9727128 Homer Walker Large-scale systems of linear and nonlinear equations occur almost ubiquitously in the computational modeling of physical phenomena through the discretization of partial differential equations, integral equations, etc. Krylov subspace methods make up a large class of iterative linear algebra methods that have been widely used for these large-scale linear systems. These methods have been adapted to large-scale nonlinear systems through Newton-Krylov implementations, in which they are used to solve the linear systems that characterize Newton steps. This research is a continuation of an ongoing program aimed at developing more robust and efficient Newton-Krylov methods for nonlinear problems and, concomitantly, more effective Krylov subspace methods for linear problems. The current research focusses primarily on Newton-Krylov methods and related methods for large-scale nonlinear problems. Specific objectives are to develop more robust globalizations, to implement and test additional and more effective Krylov solvers, to improve preconditioning strategies and options, and to explore new techniques such as automatic differentiation. A major extension of the previous program goals is to develop effective methods for large-scale continuation problems. The approach is based on certain ways of adapting Krylov subspace methods and Newton-Krylov methodology to the continuation setting that allow the use of existing preconditioners and other previously developed problem technology. A particular aim of this work is to develop improved methods for large-scale computational modeling of complex physical phenomena on high performance computers. Particular target applications include full-physics modeling of reactive flows (combustion and other chemically reacting flows) and related problems in computational fluid dynamics. Success in these applications will have great potential benefits in the design and operation of industrial facilit ies such as furnaces for heat and power generation and chemical vapor deposition reactors for semiconductor manufacturing. This work will build on previously established collaborative activities with researchers at Sandia National Laboratories, Lawrence Livermore National Laboratory, and the University of Utah Center for High Performance Computing, in which the methods under development have been successfully applied to a number of realistic flow problems involving heat and mass transport on massively parallel computers. Likely future collaborations are with researchers at the University of Utah on modeling accidental fires and explosions and at Lawrence Livermore National Laboratory on modeling contaminant transport in aquifers, with applications to environmental management and bioremediation.
