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Comparison of Radau and Lobatto methods with a novel adaptive time stepping for fluid dynamics problems

Paskevicius, Tadas LU (2023) In Bachelor's Theses in Mathematical Sciences NUMK11 20221
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract (Swedish)
Fully implicit Runge-Kutta methods play an important role in the numerical integration of stiff differential equations. Radau IIA is the most commonly used fully implicit method due to its good theoretical performance. However, Radau IA and Lobatto IIIC might also be suitable for different problems. These three methods are constructed using simplified order conditions that ensures specific integration properties. The theoretical performance is derived from various stability and order concepts. Specifically, Lobatto IIIC has a lower order, and Radau IA has a lower stiff order than Radau IIA.

A new adaptive time-stepping method is proposed and used. The order of convergence and properties of the adaptive time-stepping are examined for... (More)
Fully implicit Runge-Kutta methods play an important role in the numerical integration of stiff differential equations. Radau IIA is the most commonly used fully implicit method due to its good theoretical performance. However, Radau IA and Lobatto IIIC might also be suitable for different problems. These three methods are constructed using simplified order conditions that ensures specific integration properties. The theoretical performance is derived from various stability and order concepts. Specifically, Lobatto IIIC has a lower order, and Radau IA has a lower stiff order than Radau IIA.

A new adaptive time-stepping method is proposed and used. The order of convergence and properties of the adaptive time-stepping are examined for three distinct problems: The linear test equation (ODE), a combustion equation (ODE), and an advection-diffusion equation (PDE). The compressible Euler equations (PDE) are also assessed, albeit to a lesser extent due to computational constraints. Both the advection-diffusion equation and the Euler equations pertain to fluid dynamics, which is a primary area of interest for implicit Runge-Kutta methods. For the two last problems DUNE library \cite{BASTIAN202175} is used for spatial discretization. Numerical we see that Radau IIA best overall performing, Lobatto IIIC being reasonable good performing and Radau IA being not as good when the problem is stiff. (Less)
Please use this url to cite or link to this publication:
author
Paskevicius, Tadas LU
supervisor
organization
course
NUMK11 20221
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4051-2023
ISSN
1654-6229
other publication id
2023:K26
language
English
id
9140084
date added to LUP
2024-04-15 17:00:07
date last changed
2024-04-15 17:00:07
@misc{9140084,
  abstract     = {{Fully implicit Runge-Kutta methods play an important role in the numerical integration of stiff differential equations. Radau IIA is the most commonly used fully implicit method due to its good theoretical performance. However, Radau IA and Lobatto IIIC might also be suitable for different problems. These three methods are constructed using simplified order conditions that ensures specific integration properties. The theoretical performance is derived from various stability and order concepts. Specifically, Lobatto IIIC has a lower order, and Radau IA has a lower stiff order than Radau IIA.

A new adaptive time-stepping method is proposed and used. The order of convergence and properties of the adaptive time-stepping are examined for three distinct problems: The linear test equation (ODE), a combustion equation (ODE), and an advection-diffusion equation (PDE). The compressible Euler equations (PDE) are also assessed, albeit to a lesser extent due to computational constraints. Both the advection-diffusion equation and the Euler equations pertain to fluid dynamics, which is a primary area of interest for implicit Runge-Kutta methods. For the two last problems DUNE library \cite{BASTIAN202175} is used for spatial discretization. Numerical we see that Radau IIA best overall performing, Lobatto IIIC being reasonable good performing and Radau IA being not as good when the problem is stiff.}},
  author       = {{Paskevicius, Tadas}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Comparison of Radau and Lobatto methods with a novel adaptive time stepping for fluid dynamics problems}},
  year         = {{2023}},
}