计算与应用数学研讨会 | 2026年1月

应文俊

A Cartesian Grid-based Boundary Integral Method for Acoustic Scattering

上海交通大学

We present a Cartesian grid method for homogeneous and inhomogeous scattering problems on complex domains. The method is a generalization of the traditional boundary integral method. It solves the scattering problem in the framework of boundary integral method but avoids direct evaluation of boundary and volume integrals. The evaluation is done by indirectly solving equivalent interface problems on Cartesian grids with fast solvers. For (exterior) problems on unbounded domains, we introduce an artificial circle or sphere to accelerate the solution of interface problems, while preserving accuracy. In the talk, we shall also present numerical examples to demonstrate the method.

郑海标

Stokes-Darcy模型有限元-广义多尺度有限元耦合框架

华东师范大学

在实际应用中，许多工业和科学问题都呈现出多尺度特征，如复合材料的电导率、集成电路的设计以及多孔介质中的流体研究等。本研究重点探讨了广义多尺度有限元方法（GMsFEM）在Stokes-Darcy 问题中的应用。我们假设 Darcy 区域具有高对比度特征，提出了Stokes-Darcy 模型有限元（FEM）-广义多尺度有限元（GMsFEM）的耦合框架。该算法包括在线和离线两个步骤。对于稳态情况，离线阶段利用 GMsFEM 构建与渗透率相关的广义多尺度基函数，通过并行处理提高计算效率；在线阶段，采用 Robin-Robin区域分解算法求解耦合模型。针对非稳态情况，假设渗透率不随时间变化，离线阶段的基函数生成方法与稳态情况类似；在线阶段，采用隐式-显式方式解耦 Stokes-Darcy 方程。

马敬堂

Optimal investment under non-Markovian models via BSPDEs and deep learning

西南财经大学

We study optimal investment (utility maximization problems) in non-Markovian settings, where the dynamic programming principle (DPP) fails and Hamilton-Jacobi-Bellman (HJB) equations are inapplicable. Instead, backward stochastic partial differential equations (BSPDEs) can characterize the random field values of such problems. We propose iterative deep learning algorithms to solve these fully nonlinear BSPDEs and establish their convergence. Numerical experiments on rough volatility models validate the theoretical convergence of our metholds. (This is joint work with Haofei Wu and Harry Zheng)

杨海建

Parallel-in-time (PinT) algorithms for large-scale flow problems

湖南大学

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for large-scale simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this talk, we propose a family of parallel-in-time (PinT) algorithms for solving some large-scale flow problems from computational fluid dynamics or reservoir simulation, to fully exploit the parallelism of supercomputers.

邱常新

High-order SDC-Ensemble method for multirealization flows in stochastic dual-porosity-Navier-Stokes systems

宁波大学

We introduce an efficient high-order ensemble algorithm incorporating the semi-implicit spectral deferred correction (SDC-Ensemble algorithm) tailored to simulate flows exhibiting multiple realizations within the stochastic dual-porosity-Navier -Stokes system. This framework accommodates uncertainties stemming from initial conditions, forcing terms, interface boundary conditions and the hydraulic conductivity tensor. By consolidating all realizations into a unified coefficient matrix at each time step, the SDC-Ensemble algorithm significantly diminishes computational overhead compared to traditional methods that treat each realization independently, while preserving high-order accuracy. Furthermore, it disentangles the dual-porosity-Navier- Stokes system into three manageable sub-physics problems, thereby reducing the dimensionality of linear systems and facilitating parallel computation. Numerical experiments corroborate the theoretical findings and illustrate the applicability of the algorithm and its features to flow problems in multistage fractured horizontal wellbore with appropriate boundary/interface conditions.

