  Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the
relaxation term significantly varies in space. Using examples from rarefied
gases and the general form of the underlying balance law model, we perform a
detailed spectral analysis of the semi-discrete model that reveals the spectral
gaps. Based on that, we show the inefficiency of standard time integration
schemes expressed by a severe restriction of the CFL number. We then develop
the first spatially adaptive projective integration schemes to overcome the
prohibitive time step constraints of standard time integration schemes. The new
schemes use different time integration methods in different parts of the
computational domain, determined by the spatially varying value of the
relaxation time. We use our analytical results to derive accurate stability
bounds for the involved parameters and show that the severe time step
constraint can be overcome. The new adaptive schemes show good accuracy in a
numerical test case and can obtain a large speedup with respect to standard
schemes.
