非线性退化扩散问题的高阶松弛格式

基于正则化的高度退化扩散-反应生物膜模型的时间自适应数值解法

Time adaptive numerical solution of a highly degenerate diffusion-reaction biofilm model based on regularisation

本文基于正则化方法解决高度退化扩散-反应生物膜模型中的时间自适应数值解问题，克服了传统固定时间步长方法的局限性，研究信号扩散对生物膜中群体感应的影响。

关于退化分数阶对流扩散方程的数值方法和误差估计

On numerical methods and error estimates for degenerate fractional convection-diffusion equations

针对一大类非线性非局部可能退化的对流扩散方程，提出了一种收敛的数值方法，并开发了新的Kuznetsov型理论，得到了主导数阶数在1到2之间的最优误差估计。

Doubly Degenerate Diffuse Interface Models of Anisotropic Surface Diffusion

We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and strong anisotropies and demonstrate the capabilities of the approach for these cases numerically. The proposed model provides a variational and energy dissipative approach for anisotropic surface diffusion, enabling large scale simulations with material-specific parameters.

Approximation of surface diffusion flow: a second order variational Cahn--Hilliard model with degenerate mobilities

This paper tackles the approximation of surface diffusion flow using a Cahn--Hilliard-type model. We introduce and analyze a new second order variational phase field model which associates the classical Cahn--Hilliard energy with two degenerate mobilities. This association allows to gain an order of approximation of the sharp limit. In a second part, we propose some simple and efficient numerical schemes to approximate the solutions, and we provide numerical 2D and 3D experiments that illustrate the interest of our model in comparison with other Cahn--Hilliard models.

使用狄利克雷-内曼和内曼-内曼波形松弛算法对子扩散模型进行最优控制

Optimal Control of a Sub-diffusion Model using Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms

本文提出使用狄利克雷-内曼和内曼-内曼波形松弛算法解决子扩散模型的最优控制问题，并研究扩散系数对算法收敛性的影响。

欧拉-马尔可夫方法对具有不连续漂移和退化扩散系数的多维SDEs的收敛性

Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient

研究了具有不连续漂移和退化扩散系数的多维随机微分方程的欧拉-马尔可夫方法的强收敛性，证明了对于任意小的ε>0，收敛阶为1/4-ε，方法通过估计欧拉-马尔可夫方案与另一种数值方法之间的差异来实现。

具有多个子域的反应扩散方程含时间延迟的狄利克雷-Neumann波形松弛方法

Dirichlet-Neumann Waveform Relaxation Method with Multiple Subdomains for Reaction-Diffusion Equation with a Time Delay

本文提出了一种用于具有时间延迟的反应扩散方程的狄利克雷-Neumann波形松弛方法，通过多个子域的安排和数值实验评估其效率和有效性。

Heaviside函数驱动的退化扩散方程的数值研究

A numerical study of an Heaviside function driven degenerate diffusion equation

本文研究扩散系数为Heaviside函数的退化扩散方程，证明其等价于以目标为障碍的演化变分不等式，并采用有限差分法进行数值求解与测试。

Doubly Degenerate Diffuse Interface Models of Surface Diffusion

We discuss two doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion. Degeneracy is introduced in both the mobility function and a restriction function associated to the chemical potential. Our computational results suggest that the restriction functions yield more accurate approximations of surface diffusion. We consider a slight generalization of a model that has appeared before, which is non-variational, meaning there is no clear energy that is dissipated along the solution trajectories. We also introduce a new variational and, more precisely, energy dissipative mode…

完全非线性退化扩散的数值方法

Numerical methods for fully nonlinear degenerate diffusions

针对退化完全非线性椭圆型方程及相关传输型自由边界问题，提出有限差分方法，通过正则化程序克服困难并证明收敛性，经数值实验支撑，方法可扩展至更广泛非变分问题。

A numerical analysis focused comparison of several Finite Volume schemes for an Unipolar Degenerated Drift-Diffusion Model

In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species $c$ and the chemical potential $h$ is $h(c)=\log \frac{c}{1-c}$. We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.

相场模型中退化扩散性不足以捕捉表面扩散

Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion

研究指出，退化Cahn-Hilliard方程在长时行为中出现粗化现象，揭示了相场模型在表面扩散建模中的局限性。

A sturcture-preserving, upwind-SAV scheme for the degenerate Cahn--Hilliard equation with applications to simulating surface diffusion

This paper establishes a structure-preserving numerical scheme for the Cahn--Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn--Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under $H^{-1}$-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly …