Implicit solvers for large-scale nonlinear problems

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Computational scientists are grappling with increasingly complex, multi-rate applications that couple such physical phenomena as fluid dynamics, electromagnetics, radiation transport, chemical and nuclear reactions, and wave and material propagation in inhomogeneous media. Parallel computers with large storage capacities are paving the way for high-resolution simulations of coupled problems; however, hardware improvements alone will not prove enough to enable simulations based on brute-force algorithmic approaches. To accurately capture nonlinear couplings between dynamically relevant phenomena, often while stepping over rapid adjustments to quasi-equilibria, simulation scientists are increasingly turning to implicit formulations that require a discrete nonlinear system to be solved for each ... continued below

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12 p. (0.2 MB)

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Keyes, D E; Reynolds, D & Woodward, C S July 13, 2006.

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Computational scientists are grappling with increasingly complex, multi-rate applications that couple such physical phenomena as fluid dynamics, electromagnetics, radiation transport, chemical and nuclear reactions, and wave and material propagation in inhomogeneous media. Parallel computers with large storage capacities are paving the way for high-resolution simulations of coupled problems; however, hardware improvements alone will not prove enough to enable simulations based on brute-force algorithmic approaches. To accurately capture nonlinear couplings between dynamically relevant phenomena, often while stepping over rapid adjustments to quasi-equilibria, simulation scientists are increasingly turning to implicit formulations that require a discrete nonlinear system to be solved for each time step or steady state solution. Recent advances in iterative methods have made fully implicit formulations a viable option for solution of these large-scale problems. In this paper, we overview one of the most effective iterative methods, Newton-Krylov, for nonlinear systems and point to software packages with its implementation. We illustrate the method with an example from magnetically confined plasma fusion and briefly survey other areas in which implicit methods have bestowed important advantages, such as allowing high-order temporal integration and providing a pathway to sensitivity analyses and optimization. Lastly, we overview algorithm extensions under development motivated by current SciDAC applications.

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12 p. (0.2 MB)

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PDF-file: 12 pages; size: 0.2 Mbytes

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  • Presented at: SciDAC PI Meeting, Denver, CO, United States, Jun 25 - Jun 29, 2006

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  • Report No.: UCRL-CONF-222859
  • Grant Number: W-7405-ENG-48
  • Office of Scientific & Technical Information Report Number: 900076
  • Archival Resource Key: ark:/67531/metadc879739

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Office of Scientific & Technical Information Technical Reports

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  • July 13, 2006

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  • Sept. 22, 2016, 2:13 a.m.

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  • April 13, 2017, 3:59 p.m.

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Keyes, D E; Reynolds, D & Woodward, C S. Implicit solvers for large-scale nonlinear problems, article, July 13, 2006; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc879739/: accessed November 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.