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Implementation of Singly Diagonally Implicit Runge-Kutta Methods with Constant Step Sizes

Stål, Josefine LU (2015) In Bachelor’s Theses in Mathematical Sciences NUMK01 20141
Mathematics (Faculty of Technology) and Numerical Analysis
Abstract
Runge–Kutta methods can be used for solving ordinary differential equations of the form y0 = f(t, y) with initial condition y(t0) = y0 and where f : R x R^m -> R^m. The idea is to find a method that is efficient to implement. But it is also important for the method to be of high order and be stable. Diagonally Implicit RK-methods reduces an sm x sm matrix to s systems of m x m linear equations. Singly Diagonally Implicit RK-methods have only a single eigenvalue, which results in a reduction to only one LU-decomposition per time step. Combining the two methods, we get Singly Diagonally Implicit RK-methods.
Popular Abstract
Many problems in engineering science, natural science and even social and life sciences can be better understood by mathematical simulations. There are often several methods that can solve the same problem with. When choosing which method to use there are properties that needs to be considered, for instance time efficency and accuracy. These properties depend on the problem to solve and what criterias there are on the solution.

In this thesis only a specific class of methods are considered, for solving
differential equations. These methods are called Runge–Kutta methods.
Further on, some properties are described and also problems that can appear
when implementing the method. A specific subclass of Runge–Kutta methods,
namely Singly... (More)
Many problems in engineering science, natural science and even social and life sciences can be better understood by mathematical simulations. There are often several methods that can solve the same problem with. When choosing which method to use there are properties that needs to be considered, for instance time efficency and accuracy. These properties depend on the problem to solve and what criterias there are on the solution.

In this thesis only a specific class of methods are considered, for solving
differential equations. These methods are called Runge–Kutta methods.
Further on, some properties are described and also problems that can appear
when implementing the method. A specific subclass of Runge–Kutta methods,
namely Singly Diagonally Implicit Runge–Kutta methods, is constructed
such that these common implementation issues are not a problem. The benefits
from implementing and using these methods are described further and some tests are made to verify these statements. (Less)
Please use this url to cite or link to this publication:
author
Stål, Josefine LU
supervisor
organization
course
NUMK01 20141
year
type
M2 - Bachelor Degree
subject
keywords
Implicit Runge-Kutta methods, SDIRK, Implementation, Python
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFNA-4006-2015
ISSN
1654-6229
other publication id
2015:K12
language
English
id
7851608
date added to LUP
2015-09-09 10:09:57
date last changed
2015-12-14 13:32:13
@misc{7851608,
  abstract     = {Runge–Kutta methods can be used for solving ordinary differential equations of the form y0 = f(t, y) with initial condition y(t0) = y0 and where f : R x R^m -> R^m. The idea is to find a method that is efficient to implement. But it is also important for the method to be of high order and be stable. Diagonally Implicit RK-methods reduces an sm x sm matrix to s systems of m x m linear equations. Singly Diagonally Implicit RK-methods have only a single eigenvalue, which results in a reduction to only one LU-decomposition per time step. Combining the two methods, we get Singly Diagonally Implicit RK-methods.},
  author       = {Stål, Josefine},
  issn         = {1654-6229},
  keyword      = {Implicit Runge-Kutta methods,SDIRK,Implementation,Python},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor’s Theses in Mathematical Sciences},
  title        = {Implementation of Singly Diagonally Implicit Runge-Kutta Methods with Constant Step Sizes},
  year         = {2015},
}