# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Parallelization of the SDIRK and Newton’s method and analysis of a weighted norm for error estimations

(2016) In Master's Theses in Mathematical Sciences FMN820 20161
Mathematics (Faculty of Engineering)
Abstract
Execution time is an important issue in the field of numerical analysis. Simulations are getting more and more complex and the execution times are rapidly increasing. In the scope of this thesis a time integration library is used to solve initial value problems. The aim of the thesis is to implement and investigate two different approaches to decrease the execution time for the library. The main focus of this thesis is on the Singly Diagonal Implicit Runge Kutta method and the Jacobian Free Newton’s method. The first approach is to parallelize the library. The other approach is to implement a discrete weighted norm for the error estimation in the time integration methods instead of the currently used euclidean norm. The parallelization is... (More)
Execution time is an important issue in the field of numerical analysis. Simulations are getting more and more complex and the execution times are rapidly increasing. In the scope of this thesis a time integration library is used to solve initial value problems. The aim of the thesis is to implement and investigate two different approaches to decrease the execution time for the library. The main focus of this thesis is on the Singly Diagonal Implicit Runge Kutta method and the Jacobian Free Newton’s method. The first approach is to parallelize the library. The other approach is to implement a discrete weighted norm for the error estimation in the time integration methods instead of the currently used euclidean norm. The parallelization is tested for two well-known numerical problems and the solutions obtained are compared with the expected solutions. The implementation of the discrete weighted norm is tested by comparing solutions obtained from the different norms with each other for different tolerance levels and different test cases. The intention is to investigate if the discrete weighted norm decreases the execution time without loss of accuracy in the solutions. The parallelization of the library obtains a correct solution. It can therefore be concluded that the parallelization is correct. The results from the implementation of the discrete weighted norm are not straightforward. Comparing the results from the different test cases, one can conclude that the size of the weights and the dimension of the problem are important for the accuracy of the solutions. (Less)
author
supervisor
organization
course
FMN820 20161
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Numerical analysis, Parallel programming
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFNA-3038-2016
ISSN
1404-6342
other publication id
2016:E42
language
English
id
8889731
2016-09-12 13:48:49
date last changed
2018-10-11 16:21:19
```@misc{8889731,
abstract     = {Execution time is an important issue in the field of numerical analysis. Simulations are getting more and more complex and the execution times are rapidly increasing. In the scope of this thesis a time integration library is used to solve initial value problems. The aim of the thesis is to implement and investigate two different approaches to decrease the execution time for the library. The main focus of this thesis is on the Singly Diagonal Implicit Runge Kutta method and the Jacobian Free Newton’s method. The first approach is to parallelize the library. The other approach is to implement a discrete weighted norm for the error estimation in the time integration methods instead of the currently used euclidean norm. The parallelization is tested for two well-known numerical problems and the solutions obtained are compared with the expected solutions. The implementation of the discrete weighted norm is tested by comparing solutions obtained from the different norms with each other for different tolerance levels and different test cases. The intention is to investigate if the discrete weighted norm decreases the execution time without loss of accuracy in the solutions. The parallelization of the library obtains a correct solution. It can therefore be concluded that the parallelization is correct. The results from the implementation of the discrete weighted norm are not straightforward. Comparing the results from the different test cases, one can conclude that the size of the weights and the dimension of the problem are important for the accuracy of the solutions.},
author       = {Hjälle, Matilda},
issn         = {1404-6342},
keyword      = {Numerical analysis,Parallel programming},
language     = {eng},
note         = {Student Paper},
series       = {Master's Theses in Mathematical Sciences},
title        = {Parallelization of the SDIRK and Newton’s method and analysis of a weighted norm for error estimations},
year         = {2016},
}

```