Explain the increased positional-grid fill in the transitional BGK regime

Determine whether the increased fill of the adaptive positional grid in the transitional-regime two-dimensional spatial, two-dimensional velocity BGK simulation with collision frequency \(\nu=1\) is caused by the initial contact discontinuity being advected into phase space and smoothed less aggressively than in the \(\nu=10\) case.

Background

The paper compares adaptive sparse-grid discontinuous Galerkin simulations of a radial Sod shock tube for collision frequencies $\nu=10$ and $\nu=1$. In the lower-collision-frequency case, the authors observe a denser positional grid despite similar initial conditions. They attribute this behavior to the phase-space advection of the initial contact discontinuity and its weaker collisional smoothing, but explicitly present that explanation as a conjecture rather than an established result.

The proposed explanation has implications for adaptive-grid complexity: resolving a discontinuity in phase space while maintaining ancestor completeness may force additional spatial degrees of freedom. Establishing whether this mechanism is in fact responsible would clarify the interaction between phase-space refinement, positional-grid refinement, and collision frequency in the adaptive sparse-grid BGK method.