Stability analysis of discrete Boltzmann simulation for supersonic flows: Influencing factors, coupling mechanisms, and optimization strategies

Supersonic flow simulations encounter challenges in threefold: trans-scale modeling, numerical stability and complex field analysis, which arise from inherent nonlinear, nonequilibrium, and multiscale characteristics. The discrete Boltzmann method (DBM) provides a multiscale kinetic modeling framework and analysis tool for capturing complex discrete/nonequilibrium states and effects. Despite its fundamental role in DBM simulations, a comprehensive stability analysis is still lacking. Similar to the lattice Boltzmann method, complexity in DBM lies mainly in the intrinsic coupling between velocity and spatiotemporal discretizations, which distinguishes it from traditional computational fluid dynamics. This study conducts von Neumann stability analysis to examine factors influencing DBM simulation stability, including approaches for determining equilibrium distribution functions, thermodynamic nonequilibrium (TNE) levels, spatiotemporal discretization schemes, initial conditions, and model parameters. Key findings include: (i) among the equilibrium distribution discretization methods considered, the moment-matching approach outperforms the expansion- and weighting-coefficient-based methods in the test simulations. (ii) Increased TNE intensity/Knudsen number enhances the system's nonlinear behavior and the intrinsic nonlinearity embedded in the matching model equation, thereby amplifying the instabilities in simulations. (iii) Although additional viscous dissipation based on distribution functions improves stability, it distorts flow fields and alters constitutive relations, highlighting the need for careful trade-offs between stability and accuracy. (iv) Larger Courant–Friedrichs–Lewy numbers and relative time steps significantly degrade stability, necessitating appropriate time-stepping strategies. To assess the stability regulation capability of DBMs across different TNE levels, stability-phase diagrams and stability probability curves are constructed within the moment-matching framework using morphological analysis. These diagrams identify common stable parameter regions applicable across various TNE orders. Finally, the effects of discrete velocity configurations on achieving both physical functionality and numerical stability are assessed through comparisons between numerical and analytical TNE solutions, as well as statistical properties of distribution functions. This study reveals key factors and coupling mechanisms governing numerical stability in DBM simulation and proposes general strategies for optimizing equilibrium distribution function discretization, discrete velocity design, and stability parameter selection across supersonic regimes.