G. D. Lyng

G. D. Lyng

Applied mathematician, data scientist, aspiring coffee snob, recovering IU basketball fan, wannabe ski bum.

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publications

Low frequency stability of planar multi-D detonations

Published in Oberwolfach Reports, 2004

Recommended citation: H. Kristian Jenssen, Gregory Lyng, & Mark Williams, Low frequency stability of planar multi-D detonations, Oberwolfach Reports, Volume 1, Issue 2, 2004, report No. 18/2004, 927–928.

Recommended citation: H. Kristian Jenssen, Gregory Lyng, & Mark Williams, Low frequency stability of planar multi-D detonations, Oberwolfach Reports, Volume 1, Issue 2, 2004, report No. 18/2004, 927–928.

A stability index for detonation waves in Majda’s model for Reacting flow

Published in Physica D, 2004

Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [SIAM J. Math. Anal. 32 (2001) 929; Commun. Pure Appl. Math. 51 (7) (1998) 797], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier–Stokes equations of reacting flow in [G. Lyng, One dimensional stability of detonation waves, Doctoral Thesis, Indiana University, 2002; G. Lyng, K. Zumbrun, Stability of detonation waves, Preprint, 2003]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [SIAM J. Sci. Statist. Comput. 7u (1986) 1059] and analytical results of [Commun. Math. Phys. 204 (3) (1999) 551; Commun. Math. Phys. 202 (3) (1999) 547] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, “bump-type” ignition function) deflagration profiles, as discussed in [SIAM J. Math. Anal. 24 (1993) 968; SIAM J. Appl. Math. 55 (1995) 175] for the full equations.

Recommended citation: Gregory Lyng & Kevin Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow. Physica D, 194 (2004): 1–29. https://dx.doi.org/10.1016/j.physd.2004.01.036

One-Dimensional Stability of Viscous Strong Detonation Waves

Published in Archive for Rational Mechanics and Analysis, 2004

Building on Evans-function techniques developed to study the stability of viscous shocks, we examine the stability of strong-detonation-wave solutions of the Navier-Stokes equations for reacting gas. The primary result, following [1, 17], is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the Zeldovich-von Neumann-Döring limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided that the underlying shock is stable. Finally, for completeness, we include the calculation of the stability index for a viscous shock solution of the Navier-Stokes equations for a nonreacting gas.

Recommended citation: Gregory Lyng & Kevin Zumbrun, One-dimensional stability of viscous strong detonation waves. Archive for Rational Mechanics and Analysis, 173 (2004): 213–277. https://dx.doi.org/10.1007/s00205-004-0317-6

Equivalence of low frequency stability conditions for multidimensional detonations in three models of combustion

Published in Indiana University Mathematics Journal , 2004

We use the classical normal mode approach of hydrodynamic stability theory to define stability determinants (Evans functions) for multidimensional strong detonations in three commonly studied models of combustion: the full reactive Navier-Stokes ($\RNS$) model, and the simpler Zeldovich-von Neumann-D\"oring ($\ZND$) and Chapman-Jouguet (CJ) models. The determinants are functions of frequencies $(\lambda,\eta)$, where $\lambda$ is a complex variable dual to the time variable, and $\eta\in\bR^{d-1}$ is dual to the transverse spatial variables. The zeros of these determinants in $\Re\lambda>0$ correspond to perturbations that grow exponentially with time.

Recommended citation: Helge Kristian Jenssen, Gregory Lyng, Mark Williams, Equivalence of low frequency stability conditions for multidimensional detonations in three models of combustion,   Indiana Univ. Math. J.  54 (2005),  1-64. https://dx.doi.org/10.1512/iumj.2005.54.2685

Evaluation of the Lopatinski condition for gas dynamics

Published in Handbook of Mathematical Fluid Dynamics III, 2004

The purpose of this appendix is to calculate the Lopatinski determinant, or “stability function,” for the Euler equations of compressible gas dynamics. We describe two approaches to this problem. In the first we use a change of variables to simplify the computation. In the second method we take advantage of the Galilean invariance of the Euler equations and exploit a relationship between the eigenvectors of a particular pair of matrices. As a preliminary step for the first technique, we discuss how the Lopatinski determinant behaves under a change of coordinates.

Recommended citation: H. Kristian Jenssen & Gregory Lyng, Evaluation of the Lopatinski condition for gas dynamics, appendix to K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier–Stokes equations, Handbook of Mathematical Fluid Dynamics III, S. Friedlander & D. Serre eds., North-Holland, Amsterdam, 2004.

Equivalence of low frequency stability conditions for multidimensional detonations in three models of combustion

Published in Indiana University Mathematics Journal , 2005

We use the classical normal mode approach of hydrodynamic stability theory to define stability determinants (Evans functions) for multidimensional strong detonations in three commonly studied models of combustion: the full reactive Navier-Stokes (RNS) model, and the simpler Zeldovich-von Neumann-Doering (ZND) and Chapman-Jouguet (CJ) models. The determinants are functions of frequencies $(\lambda,\eta)$, where $\lambda$ is a complex variable dual to the time variable, and $\eta\in\mathbb{R}^{d-1}$ is dual to the transverse spatial variables. The zeros of these determinants in $\Re\lambda>0$ correspond to perturbations that grow exponentially with time.

