Kinetic Linear Stability Theory for High-Speed Compressible Flows: A High Performance Computing Framework

Abstract

Shock waves in high-speed compressible flows contain finite-thickness, high-gradient regions where the continuum assumption becomes questionable and translational non-equilibrium arises, including non-Maxwellian micro-velocity distributions. Classical shock stability analyses rely on Navier-Stokes or moment closures and cannot retain bi-modal velocity distributions inside the shock. We develop and apply, for the first time, a kinetic linear stability theory (kLST) for one-dimensional normal shocks by linearizing the Boltzmann-BGK equation about kinetic BE-BGK base flows. Perturbations are posed in reduced distribution functions, with macroscopic fields recovered by velocity-space moments, so the stability operator acts on the VDF rather than a closed continuum system. Verified against compressible Couette eigenvalue benchmarks near continuum, the framework is applied to argon shocks at $M_\infty=1.2$, $3.0$, and $4.0$. At low Mach number, where BE-BGK and Gilbarg-Paolucci profiles nearly coincide, the spectra recover stable continuous branches. At higher Mach number, comparing Maxwellian and non-equilibrium VDF-based eigenspectra shows that kinetic effects shift the spectrum toward less stable regions, so continuum predictions can miss important changes even when macroscopic profiles appear well resolved. For large high-Mach matrices--$O(10^5)$ unknowns and up to billions of nonzeros--we develop a parallel SLEPc/PETSc infrastructure using shift-and-invert Arnoldi with MUMPS LU for moderate sizes and Jacobi-Davidson (JD) with block-Jacobi ILU for the largest systems. Coupled spatial/micro-velocity sparsity causes severe LU fill-in, making direct solvers memory-limited and motivating JD. We compute kLST spectra for an $M_\infty=4.0$ shock with 281088 unknowns, to our knowledge the highest-Mach kinetic linear stability calculation reported for isolated finite-thickness shock layers.