中科院数学与系统科学研究院

数学研究所

动力系统研讨班报告

报告人：  Dr. Molei Tao

题  目：Explicit high-order symplectic integration of nonseparable Hamiltonians: algorithms and long time performance

时  间：2018.07.11（星期三）, 10:30-11:30

地  点：数学院南楼N913室

摘  要：Symplectic integrators preserve the phase-space volume and have favorable performances in long time simulations. Methods for an explicit symplectic integration have been extensively studied for separable Hamiltonians (i.e., H(q,p)=K(p)+V(q)), and they correspond to both accuracy and efficiency. However, nonseparable Hamiltonians also model important problems, such as non-Newtonian mechanics and nearly integrable systems in action-angle coordinates. Unfortunately, implicit methods had been the only available symplectic approach for general nonseparable systems. This talk will describe a recent result that constructs explicit and arbitrary high-order symplectic integrators for arbitrary Hamiltonians. These integrators are based on a mechanical restraint that binds two copies of phase space together. Based on backward error analysis, KAM theory, and some additional multiscale analysis, a pleasant error bound is established for integrable systems, and demonstrated on a conceptual example and the Schwarzschild geodesics problem. For nonintegrable systems, some numerical experiments with the nonlinear Schr？dinger equation will be discussed.