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The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in…
A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, which is second-order accurate in…
Fluid dynamics accompanies with the entropy production thus increases the local temperature, which plays an important role in charged systems such as the ion channel in biological environment and electrodiffusion in capacitors/batteries. In…
An asymptotic preserving and energy stable scheme for the Euler-Poisson system under the quasineutral scaling is designed and analysed. Correction terms are introduced in the convective fluxes and the electrostatic potential, which lead to…
In this paper, we present a first-order finite element scheme for the viscoelastic electrohydrodynamic model. The model incorporates the Poisson-Nernst-Planck equations to describe the transport of ions and the Oldroyd-B constitutive model…
We establish three partial differential equation models describing the thermodynamics of the fluid, by combining the energetic variational approach, appropriate constitutive relations, and classical thermodynamics laws. What is more, by…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide…
We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with…
To determine the electron heat flux density on macroscopic scales, the most widely used approach is to solve a diffusion equation through a multi-group technique. This method is however restricted to transport induced by temperature…
The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the…
A normal ordered exponential parametrization is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic…
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness…
We present and analyze a linearized finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor--Hood type elements…
The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non-local boundary conditions, which leads to a stochastic interpretation of the…
We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the…
Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable…
A diffusion interface two-phase magnetohydrodynamic model has been used for matched densities in our previous work [1,2], which may limit the applications of the model. In this work, we derive a thermodynamically consistent diffuse…
We consider a scalar diffusion equation with a sign-changing coefficient in its principle part. The well-posedness of such problems has already been studied extensively provided that the contrast of the coefficient is non-critical.…
Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential…
We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of…