Comparative Analysis of Adaptive Domain Decomposition Algorithms for a Time-Spectral Method

Sammanfattning: Time-spectral solvers for partial differential equations (PDE) have been explored in various forms during the last few decades. The generalized weighted residual method (GWRM) is one such method with a high accuracy and efficiency. The GWRM has so far been implemented almost exclusively with a uniform grid of subdomains in the spatial domain. Recent research has indicated that an adaptive grid can yield a significant improvement in accuracy and efficiency of the GWRM. In this thesis a comparison is performed between a uniform grid and three different adaptive grid decomposition methods. Three initial- value PDEs are used to benchmark these methods; the one-dimensional Burger’s equation, the 4th order Fisher-Kolmogorov equation and the non-linear Schrödinger equation. It was found that the average adaptive algorithm is the most efficient out of the algorithms evaluated in this thesis. The average adaptive algorithms solution time was up to 1.6 times faster than the uniform algorithm when solving the Fisher-Kolmogorov equation and with an error up to a factor of 22.5 smaller than the uniform algorithm when solving the one- dimensional Burger’s equation. The uniform algorithm needed 25 spatial subdomains to get errors of the same order of magnitude as the average adaptive algorithm got using only 12 spatial subdomains. The average subdomain decomposition algorithm is a fast, robust and efficient method, which can be applied to a variety of different problems to further increase the efficiency of the GWRM.