Stability of strong ideal-gas shock layers

Abstract

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 endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball $|\lambda |\le \Lambda$. 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.

References in corpus (5)

• An efficient shooting algorithm for Evans function calculations in large systems
• Stability of isentropic viscous shock profiles in the high-Mach number limit
• Stability of viscous shocks in isentropic gas dynamics
• Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers
• Viscous Boundary Value Problems for Symmetric Systems with Variable Multiplicities

Cited by in corpus (27)

• Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
• One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamic
• Existence and stability of noncharacteristic boundary-layers for the compressible Navier-Stokes and viscous MHD equations
• Metastability of solitary roll wave solutions of the St. Venant equations with viscosity
• Multidimensional stability of large-amplitude Navier-Stokes shocks
• Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization
• Existence and stability of viscoelastic shock profiles
• Viscous hyperstabilization of detonation waves in one space dimension
• Efficient numerical stability analysis of detonation waves in ZND
• Stability of large-amplitude viscous shock under periodic perturbation for 1-d isentropic Navier-Stokes equations
• High-frequency asymptotics and 1-D stability of ZND detonations in the small-heat release and high-overdrive limits
• Stability of viscous detonations for Majda's model
• Stability of hydraulic shock profiles
• A numerical stability investigation of strong ZND detonations for Majda's model
• Spectral stability of weak dispersive shock profiles for quantum hydrodynamics with nonlinear viscosity
• Long-time stability of multi-dimensional noncharacteristic viscous boundary layers
• Computing Evans functions numerically via boundary-value problems
• Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions
• Stability of detonations in the ZND limit
• The WKB approximation of semiclassical eigenvalues of the Zakharov-Shabat problem
• Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic--parabolic systems
• Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability
• Euler vs. Lagrange: The role of coordinates in practical Evans-function computations
• On asymptotic stability of noncharacteristic viscous boundary layers
• Stability of noncharacteristic boundary layers in the standing shock limit
• Computing the refined stability condition
• A Galilei Invariant Version of Yong's Model