Convergent Difference Schemes for Hamilton-Jacobi equations

Handle URI:
http://hdl.handle.net/10754/627772
Title:
Convergent Difference Schemes for Hamilton-Jacobi equations
Authors:
Duisembay, Serikbolsyn ( 0000-0003-0467-4750 )
Abstract:
In this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.
Advisors:
Gomes, Diogo A. ( 0000-0002-3129-3956 )
Committee Member:
Alouini, Mohamed-Slim ( 0000-0003-4827-1793 ) ; Parsani, Matteo ( 0000-0001-7300-1280 )
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
7-May-2018
Type:
Thesis
Appears in Collections:
Theses

Full metadata record

DC FieldValue Language
dc.contributor.advisorGomes, Diogo A.en
dc.contributor.authorDuisembay, Serikbolsynen
dc.date.accessioned2018-05-07T13:34:34Z-
dc.date.available2018-05-07T13:34:34Z-
dc.date.issued2018-05-07-
dc.identifier.urihttp://hdl.handle.net/10754/627772-
dc.description.abstractIn this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.en
dc.language.isoenen
dc.subjectHamilton-Jacobi equationsen
dc.subjectdifference schemesen
dc.subjectViscosity solutionsen
dc.subjectnumerical methodsen
dc.titleConvergent Difference Schemes for Hamilton-Jacobi equationsen
dc.typeThesisen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen
dc.contributor.committeememberAlouini, Mohamed-Slimen
dc.contributor.committeememberParsani, Matteoen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameMaster of Scienceen
dc.person.id149367en
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