  This work is focused on the extension and assessment of the
monotonicity-preserving scheme in [3] and the local bounds preserving scheme in
[5] to hierarchical octree adaptive mesh refinement (AMR). Whereas the former
can readily be used on this kind of meshes, the latter requires some
modifications. A key question that we want to answer in this work is whether to
move from a linear to a nonlinear stabilization mechanism pays the price when
combined with shock-adapted meshes. Whereas nonlinear (or shock-capturing)
stabilization leads to improved accuracy compared to linear schemes, it also
negatively hinders nonlinear convergence, increasing computational cost. We
compare linear and nonlinear schemes in terms of the required computational
time versus accuracy for several steady benchmark problems. Numerical results
indicate that, in general, nonlinear schemes can be cost-effective for
sufficiently refined meshes. Besides, it is also observed that it is better to
refine further around shocks rather than use sharper shock capturing terms,
which usually yield stiffer nonlinear problems. In addition, a new refinement
criterion has been proposed. The proposed criterion is based on the graph
Laplacian used in the definition of the stabilization method. Numerical results
show that this shock detector performs better than the well-known Kelly
estimator for problems with shocks or discontinuities.
