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King Abdullah University of Science and Technology

1. Hadjimichael, Yiannis. Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs.

Degree: 2017, King Abdullah University of Science and Technology

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.

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…

Advisors/Committee Members: Ketcheson, David I., Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, Keyes, David E., Samtaney, Ravi, Tzavaras, Athanasios, Higueras, Inmaculada.

Subjects/Keywords: time-integration; strong stability preservation; Runge-Kutta; linear multistep methods; Hyperbolic PDE

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APA (6th Edition):

Hadjimichael, Y. (2017). Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/625526

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Hadjimichael, Yiannis. “Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs.” 2017. Thesis, King Abdullah University of Science and Technology. Accessed October 18, 2017. http://hdl.handle.net/10754/625526.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Hadjimichael, Yiannis. “Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs.” 2017. Web. 18 Oct 2017.

Vancouver:

Hadjimichael Y. Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2017. [cited 2017 Oct 18]. Available from: http://hdl.handle.net/10754/625526.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Hadjimichael Y. Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs. [Thesis]. King Abdullah University of Science and Technology; 2017. Available from: http://hdl.handle.net/10754/625526

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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