Xiaoming Wang, Professor, Eastern Institute of Technology, Ningbo

Haijun Yu, Professor

A Highly Efficient Second-Order Scheme for Capturing Long-Time Statistical Behavior in Geophysical Fluid Models

10:00-11:00 May 23(Friday), N219

Since Kolmogorov’s foundational work, it has been understood that the physical laws governing turbulent and chaotic systems often emerge through their statistical properties. Invariant measures, when they exist, encode the long-time statistical behavior of such systems. However, accurately and efficiently approximating these measures—especially in high-dimensional settings—remains a major computational challenge. Many classical numerical methods that excel in finite-time simulations fail to preserve the correct long-time dynamics.

In this talk, I will introduce a novel, highly efficient, second-order accurate numerical scheme that combines a mean-reverting Scalar Auxiliary Variable (mr-SAV) formulation with the BDF2-Gear approach. The scheme is designed for a class of nonlinear models arising in geophysical fluid dynamics. Notably, it requires solving only a fixed symmetric positive definite linear system at each time step and guarantees uniform boundedness of the numerical solution under bounded external forcing.

We show that for a family of finite-dimensional nonlinear geophysical models—including the barotropic quasi-geostrophic model in a channel—the proposed scheme is a small perturbation of the classical IMEX-BDF2-Gear method in the forcing term. This induces only a modest shift in the generalized Grashof number and allows us to prove convergence of the global attractors and invariant measures of the discrete system to those of the continuous one, thereby preserving its long-time statistical structure.

Applications to the Lorenz 96 model highlight the long transients often required to reach statistical equilibrium—an issue of central importance in climate modeling. Further application to the 2D Navier–Stokes equations at moderate Reynolds numbers illustrates intermittent behavior, shedding light on challenges in representing variability in geophysical flows.

报告人简介：王晓明教授本科及硕士阶段就读于复旦大学数学系，后赴美在印第安纳大学伯明顿分校获得应用数学博士学位，并在纽约大学库朗数学科学研究所完成博士后研究。2024年起，王教授加入东方理工大学，担任创校讲席教授。在此之前，他曾在多所高校担任终身教职，包括密苏里科技大学首任 Havener 讲席系主任、南方科技大学讲席教授、复旦大学特聘教授、佛罗里达州立大学教授，以及爱荷华州立大学终身副教授。王教授同时也是国家特聘专家。

王晓明教授长期致力于现代应用数学前沿研究，主要方向包括流体动力学、地下水流动、地球物理流体力学、湍流与气候变化等。已在《Communications on Pure and Applied Mathematics》（CPAM）等高质量期刊发表学术论文一百余篇，并由剑桥大学出版社出版学术专著一本。