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

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.