The FEDVR method is an efficient approach to discretizing partial differential equations containing second-order or lower derivatives in space. The piecewise continuous nature of this representation, leading to sparse and structured matrices, combined with its ability to accurately represent matrix elements of local operators as their values on the grid, make it an extremely efficient spectral-element method. When combined with a time-propagation technique such as the short iterative Lanczos or real space propagation method, it is possible to parallelize the solution for the TDSE in a manner which scales linearly with the time dimension ( i.e. the H_2 molecule in an intense laser field) and applications have been made demonstrating this linear scaling on a number of NSF supercomputers. The method will be described in some detail in the talk and will conclude with one or two illustrative examples.
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