北京雁栖湖应用数学研究院

This work is concerned with the uniform accuracy of implicit-explicit methods for general linear hyperbolic relaxation systems satisfying the structural stability condition . We prove the uniform stability and accuracy of a class of implicit-explicit backward differentiation formulas (IMEX-BDF) and implicit-explicit Runge-Kutta (IMEX-RK) schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory.

任意拉格朗日-欧拉（ALE）方法在多介质流数值模拟中有着广泛的应用，设计高精度且保物理特性的ALE方法是计算流体力学中具有挑战性的重要课题。间接ALE方法包括三个步骤：拉格朗日步、网格重分步和物理量重映步。本报告中，我们将介绍有限体积和间断有限元框架下的高阶保正守恒重映方法的研究进展。进一步结合保持界面的网格重分策略与高精度保正拉格朗日格式，构建了高阶保正守恒ALE方法。数值算例验证了上述方法的有效性。

We shall discuss the methodology of heterpgeneous multiscale metod (HMM) applying to multiscale partial differential equations, which is a framework for constructing and analyzing multiscale method. The focus is to design an efficient method to achieve high accuracy, which is based on an online-offline strategy. Arbitrary high accuracy may be achieved for deriving the macroscopic information. We shall also discuss HMM for the strain gradient elasticity model for heterogeneous media, which is a representative for the higher order elliptic system. This is a joint work with Yulei Liao (National University of Singapore) and Si Qi Song (AMSS).

We propose a series of temporal high-order parametric finite element methods to simulate geometric flows. Particularly, for those flows with multiple geometric structures, e.g., surface diffusion which decreases the area and preserves the volume, we propose a type of structure-preserving methods by incorporating two scalar Lagrange multipliers and two evolution equations involving the area and volume, respectively. These schemes can effectively preserve the structure at a fully discrete level. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of the surface diffusion.

北太天元科学计算与系统仿真软件是在北京大学数学科学学院、北京大学大数据分析与应用技术国家工程实验室、北京大学重庆大数据研究院的指导和支持下，由北京大学重庆大数据研究院基础软件科学研究中心自主研发的国内首款具有完全自主知识产权的科学计算平台。该软件聚焦科学计算领域"卡脖子"问题的解决，实现了科学计算领域根技术的突破。其具备强大的底层数学函数库，可提供科学计算、可视化、交互式程序设计功能，支持数值计算、数据分析、数据可视化、数据优化、算法开发等场景，并通过SDK与API接口，扩展支持各类学科与行业应用。目前软件已发布2025版本，已有300多所高校开展试用，提供6次学会大型赛事支持，获得中央电视台、重庆新闻联播等媒体宣传报道。本次报告将介绍其核心算法、技术与功能，并探讨未来可能的发展方向。