Implementation of Singly Diagonally Implicit RungeKutta Methods with Constant Step Sizes
(2015) In Bachelor’s Theses in Mathematical Sciences NUMK01 20141Mathematics (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 RKmethods reduces an sm x sm matrix to s systems of m x m linear equations. Singly Diagonally Implicit RKmethods have only a single eigenvalue, which results in a reduction to only one LUdecomposition per time step. Combining the two methods, we get Singly Diagonally Implicit RKmethods.
 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:
http://lup.lub.lu.se/studentpapers/record/7851608
 author
 Stål, Josefine ^{LU}
 supervisor

 Claus Führer ^{LU}
 organization
 course
 NUMK01 20141
 year
 2015
 type
 M2  Bachelor Degree
 subject
 keywords
 Implicit RungeKutta methods, SDIRK, Implementation, Python
 publication/series
 Bachelor’s Theses in Mathematical Sciences
 report number
 LUNFNA40062015
 ISSN
 16546229
 other publication id
 2015:K12
 language
 English
 id
 7851608
 date added to LUP
 20150909 10:09:57
 date last changed
 20151214 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 RKmethods reduces an sm x sm matrix to s systems of m x m linear equations. Singly Diagonally Implicit RKmethods have only a single eigenvalue, which results in a reduction to only one LUdecomposition per time step. Combining the two methods, we get Singly Diagonally Implicit RKmethods.}, author = {Stål, Josefine}, issn = {16546229}, keyword = {Implicit RungeKutta methods,SDIRK,Implementation,Python}, language = {eng}, note = {Student Paper}, series = {Bachelor’s Theses in Mathematical Sciences}, title = {Implementation of Singly Diagonally Implicit RungeKutta Methods with Constant Step Sizes}, year = {2015}, }