Implicit solvers for large-scale nonlinear problems

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  • Reynolds D
  • Woodward C
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Abstract

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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