44 citations · 83 across the 10 of their papers we have counts for
10 papers
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…
Three-dimensional Multiscale Model of Deformable Platelets Adhesion to Vessel Wall in Blood Flow
Ziheng Wu, Zhiliang Xu, Oleg Kim +1
When a blood vessel ruptures or gets inflamed, the human body responds by rapidly forming a clot to restrict the loss of blood. Platelets aggregation at the injury site of the bloo…
Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes
Zhiliang Xu, Dinshaw Balsara
In this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes.…
Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes -- Comparison with Runge-Kutta Methods
Dinshaw S. Balsara, Chad Meyer, Michael Dumbser +2
ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update ca…