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Open Access Publications from the University of California

Quantification of the upstream-to-downstream influence in the Muskingum method and implications for speedup in parallel computations of river flow


The mathematical formulation of the Muskingum method, like that of many numerical schemes used for river routing, requires that all upstream river reaches be updated prior to updating the flow rate of any given reach. Due to this topological constraint, such numerical schemes have traditionally been solved in an upstream-to-downstream manner which imposes inherent limitations on the speedup that can be achieved in a parallel computing environment because each computing core has to wait for completion of all cores addressing upstream subbasins prior to starting its own subbasin. The research presented in this paper quantifies the exact influence among river reaches during the update step of the Muskingum method and shows that the influence decreases with increasing distance between two reaches until it becomes too small to be accounted for by floating-point arithmetic. A formal definition of the minimal distance from which the relative influence becomes numerically inexistent—the radius of influence—is presented. Based on this distance, expressed as a number of river reaches, a new estimate of the maximum theoretical speedup that can be achieved by the Muskingum method or by similar numerical schemes is presented and implies large potential gains in computing time when domains are much larger than the radius of influence. An application to the approximately 180,000 river reaches of the Upper Mississippi River Basin at a 15 min time step over 2004 shows a radius of influence on the order of 150 river reaches. The speedup obtained for this application is much higher than previously thought possible but also much lower than could be attained, suggesting that further investigations are necessary.

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