Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs

Handle URI:
http://hdl.handle.net/10754/625526
Title:
Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs
Authors:
Hadjimichael, Yiannis ( 0000-0003-3517-8557 )
Abstract:
A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases it is difficult to satisfy certain physical properties while maintaining high order of accuracy. In this thesis, we develop high-order time-stepping methods that are capable of maintaining stability constraints of the solution, when coupled with suitable spatial discretizations. Such methods are called strong stability preserving (SSP) time integrators, and we mainly focus on perturbed methods that use both upwind- and downwind-biased spatial discretizations. Firstly, we introduce a new family of third-order implicit Runge–Kuttas methods with arbitrarily large SSP coefficient. We investigate the stability and accuracy of these methods and we show that they perform well on hyperbolic problems with large CFL numbers. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain augmented monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding non-additive SSP linear multistep methods. Furthermore, we develop the first SSP linear multistep methods of order two and three with variable step size, and study their optimality. We describe an optimal step-size strategy and demonstrate the effectiveness of these methods on various one- and multi-dimensional problems. Finally, we establish necessary conditions to preserve the total variation of the solution obtained when perturbed methods are applied to boundary value problems. We implement a stable treatment of nonreflecting boundary conditions for hyperbolic problems that allows high order of accuracy and controls spurious wave reflections. Numerical examples with high-order perturbed Runge–Kutta methods reveal that this technique provides a significant improvement in accuracy compared with zero-order extrapolation.
Advisors:
Ketcheson, David I. ( 0000-0002-1212-126X )
Committee Member:
Keyes, David E. ( 0000-0002-4052-7224 ) ; Samtaney, Ravi ( 0000-0002-4702-6473 ) ; Tzavaras, Athanasios ( 0000-0002-1896-2270 ) ; Higueras, Inmaculada
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
30-Sep-2017
Type:
Dissertation
Appears in Collections:
Dissertations

Full metadata record

DC FieldValue Language
dc.contributor.advisorKetcheson, David I.en
dc.contributor.authorHadjimichael, Yiannisen
dc.date.accessioned2017-10-01T05:23:43Z-
dc.date.available2017-10-01T05:23:43Z-
dc.date.issued2017-09-30-
dc.identifier.urihttp://hdl.handle.net/10754/625526-
dc.description.abstractA plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases it is difficult to satisfy certain physical properties while maintaining high order of accuracy. In this thesis, we develop high-order time-stepping methods that are capable of maintaining stability constraints of the solution, when coupled with suitable spatial discretizations. Such methods are called strong stability preserving (SSP) time integrators, and we mainly focus on perturbed methods that use both upwind- and downwind-biased spatial discretizations. Firstly, we introduce a new family of third-order implicit Runge–Kuttas methods with arbitrarily large SSP coefficient. We investigate the stability and accuracy of these methods and we show that they perform well on hyperbolic problems with large CFL numbers. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain augmented monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding non-additive SSP linear multistep methods. Furthermore, we develop the first SSP linear multistep methods of order two and three with variable step size, and study their optimality. We describe an optimal step-size strategy and demonstrate the effectiveness of these methods on various one- and multi-dimensional problems. Finally, we establish necessary conditions to preserve the total variation of the solution obtained when perturbed methods are applied to boundary value problems. We implement a stable treatment of nonreflecting boundary conditions for hyperbolic problems that allows high order of accuracy and controls spurious wave reflections. Numerical examples with high-order perturbed Runge–Kutta methods reveal that this technique provides a significant improvement in accuracy compared with zero-order extrapolation.en
dc.language.isoenen
dc.subjecttime-integrationen
dc.subjectstrong stability preservationen
dc.subjectRunge-Kuttaen
dc.subjectlinear multistep methodsen
dc.subjectHyperbolic PDEen
dc.titlePerturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEsen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen
dc.contributor.committeememberKeyes, David E.en
dc.contributor.committeememberSamtaney, Ravien
dc.contributor.committeememberTzavaras, Athanasiosen
dc.contributor.committeememberHigueras, Inmaculadaen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id115187en
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