Special Lecture in Celebration of Marsha Berger and Margaret Wright

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Introduced by Marsha Berger in 1982, adaptive mesh refinement ("AMR") is now a widely used computational technique for improving the efficiency of solvers for logically Cartesian meshes.  While far from simple to implement, the idea behind Berger's original AMR scheme is elegant.  A coarse computational, Cartesian mesh is "enhanced" with layers of finer meshes in regions where the solution exhibits sharp, localized features, such as a shock or other discontinuity.  These finer meshes are allowed to dynamically evolve to follow the shock or discontinuity as it moves through space and regions with less demanding solution structure are resolved only on coarser level meshes.  Local time stepping contributes to performance gains and accuracy by allowing large time steps to be taken on coarser meshes and smaller time steps on finer meshes.  Forty years later, these original mesh generation and time stepping ideas are still the most widely used algorithms in the AMR community.  This talk will present the technical and numerical challenges that arise  in developing AMR software, some of the more recent advances in AMR technology and several applications in geosciences and hazards modeling.