Efficient Time Stepping Methods and Sensitivity Analysis for Large Scale Systems of Differential Equations

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dc.contributor.authorZhang, Hongen
dc.contributor.committeechairSandu, Adrianen
dc.contributor.committeememberCao, Yangen
dc.contributor.committeememberSpiteri, Raymond Johnen
dc.contributor.committeememberLin, Taoen
dc.contributor.committeememberRibbens, Calvin J.en
dc.contributor.committeememberIliescu, Traianen
dc.contributor.departmentComputer Scienceen
dc.date.accessioned2014-09-10T08:00:11Zen
dc.date.available2014-09-10T08:00:11Zen
dc.date.issued2014-09-09en
dc.description.abstractMany fields in science and engineering require large-scale numerical simulations of complex systems described by differential equations. These systems are typically multi-physics (they are driven by multiple interacting physical processes) and multiscale (the dynamics takes place on vastly different spatial and temporal scales). Numerical solution of such systems is highly challenging due to the dimension of the resulting discrete problem, and to the complexity that comes from incorporating multiple interacting components with different characteristics. The main contributions of this dissertation are the creation of new families of time integration methods for multiscale and multiphysics simulations, and the development of industrial-strengh tools for sensitivity analysis. This work develops novel implicit-explicit (IMEX) general linear time integration methods for multiphysics and multiscale simulations typically involving both stiff and non-stiff components. In an IMEX approach, one uses an implicit scheme for the stiff components and an explicit scheme for the non-stiff components such that the combined method has the desired stability and accuracy properties. Practical schemes with favorable properties, such as maximized stability, high efficiency, and no order reduction, are constructed and applied in extensive numerical experiments to validate the theoretical findings and to demonstrate their advantages. Approximate matrix factorization (AMF) technique exploits the structure of the Jacobian of the implicit parts, which may lead to further efficiency improvement of IMEX schemes. We have explored the application of AMF within some high order IMEX Runge-Kutta schemes in order to achieve high efficiency. Sensitivity analysis gives quantitative information about the changes in a dynamical model outputs caused by caused by small changes in the model inputs. This information is crucial for data assimilation, model-constrained optimization, inverse problems, and uncertainty quantification. We develop a high performance software package for sensitivity analysis in the context of stiff and nonstiff ordinary differential equations. Efficiency is demonstrated by direct comparisons against existing state-of-art software on a variety of test problems.en
dc.description.degreePh. D.en
dc.format.mediumETDen
dc.identifier.othervt_gsexam:3646en
dc.identifier.urihttp://hdl.handle.net/10919/50492en
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectTime Steppingen
dc.subjectGeneral Linear Methodsen
dc.subjectImplicit-expliciten
dc.subjectSensitivity Analysisen
dc.titleEfficient Time Stepping Methods and Sensitivity Analysis for Large Scale Systems of Differential Equationsen
dc.typeDissertationen
thesis.degree.disciplineComputer Science and Applicationsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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