蒋凯

丢番图误差的算法和理论

湘潭大学

实数是数学研究的基石，亦是理解自然现象的基础。实数由零测度的有理数集和满测度的无理数集共同构成。由此推断，无理数起主导作用的体系理应更为普遍。然而，由于计算机难以存储和表示无理数，在对无理数起主导作用体系进行数值计算时会产生丢番图误差，即有理数逼近无理数之误差。这类误差会对计算结果的准确性可能起决定性影响。在本报告中，我们将分析丢番图误差机理及其对数值计算的影响，建立任意维丢番图频率准周期函数的逼近理论；进而提出能避免丢番图误差的新型算法--投影法和有限点恢复法，将高精度数值计算的范围从有理数域拓展到实数域，并利用无理数的遍历性和算术性质，建立了算法的数学理论。同时，我们将展示这些算法在一些科学问题上的应用，包括准晶及其相变、准周期薛定谔算子、界面结构、准周期多尺度问题等。

王坤

Linear Maximum Bound Principle Preserving Finite Difference Schemes for the Convective Allen-Cahn Equation

重庆大学

The convective Allen-Cahn equation generalizes the classical Allen-Cahn equation by introducing an additional convective term associated with a solenoidal velocity field while maintaining the maximum bound principle (MBP). However, developing high-order numerical schemes that are accurate in both time and space and preserve the MBP unconditionally has remained a significant challenge. In this paper, we address this by first defining new auxiliary variables to reformulate the interaction of the velocity with the phase field. We then transform the convective Allen-Cahn equation into a generalized Fokker-Planck form using an exponential transformation, enabling the development of MBP-preserving linear schemes. Subsequently, we propose first- and second-order in time schemes for the reformulated equations with a second-order quasi-symmetric finite difference discretization in space. In this approach, the auxiliary variables are replaced with known functions related to the velocity, simplifying the numerical implementation. For the first-order in time scheme, we derive its optimal error estimate and its unconditional MBP-preservation. For the second-order in time scheme, we show its MBP-preservation under mild constraints on the mesh and time step sizes. Some numerical experiments in two and three dimensions are also presented to validate the theoretical findings and illustrate the accuracy and efficiency of our proposed schemes.

蒋维

数值模拟几何流的高精度保结构参数有限元方法

武汉大学

科学和工程计算中大量的界面问题都与几何流问题相关，例如平均曲率流和表面扩散流等。几何流问题一直是纯数学（尤其几何分析）领域中的研究热点，关于它的数值模拟和算法分析给传统计算数学领域带来了诸多新的机遇和挑战。本报告主要简述近年来在数值模拟几何流的高精度保结构参数有限元方法的一些进展，包括时间高阶保结构格式的设计、形状度量（流形距离）的提出思想、基于Lagrange乘子的保结构格式和参数化 LDG 格式的设计等方面。

易年余

RINN: 偏微分方程的秩启发神经网络方法

湘潭大学

PINNs和ELM等神经网络方法具有通用性强、用户友好、代码生态完善等优势，但可解释性差、存在参数敏感等不确定性。在本报告中，我们介绍基于秩启发的神经网络方法，以解决物理信息极值学习方法求解效果对权值初始化敏感的问题。秩启发神经网络算法的训练过程分两个阶段，第一阶段进行非线性优化过程训练隐藏层权值参数，使最后一层隐藏层输出函数满足正交性约束,进而增强所张成线性空间的函数表示和逼近能力；第二阶段冻结隐藏层参数，利用最小二乘法确定输出层参数来求解偏微分方程。大量的数值实验表明，与物理信息极值学习方法相比，秩启发神经网络方法显著降低了由参数初始化引起的性能差异，能保持求解的高精度。

赵泉

A stable velocity-splitting parametric finite element method for Willmore flow

中国科学技术大学

In this talk, I will introduce a new parametric finite element method for Willmore flow of hypersurfaces in a unified framework. The method is linear and employs a splitting of the normal and tangential velocity of the flow. The normal velocity is approximated via an evolution equation for the curvature, and follows the arbitrary Lagrangian-Eulerian approach. This enables an unconditional energy stability with respect to the discrete energy. We also incorporate tangential velocity through a curvature identity to preserve the mesh quality. We show various numerical examples to demonstrate the favorite properties of the introduced method.