Stable difference methods for block-structured adaptive grids

The time-dependent Schrödinger equation describes quantum dynamical phenomena. Solving it numerically, the small-scale interactions that are modeled require very fine spatial resolution. At the same time, the solutions are localized and confined to small regions in space. Using the required resolution over the entire high-dimensional domain often makes the model problems intractable due to the prohibitively large grids that result from such a discretization. In this paper, we present a block-structured adaptive mesh refinement scheme, aiming at efficient adaptive discretization of high-dimensional partial differential equations such as the timedependent Schrödinger equation. Our framework allows for anisotropic grid refinement in order to avoid unnecessary refinement. For spatial discretization, we use standard finite difference stencils together with summation-by-parts operators and simultaneous-approximation-term interface treatment. We propagate in time using exponential integration with the Lanczos method. Our theoretical and numerical results show that our adaptive scheme is stable for long time integrations. We also show that the discretizations meet the expected convergence rates.