Computational solutions of differential equations are essential in natural sciences and technology. This project focuses on deep connections between two rather complementary approaches to geometric integration methods by combining advanced differential geometry and modern algebraic combinatorics. The partnership will develop algorithms and clarify mathematical theories. The University of Bergen, donor partner, and the Carlos III University of Madrid (ICMAT Institute of Mathematical Sciences Department), are both internationally well recognized for their expertise in mathematical research. The donor partner will contribute its know-how and many years of experience in algebraic combinatorics and research in discrete mechanics. The methods and techniques that are going to be combined in this interaction between the geometric, algebraic and computational approaches will most likely lead to significant advances in research on geometric integration methods. Through seminars, advanced courses and writing of a research monograph, new results will be communicated to a broader research community in the two countries.
Summary of project results
In the last years, the collaboration between the University of Bergen and the ICMAT has been strengthened. K. Ebrahimi-Fard (ICMAT) and H. Munthe-Kaas (Bergen) have cosupervised together one PhD thesis. Both researchers are further developing in an ongoing collaboration the algebraic, geometric and combinatorial structures underlying general Lie Group integrators. Apart from this strong link, D. Martín de Diego and M. Barbero Liñán have started to work in geometric integrators for nonholonomic systems and control systems. On the other hand the group at University of Bergen is well-known for developing Runge-Kutta-Munthe-Kaas methods to integrate differential equations in the framework of modern algebraic combinatorics. To work together on this research line will definitely have a great impact on the development of geometric integrators. As a result of the collaboration, several seminars have been held and various papers have been prepared and submitted to international journals. Participants are preparing a volume of research papers that will be published in 2016 in Springer’s prestigious Proceedings in Mathematics and Statistics. The volume includes both papers by researchers at University of Bergen and at the ICMAT, plus high quality contributions from several other participants of the “Brainstorming Workshops on New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series”, partially funded also by NILS programme. The research has focused on constructing intrinsically Lie-Poisson integrators that preserve the integrability of the systems in the discrete setting. One of the papers in the above mentioned volume studies this topic generalizing the construction by Y. Suris in his book in 2003. Two other papers in the volume are research surveys on Lie-Poisson integrators and on the geometry and algebra of Butcher and Lie-Butcher series. The fourth paper by the members of this research project is on the Magnus expansion and Post-Lie flow equations. Moreover, the following research lines are ongoing projects, and will eventually lead to further publications. Among others: K. Follesdal and H. Munthe-Kaas are preparing a series of articles on algebraic and geometric structures of symmetric spaces and related aspects in numerical methods. E. Celledoni, E.H.Hoiseth (NTNU), M. Farré Puiggalí and D. Martin de Diego (ICMAT) have prepared a paper on energy-preserving integrators for noholomic mechanical systems.
Summary of bilateral results
This project has strengthened the links between the Norwegian and Spanish institutions, not only the ones involved in the project. Developments during recent years made it clear, that there exist common research interests between the institutions from both countries on geometric integrators, algebraic combinatorics and discrete mechanics. Collaborations with the University of Bergen and the Norwegian University of Science and Technology are going to be very fruitful, due to the fact that the latter institutions already have ongoing collaborations with industrial partners focusing on a wide range of real world applications. The Spanish partner may contribute to these activities with a thorough understanding of the underlying theoretical notions, which might lead to improve as well as more efficient geometric integrator methods applicable in the context of applications relevant to real world problems. The Project results will have a great impact on developing efficient geometric integrators. It is already known in the literature that constructing numerical methods, preserving geometric properties, guarantees the good performance of those methods. The main beneficiaries of this project are research engineers at universities and companies in need of efficient numerical methods to solve real-world problems. The algorithms to be obtained and the mathematical theories will lead to the development of useful software packages. Partners expect to continue collaborating, among others, as follows: - University of Bergen (Norway), ICMAT (Spain) on Lie Poisson integrators (in preparation), on Lie triple systems/symmetric spaces. - Norwegian University of Science and Technology (Norway), ICMAT (Spain) on geometric integrators and stochastic dynamics. - Universidad Carlos III de Madrid (Spain), University of the Basque Country, San Sebastian (Spain), ICMAT (Spain) on stochastic differential equations and formal series expansions. - Old Dominion University, ICMAT (Spain) on discrete-time Fliess operators and efficient approximation methods.