DG Boltzmann Solver Shows Anisotropic Gaussian Persists in Affine Far-from-Equilibrium Flows

The study reduces the Boltzmann equation to a velocity-space problem under four affine deformations—simple shear, pressure shear, bidirectional shear, and vortex flow—then solves it with tensor-product Lagrange DG elements without moment closure. This yields direct evolution of the distribution function for the first time in such regimes. The principal numerical result is that an anisotropic Gaussian form approximates the solution across all cases despite strong departure from equilibrium.

In the collisionless limit the covariance tensor exactly equals the inverse right Cauchy-Green deformation tensor; numerical solutions track this prediction closely at early times then diverge as particle collisions become significant. The gap quantifies the thermalization rate and highlights the role of collisions in isotropizing the distribution.

Related work on non-equilibrium kinetic theory, including Grad’s moment methods and DSMC validations of shear flows, shows similar short-time anisotropy but lacks the closure-free resolution achieved here. The approach therefore bridges continuum mechanics and kinetic theory, with direct implications for modeling polymer processing, high-strain-rate materials, and rarefied gas dynamics.

Next steps include extending the framework to non-affine flows and coupling with boundary conditions to test whether the Gaussian approximation survives in realistic geometries.