spectral methods have proven a powerful tool in simulation of incompressible turbulent flows, and recent advances allow the application of spectral methods to compressible reactive flows. We review the recent advances in the theory and application of spectral methods which allow stable computations of discontinuous phenomena, and the recovery of high order information via postprocessing, and present applications of high Mach number reactive flows.

David Gottlieb ^{1};
Sigal Gottlieb ^{2}

@article{CRMECA_2005__333_1_3_0, author = {David Gottlieb and Sigal Gottlieb}, title = {Spectral methods for compressible reactive flows}, journal = {Comptes Rendus. M\'ecanique}, pages = {3--16}, publisher = {Elsevier}, volume = {333}, number = {1}, year = {2005}, doi = {10.1016/j.crme.2004.09.013}, language = {en}, }

David Gottlieb; Sigal Gottlieb. Spectral methods for compressible reactive flows. Comptes Rendus. Mécanique, Volume 333 (2005) no. 1, pp. 3-16. doi : 10.1016/j.crme.2004.09.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.09.013/

[1] A.N. Aleshin, E.I. Chebotareva, V.V. Krivets, E.V. Lazareva, S.V. Sergeev, S.N. Titov, S.G. Zaytsev, Investigation of evolution of interface after its interaction with shock waves, Report No. 06-96, LANL

[2] Padé approximants method and its applications to mechanics (J. Ehlers; K. Hepp; H.A. Weidenmüller; J. Zittartz, eds.), Lecture Notes in Phys., Springer-Verlag, New York, 1976

[3] W.S. Don, B. Costa, PseudoPack 2000: Numerical Library for Pseudospectral Differentiation

[4] W.S. Don, D. Gottlieb, High order methods for complicated flows interacting with shock waves, AIAA 97-0538

[5] W.S. Don, D. Gottlieb, J.H. Jung, A multi-domain spectral method for supersonic reactive flows, J. Comput. Phys., submitted for publication

[6] W.S. Don, D. Gottlieb, J.H. Jung, Multi-domain spectral method approach to supersonic combustion of recessed cavity flame-holders, JANNAF 38th Combustion, 26th Airbreathing Propulsion, 20th Propulsion Systems Hazards, and 2nd Modeling and Simulation Subcommittees Joint Meeting, held in Destin, FL 8–12 April 2002

[7] W.S. Don, D. Gottlieb, C.W. Shu, High order numerical methods for the two dimensional Richtmyer–Meshkov Instability, Part I, in: Conference Proceeding for the International Workshop for the Physics of Compressible Turbulence Mixing, Laser and Particle Beams, in press

[8] Numerical Analysis of Spectral Methods: Theory and Applications, CBMS Conf. Ser. in Appl. Math., vol. 26, SIAM, 1977

[9] Introduction: theory and applications of spectral methods (R. Voigt; D. Gottlieb; M.Y. Hussaini, eds.), Spectral Methods for Partial Differential Equations, SIAM, Philadelphia, 1984, pp. 1-54

[10] On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function, J. Comput. Appl. Math., Volume 43 (1992), pp. 81-98

[11] Resolution properties of the Fourier method for discontinuous waves, Comput. Methods Appl. Mech. Engrg., Volume 116 (1994), pp. 27-37

[12] On the Gibbs phenomenon III: recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise in a sub-interval from a spectral partial sum of a piecewise analytic function, SIAM Numer. Anal., Volume 33 (1996), pp. 280-290

[13] On the Gibbs phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comput., Volume 64 (1995), pp. 1081-1095

[14] On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math., Volume 71 (1995), pp. 511-526

[15] High order methods for resolving waves: number of points per wavelength, SIAM J. Sci. Comput., Volume 15 (2000) no. 4, pp. 417-439

[16] Comparison of accurate methods for the integration of hyperbolic equations, Tellus, Volume XXIV (1972) no. 3

[17] Instability of the interface of two gases accelerated by a shock wave, Izv. Akad. Nauk. SSSR Mekh. Zhidk. Gaza, Volume 4 (1969), pp. 101-104

[18] Mémoire sur les développements en fractions continues de la fonction exponentielle puvant servir d'introduction à la théorie des fractions continues algébriques, Ann. Fac. Sci. École Norm. Sup., Volume 16 (1899), pp. 395-436

[19] Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., Volume 83 (1989) no. 1, pp. 32-78

[20] The Gibbs phenomenon and its resolution, SIAM Rev., Volume 39 (1997), pp. 644-668

[21] D. Gottlieb, C.W. Shu, General theory for the resolution of the Gibbs phenomenon, Accademia Nazionale Dei Lincey, ATTI Dei Convegni Lincey 147, 1998, pp. 39–48

[22] B.D. Shizgal, J.H. Jung, On the resolution of the Gibbs phenomenon, in press

[23] L. Emmel, S.M. Kaber, Y. Maday, Padé–Jacobi filtering for spectral approximations of discontinuous solutions, Numerical Algorithms (2002) in press

[24] T.A. Driscoll, B. Fornberg, Padé algorithm for the Gibbs phenomenon, Numerical Algorithms (2000) submitted for publication

[25] Rational trigonometric approximations using Fourier series partial sums, J. Sci. Comput., Volume 10 (1995) no. 3, pp. 325-356

[26] W.S. Don, O. Kaber, M.S. Min, in press

[27] Spectral methods for discontinuous problems (N.W. Morton; M.J. Baines, eds.), Numerical Methods for Fluid Dynamics II, Oxford University Press, 1986, pp. 128-153

[28] Accuracy and resolution in the computation of solutions of linear and nonlinear equations, Recent Advances in Numerical Analysis, Proc. Symp., Mathematical Research Center, University of Wisconsin, Academic Press, 1978, pp. 107-117

[29] Convergence of spectral methods for nonlinear conservation laws, SINUM, Volume 26 (1989), pp. 30-44

[30] Shock capturing by the spectral viscosity method, Proceedings of ICOSAHOM 89, Elsevier Science, North-Holland, IMACS, 1989

[31] Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., Volume 30 (1993), pp. 321-342

[32] H. Ma, Chebyshev–Legendre super spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., submitted for publication

[33] M. Carpenter, D. Gottlieb, C.W. Shu, On the conservation and convergence to weak solutions of global schemes, J. Sci. Comput., in press

[34] A note on the accuracy of spectral method applied to nonlinear conservation laws, J. Sci. Comput., Volume 10 (1995), pp. 357-369

[35] S. Gottlieb, D. Gottlieb, C.W. Shu, in press

[36] SISC, 2 (1981), p. 296

[37] Spectral methods for two dimensional flows, Spectral Methods for PDEs, SIAM, Philadelphia, 1984

[38] Numerical study of pseudospectral methods in shock wave applications, J. Comput. Phys., Volume 110 (1994), pp. 103-111

[39] Numerical simulation of reactive flow, Part I: resolution, J. Comput. Phys, Volume 122 (1995), pp. 244-265

[40] Spectral simulations of supersonic reactive flows, SIAM J. Numer. Anal., Volume 35 (1998) no. 6, pp. 2370-2384

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