  In this paper, we present a block-oriented scheme for adaptive mesh
refinement based on summation-by-parts (SBP) finite difference methods and
simultaneous-approximation-term (SAT) interface treatment. Since the order of
accuracy at SBP-SAT grid interfaces is lower compared to that of the interior
stencils, we strive at using the interior stencils across block-boundaries
whenever possible. We devise a stable treatment of SBP-FD junction points, i.e.
points where interfaces with different boundary treatment meet. This leads to
stable discretizations for more flexible grid configurations within the SBP-SAT
framework, with a reduced number of SBP-SAT interfaces. Both first and second
derivatives are considered in the analysis. Even though the stencil order is
locally reduced close to numerical interfaces and corner points, numerical
simulations show that the locally reduced accuracy does not severely reduce the
accuracy of the time propagated numerical solution. Moreover, we explain how to
organize the grid and how to automatically adapt the mesh, aiming at problems
of many variables. Examples of adaptive grids are demonstrated for the
simulation of the time-dependent Schr\"odinger equation and for the advection
equation.
