2007/05/29 by Blake Barker, Jeffrey Humpherys, Keith Rudd +1 · 29 citations
Engineering · Mathematics · #Adiabatic process #Bounded function #Compressibility #Computational Fluid Dynamics and Aerodynamics #Eigenvalues and eigenvectors #Gas Dynamics and Kinetic Theory #Isentropic process #Mach number #Navier-Stokes equation solutions #Shock (circulatory) #Stability (learning theory) #math.AP #math.DS #msc:35L65 #msc:35Q30 #msc:76L05
paper · pdf · doi:10.1007/s00220-008-0487-4
published in Communications in Mathematical Physics 281(1), 231-249 (Springer Science+Business Media)
arxiv created 2007/05/29 · openalex publication_date 2008/05/05 · openalex created_date 2016/06/24 · arxiv updated 2017/06/12 · openalex updated_date 2026/08/05
In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier--Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M≈ 3000 for 1 ≤ γ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.