Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting

地   点：  腾讯会议APP4()

报告人：  商晓成

单   位：  英国伯明翰大学

邀请人：  杨志国

备   注：  线上腾讯会议： 会议号:774375728 密码:771376 报告人简介： Xiaocheng Shang is a Lecturer in Mathematics and Statistics at the University of Birmingham, he is also a Fellow of The Alan Turing Institute, the UK's national institute for data science and artificial intelligence. His primary research interests lie in the optimal design of numerical methods for dissipative and stochastic dynamics with a strong emphasis on applications ranging from computational mathematics, statistics, physics, to data science.

报告摘要：  

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a framework to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.