Skip to main content

LUP Student Papers

LUND UNIVERSITY LIBRARIES

Pseudospectra and the numerical solution of differential equations

Ioannidis, Mavroudis LU (2021) In Master's Theses in Mathematical Sciences NUMM03 20211
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
Eigenvalue analysis has been a key tool to science and engineering for several
decades. Eigenvalues can predict the behaviour of many mathematical systems
of equations but alone they cannot fully explain phenomena such as stability
or stiffness. Together, eigenvalues and pseudospectra can give a better understanding
of several phenomena such as instability in nonnormal matrices or
operators.
In this thesis, the basic concepts of pseudospectra are utilized to assist in understanding
how pseudospectra can better explain the stability of PDE discretizations,
the stability of the method of lines, the stiffness of ODEs and the
GKS-stability of boundary conditions.
It has been based on the book of Lloyd N. Trefethen and Mark Embree,... (More)
Eigenvalue analysis has been a key tool to science and engineering for several
decades. Eigenvalues can predict the behaviour of many mathematical systems
of equations but alone they cannot fully explain phenomena such as stability
or stiffness. Together, eigenvalues and pseudospectra can give a better understanding
of several phenomena such as instability in nonnormal matrices or
operators.
In this thesis, the basic concepts of pseudospectra are utilized to assist in understanding
how pseudospectra can better explain the stability of PDE discretizations,
the stability of the method of lines, the stiffness of ODEs and the
GKS-stability of boundary conditions.
It has been based on the book of Lloyd N. Trefethen and Mark Embree, "Spectra
and Pseudospectra The Behavior of Nonnormal Matrices and Operators"
[Trefethen and Embree, 2005]. Despite that, it tries to explain in a more analytical
manner certain points of the book. Using also other references, we
attempt to clarify some more aspects of pseudospectra. The code for the figures
has been based on Lloyd N. Trefethen [Trefethen, 1999] and is presented in the
Appendix. For more on MATLAB codes for solving problems using spectra and
pseudospectra, see [Trefethen, 2000].
i (Less)
Please use this url to cite or link to this publication:
author
Ioannidis, Mavroudis LU
supervisor
organization
course
NUMM03 20211
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Pseudospectra, numerical solutions of differential equations, spectral differentiation matrices, Lax-stability, stability of MOL, stiffness, GKSstability
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFNA-3039-2022
ISSN
1404-6342
other publication id
2022:E83
language
English
id
9078457
date added to LUP
2024-05-13 16:36:25
date last changed
2024-05-13 16:36:25
@misc{9078457,
  abstract     = {{Eigenvalue analysis has been a key tool to science and engineering for several
decades. Eigenvalues can predict the behaviour of many mathematical systems
of equations but alone they cannot fully explain phenomena such as stability
or stiffness. Together, eigenvalues and pseudospectra can give a better understanding
of several phenomena such as instability in nonnormal matrices or
operators.
In this thesis, the basic concepts of pseudospectra are utilized to assist in understanding
how pseudospectra can better explain the stability of PDE discretizations,
the stability of the method of lines, the stiffness of ODEs and the
GKS-stability of boundary conditions.
It has been based on the book of Lloyd N. Trefethen and Mark Embree, "Spectra
and Pseudospectra The Behavior of Nonnormal Matrices and Operators"
[Trefethen and Embree, 2005]. Despite that, it tries to explain in a more analytical
manner certain points of the book. Using also other references, we
attempt to clarify some more aspects of pseudospectra. The code for the figures
has been based on Lloyd N. Trefethen [Trefethen, 1999] and is presented in the
Appendix. For more on MATLAB codes for solving problems using spectra and
pseudospectra, see [Trefethen, 2000].
i}},
  author       = {{Ioannidis, Mavroudis}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Pseudospectra and the numerical solution of differential equations}},
  year         = {{2021}},
}