44 citations · 83 across the 10 of their papers we have counts for
7 papers · 1 filter
Stability and Robustness of Time-discretization Schemes for the Allen-Cahn Equation via Bifurcation and Perturbation Analysis
Wenrui Hao, Sun Lee, Xiaofeng Xu +1
The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper in…
Local Discontinuous Galerkin Methods for Solving Convection-Diffusion and Cahn-Hilliard Equations on Surfaces
Shixin Xu, Zhiliang Xu
Local discontinuous Galerkin methods are developed for solving second order and fourth order time-dependent partial differential equations defined on static 2D manifolds. These sch…
Energetic Variational Neural Network Discretizations of Gradient Flows
Ziqing Hu, Chun Liu, Yiwei Wang +1
We present a structure-preserving Eulerian algorithm for solving -gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algor…
Thermodynamically Consistent Diffuse Interface Model for Cell Adhesion and Aggregation
Lingyue Shen, Ping Lin, Zhiliang Xu +1
A thermodynamically consistent phase-field model is introduced for simulating multicellular deformation, and aggregation under flow conditions. In particular, a Lennard-Jones type…
High-order space-time finite element methods for the Poisson-Nernst-Planck equations: Positivity and unconditional energy stability
Guosheng Fu, Zhiliang Xu
We present a novel class of high-order space-time finite element schemes for the Poisson-Nernst-Planck (PNP) equations. We prove that our schemes are mass conservative, positivity…
An Energy Stable C0 Finite Element Scheme for A Phase-Field Model of Vesicle Motion and Deformation
Lingyue Shen, Zhiliang Xu, Ping Lin +2
A thermodynamically consistent phase-field model is introduced for simulating motion and shape transformation of vesicles under flow conditions. In particular, a general slip bound…
Neural-PDE: A RNN based neural network for solving time dependent PDEs
Yihao Hu, Tong Zhao, Shixin Xu +2
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PD…