Efficient Fully Discrete Finite-Element Numerical Scheme with Second-Order Temporal Accuracy for the Phase-Field Crystal Model

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically.

A Coupled Discrete-Finite Element Method for Shear Strength Analysis of Geogrid-Reinforced Railway Ballast

This paper presents a coupled discrete-finite element method for the investigation of shear strength of geogrid-reinforced ballast by direct shear tests and pull-out tests. The discrete element method (DEM) and finite element method (FEM) are employed to simulate ballast and geogrid, respectively. Irregularly shaped ballast particles are modeled with clumps, and the nonlinear contact force model is used to calculate contact force between particles. Continuum geogrid is modeled by a two-node beam element with six degrees of freedom. A contact algorithm based on the static equilibrium is proposed at the geogrid-ballast contact surface. The simulation results indicate that shear strengths increase with the installation of geogrid. Moreover, ballast particle displacements and nominal volumetric strains are analyzed to provide a microscopic view on the mechanism of the reinforcement effect of geogrid.

A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes

We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying $$ \left\{ \begin{array}{l} \partial_t u - {\Delta} u - \nabla\cdot (u\nabla v)=0 \ \ \text{ in}\ {\Omega},\ t>0,\\ \partial_t v - {\Delta} v + v = u^p \ \ { in}\ {\Omega},\ t>0, \end{array} \right. $$ ∂ t u − Δ u − ∇ ⋅ ( u ∇ v ) = 0 in Ω , t > 0 , ∂ t v − Δ v + v = u p i n Ω , t > 0 , with p ∈ (1, 2), ${\Omega }\subseteq \mathbb {R}^{d}$ Ω ⊆ ℝ d a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn-Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity

We construct a fully-discrete finite element numerical scheme for the Cahn-Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier-Stokes equations with the Strang operator splitting method, and introduce several nonlical variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn-Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh-Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size $\sigma$ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom $N$ is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting $\sigma=N^{-1}$ leads to $O(N^3)$ operations of evaluating the temporal discretization coefficients and $O(N^2)$ memory. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only $O(\log N)$ terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of coefficients from $O(N^3)$ to $O(N^2\log N)$ and the corresponding memory from $O(N^2)$ to $O(N\log N)$. Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.