Recommended citation: Helge Kristian Jenssen, Gregory Lyng, Mark Williams, Equivalence of low frequency stability conditions for multidimensional detonations in three models of combustion,  Indiana Univ. Math. J. 54 (2005),  1-64. https://dx.doi.org/10.1512/iumj.2005.54.2685

The N-soliton of the focusing nonlinear Schrödinger equation for N large

Published in Communications on Pure and Applied Mathematics, 2006

We present a detailed analysis of the solution of the focusing nonlinear Schrödinger equation with initial condition $\psi(x, 0) = N \mathrm{sech}(x)$ in the limit $N\to\infty$. We begin by presenting new and more accurate numerical reconstructions of the $N$-soliton by inverse scattering (numerical linear algebra) for $N = 5, 10, 20$, and $40$. We then recast the inverse-scattering problem as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this problem in the large-$N$ limit. For those $(x, t)$ where results have been obtained by other authors, we improve the error estimates from $O(N^{−1/3})$ to $O(N^{−1})$. We also analyze the Fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation. We then study the N-soliton for values of $(x, t)$ where analysis has not been carried out before, and we uncover new phenomena. The main discovery of this paper is the mathematical mechanism for a secondary caustic (phase transition), which turns out to differ from the mechanism that generates the primary caustic. The mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum of the N-soliton. Moreover, these results evidently cannot be recovered from an analysis of an ostensibly similar “condensed-pole” Riemann-Hilbert problem.

Recommended citation: Gregory Lyng & Peter D. Miller, The N-soliton of the focusing nonlinear Schrödinger equation for N large. Communications on Pure and Applied Mathematics, 60 (2007): 951–1026. https://dx.doi.org/10.1002/cpa.20162

Pointwise Green function bounds and the stability of combustion waves

Published in Journal of Differential Equations, 2006

Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier–Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman–Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu–Ying and Tesei–Tan for Majda's model and the reactive Navier–Stokes equations, respectively.

Recommended citation: Gregory Lyng, Mohammadreza Raoofi, Benjamin Texier, & Kevin Zumbrun, Pointwise Green function bounds and the stability of combustion waves. Journal of Differential Equations, 233 (2007): 654–698. https://doi.org/10.1016/j.jde.2006.10.006

Existence and stability of curved multidimensional detonation fronts

Published in Indiana University Mathematics Journal , 2007

The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck. He used a normal mode analysis to define a stability function $V (\lambda, \eta)$, whose zeros in $\Re\lambda > 0$ correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. Later, he was able to prove that for large classes of steady ZND profiles, unstable zeros of $V$ always exist in the high-frequency regime, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well. In this paper we begin a rigorous study of the implications for non-linear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of $V$, one can prove the finite (but arbitrarily long) time existence of slightly curved, non-steady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, $\Delta_{\mathrm{ZND}}((\hat\lambdaλ, \hat\eta)$, which turns out to coincide with the high frequency limit of a scaled version of $V (\lambda, \eta)$ in $\Re\hat\lambda > 0$. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where $\Delta_{\mathrm{ZND}}(\hat\lambda,\hat\eta)$ is bounded away from zero in $\Re\hat\lambda > 0$. We also revisit Erpenbeck's arguments in order to simplify and complete some of the analysis in the proof of his main result.

Recommended citation: Nicola Costanzino, H. Kristian Jenssen, Gregory Lyng, & Mark Williams, Existence and stability of curved multidimensional detonation fronts. Indiana University Mathematics Journal, 56 (2007): 1405–1462. https://dx.doi.org/10.1512/iumj.2007.56.2972

Spectral stability of ideal-gas shock layers

Published in Archive for Rational Mechanics and Analysis, 2008

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real part) zeros outside a uniform ball. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant $\gamma$ in [1.2, 3] and ratio $\nu/\mu$ of heat conduction to viscosity coefficient within [0.2, 5] ($\gamma \approx 1.4$, $\nu/\mu \approx 1.47$ for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.

Recommended citation: Jeffrey Humpherys, Gregory Lyng, & Kevin Zumbrun, Spectral stability of ideal-gas shock layers. Archive for Rational Mechanics and Analysis, 194 (2009): 1029–1079. https://dx.doi.org/10.1007/s00205-008-0195-4

The Gaussian semiclassical soliton ensemble and numerical methods for the focusing nonlinear Schrödinger equation

Published in Physica D, 2012

We report on a number of careful numerical experiments motivated by the semiclassical (zero-dispersion, $\epsilon\downarrow 0$ ) limit of the focusing nonlinear Schrödinger equation. Our experiments are designed to study the evolution of a particular family of perturbations of the initial data. These asymptotically small perturbations are precisely those that result from modifying the initial-data by using formal approximations to the spectrum of the associated spectral problem; such modified data has always been a standard part of the analysis of zero-dispersion limits of integrable systems. However, in the context of the focusing nonlinear Schrödinger equation, the ellipticity of the Whitham equations casts some doubt on the validity of this procedure. To carry out our experiments, we introduce an implicit finite difference scheme for the partial differential equation, and we validate both the proposed scheme and the standard split-step scheme against a numerical implementation of the inverse scattering transform for a special case in which the scattering data is known exactly. As part of this validation, we also investigate the use of the Krasny filter which is sometimes suggested as appropriate for nearly ill-posed problems such as we consider here. Our experiments show that that the $O(\epsilon)$ rate of convergence of the modified data to the true data is propagated to positive times including times after wave breaking.

Recommended citation: Long Lee, Gregory Lyng, & Irena Vankova, The Gaussian semiclassical soliton ensemble and numerical methods for the focusing nonlinear Schrödinger equation. Physica D, 241 (2012): 1767–1781. https://doi.org/10.1016/j.physd.2012.08.006

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