韩道志 | 计算地球物理流体模型稳态统计性质的高效数值方法

Dr. Han is an Associate Professor of Mathematics at the State University of New York at Buffalo. He received his Ph.D. in Applied and Computational Mathematics from Florida State University in 2015 and then held a Zorn Postdoctoral Fellowship at Indiana University. His research focuses on numerical analysis and scientific computing for partial differential equations arising in fluid mechanics. His work has been supported by the National Science Foundation and the Simons Foundation.

We introduce a highly efficient, second-order time-marching scheme for nonlinear geophysical fluid models, designed to accurately approximate invariant measures - that is, the stationary statistical properties of the underlying dynamical system. Beyond second-order accuracy in time, the scheme is particularly well suited for long-time simulations due to two key features: (i) it requires solving only a fixed symmetric positive-definite linear system with constant coefficients at each step, and (ii) it guarantees long-time stability, producing uniformly bounded solutions in time for any bounded external forcing, regardless of initial data. We rigorously prove convergence of both global attractors and invariant measures of the discrete system to those of the continuous model in the vanishing time-step limit. Finally we discuss recent progress in the design of ensemble version of the method, and some challenges in the computation of invariant measures.