插值热方程和波动方程的积分微分方程的数值解

Numerical solution of the heat conduction problem with memory

It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conductivity processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally si…

热方程时空离散化中的小波压缩修正希尔伯特变换

Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation

本文针对热方程时空离散化中的修正希尔伯特变换，构造样条小波 Riesz 基实现时间方向多级分解与算子压缩，并通过代数张量积获得稀疏时空空间，提出线性复杂度预条件子，显著降低了自由度与计算时间。

On the discrete equation model for compressible multiphase fluid flows

The modeling of multi-phase flow is very challenging, given the range of scales as well as the diversity of flow regimes that one encounters in this context. We revisit the discrete equation method (DEM) for two-phase flow in the absence of heat conduction and mass transfer. We analyze the resulting probability coefficients and prove their local convexity, rigorously establishing that our version of DEM can model different flow regimes ranging from the disperse to stratified (or separated) flow. Moreover, we reformulate the underlying mesoscopic model in terms of an one-parameter family of PDE…

潜空间动力学识别算法在侵入式与非侵入式降阶建模中的综合评述

A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling

本文综述了潜空间动力学识别（LaSDI）算法，用于侵入式与非侵入式降阶建模，通过将PDE数据转换为ODE潜空间数据，并引入热力学约束、弱形式、主动学习及高斯过程等策略，在多个物理问题上实现高精度与数千倍加速。