Lie group and exponential integrators: Theory, implementation, and applications
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This PhD-thesis contains an introduction and six research papers sorted chronologically, of which the first four are accepted for publication. The introduction aims at giving a very brief summary of the background theory needed for the following papers. Also, some motivation of the issues addressed by the papers is given. Paper I discusses algebraic structures of ordered rooted trees and their applications to Lie group integrators. Results from Hopf algebra theory on elementary differentials for Lie group integrators are used, and applications to order analysis and backward error analysis are given. Paper II, III, IV, and V are primarily on exponential integrators, a class of numerical schemes tailored the solution of stiff systems of systems of ordinary differential equations. Paper II discusses classical order analysis and gives some theoretical results on the form of the integrators, applicable for the construction of high order exponential integrators. Paper III is on an implementation of exponential integrators on computers, and source code, available electronically, accompanies the paper. Paper IV includes an analytical and numerical study of the performance of two classes of exponential integrators on the nonlinear Schrödinger equation. Paper V is a numerical study of behaviour over long integration invervals on the nonlinear Schrödinger equation, using nonlinear spectral theory for determining validity of the numerical solution and thereby jugdging the numerical integrators. At last, in Paper VI, properties of a class of exponential like functions, essential in exponential integrators, are derived, using an approach based on Lie group theory.
Has partsBerland, Håvard; Owren, Brynjulf. Algebraic structures on ordered rooted trees and their significance to Lie group integrators. Group Theory and Numerical Analysis - CRM Proceedings & Lecture Notes: 49-63, 2005.
Berland, Håvard; Owren, Brynjulf; Skaflestad, Bård. B-series and Order Conditions for Exponential Integrators. SIAM Journal on Numerical Analysis. 43(4): 1715-1727, 2005.
Berland, Håvard; Skaflestad, Bård; Wright, Will M. EXPINT---A MATLAB package for exponential integrators. ACM Transactions on Mathematical Software (TOMS). 33(1), 2007.
Berland, Håvard; Owren, Brynjulf; Skaflestad, Bård. Solving the nonlinear Schrödinger equation using exponential integrators. Modeling, Identification and Control (MIC). 27(4): 201-218, 2006.
Berland, Håvard; Islas, Alvaro L; Schober, Constance M. Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation. Journal of Computational Physics. 225(1): 284-299, 2007.
Berland, Håvard. Generalized affine groups in exponential integrators. Preprints Numerics no. 2/06, Department of Mathematical Sciences, Norwegian University of Science and Technology(2), 2006.