The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学重点实验室）
Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method
Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（法国国家信息与自动化研究院南锡-大东部中心）；Inria Paris（法国国家信息与自动化研究院巴黎中心）；Université Paris-Est, CERMICS (ENPC)（巴黎东大学，CERMICS（国立路桥学校））；Laboratoire de Mathématiques Raphaël Salem, Université de Rouen（鲁昂大学拉斐尔·萨莱姆数学实验室）
