Nonlinear Preconditioning and its Application in Multicomponent Problems

Handle URI:
http://hdl.handle.net/10754/583375
Title:
Nonlinear Preconditioning and its Application in Multicomponent Problems
Authors:
Liu, Lulu ( 0000-0002-0357-1322 )
Abstract:
The Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) algorithm is presented as a complement to Additive Schwarz Preconditioned Inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. The ASPIN framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this dissertation, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size. We consider the additive and multiplicative types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Moreover, we provide the convergence analysis of the MSPIN algorithm. Under suitable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be obtained when the forcing terms are picked suitably. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations, and that MSPIN can be more robust than ASPIN, and maintain fast convergence even for challenging problems, such as high-Reynolds number Navier-Stokes equations.
Advisors:
Keyes, David E. ( 0000-0002-4052-7224 )
Committee Member:
Sun, Shuyu ( 0000-0002-3078-864X ) ; Samtaney, Ravi ( 0000-0002-4702-6473 ) ; Krause, Rolf
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program
Program:
Applied Mathematics and Computational Science
Issue Date:
7-Dec-2015
Type:
Dissertation
Appears in Collections:
Applied Mathematics and Computational Science Program; Dissertations; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.advisorKeyes, David E.en
dc.contributor.authorLiu, Luluen
dc.date.accessioned2015-12-08T07:42:37Zen
dc.date.available2015-12-08T07:42:37Zen
dc.date.issued2015-12-07en
dc.identifier.urihttp://hdl.handle.net/10754/583375en
dc.description.abstractThe Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) algorithm is presented as a complement to Additive Schwarz Preconditioned Inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. The ASPIN framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this dissertation, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size. We consider the additive and multiplicative types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Moreover, we provide the convergence analysis of the MSPIN algorithm. Under suitable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be obtained when the forcing terms are picked suitably. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations, and that MSPIN can be more robust than ASPIN, and maintain fast convergence even for challenging problems, such as high-Reynolds number Navier-Stokes equations.en
dc.language.isoenen
dc.subjectnon-linear equationsen
dc.subjectnonlinear preconditioningen
dc.subjectfield splittingen
dc.subjectNewton methoden
dc.subjectmultiplicative schwarzen
dc.subjectlocal convergenceen
dc.titleNonlinear Preconditioning and its Application in Multicomponent Problemsen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberSun, Shuyuen
dc.contributor.committeememberSamtaney, Ravien
dc.contributor.committeememberKrause, Rolfen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id113342en
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