  When using Laguerre and Hermite spectral methods to numerically solve PDEs in
unbounded domains, the number of collocation points assigned inside the region
of interest is often insufficient, particularly when the region is expanded or
translated to safely capture the unknown solution. Simply increasing the number
of collocation points cannot ensure a fast convergence to spectral accuracy. In
this paper, we propose a scaling technique and a moving technique to adaptively
cluster enough collocation points in a region of interest in order to achieve a
fast spectral convergence. Our scaling algorithm employs an indicator in the
frequency domain that is used to determine when scaling is needed and informs
the tuning of a scaling factor to redistribute collocation points to adapt to
the diffusive behavior of the solution. Our moving technique adopts an
exterior-error indicator and moves the collocation points to capture the
translation. Both frequency and exterior-error indicators are defined using
only the numerical solutions. We apply our methods to a number of different
models, including diffusive and moving Fermi-Dirac distributions and nonlinear
Dirac solitary waves, and demonstrate recovery of spectral convergence for
time-dependent simulations. Performance comparison in solving a linear
parabolic problem shows that our frequency scaling algorithm outperforms the
existing scaling approaches. We also show our frequency scaling technique is
able to track the blowup of average cell sizes in a model for cell
proliferation.
