PhD Dissertation Defense

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PhD Defense| Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs

Start Date: September 17, 2017
End Date: September 17, 2017

By Yiannis Hadjimichael, PhD candidate of Professor David Ketcheson (KAUST)   
 
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. 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. 
 
 
Biography: Yiannis Hadjimichael is a Ph.D. candidate in the Applied Mathematics and Computation Science program at King Abdullah University of Science and Technology (KAUST). He is working under the supervision of Prof. David Ketcheson and he is a member of the Numerical Mathematics group at KAUST. He has obtained his BSc degree in Mathematics from the University of Cyprus in 2007. He then received a master in Mathematics and the Foundations of Computer Science (2008) and a master in Mathematical Modeling and Scientific Computing (2010), both from University of Oxford. His main research interests are the numerical analysis of ordinary and partial differential equations, strong stability preservation, and developing optimal time integrators for hyperbolic PDEs. His research work has resulted in peer-reviewed publications in high-quality journals.
 

More Information:

For more info contact:  Professor. David Ketcheson: email: david.ketcheson@kaust.edu.sa
 
Date: Sunday 17th Sep 2017
Time:03:00 PM - 05:00 PM
Location: Buillding 3, Level 5, Room 5209
Refreshments: Will be available from 2:45 PM.