In this paper, a linear second order numerical scheme is developed and investigated for the Allen–Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for the Allen–Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

This paper investigates the critical buckling behavior of axially functionally graded (FG) material beams with three end support conditions. The FG materials are assumed to have continuously graded based on a power-law function of the volume fractions of the constituents. The governing equation for buckling is derived and solved using the differential quadrature (DQ) method. A comparison between the results obtained from the DQ method and the analytical approach reveals excellent agreement. The effects of various parameters, such as the gradient index and boundary conditions, on the critical buckling load is thoroughly analyzed. The findings highlight the efficiency and accuracy of the DQ method for analyzing functionally graded beams. Moreover, the insights gained from this study can inform the design and optimization of functionally graded structures.

Unstructured finite volume methods are typically categorized based on control volumes into the node- and cell-centered types. Because of certain inherent geometric properties, the second-order Linear-Galerkin scheme, favored for its simplicity and ability to preserve second-order solution accuracy, is predominantly applied to node-centered control volumes. However, when directly applied to cell-centered control volumes, the designed solution accuracy can be lost, particularly on irregular grids. In this paper, the least-square based Linear-Galerkin discretization is extended to arbitrary cell-centered elements to ensure the second-order accuracy can always be achieved. Altogether four formulations of corrected schemes, with one being fully equivalent to a conventional second-order finite volume scheme, are proposed and examined by problems governed by the linear convective, Euler and Navier-Stokes equations. The results demonstrate that the second-order accuracy lost by the original Linear-Galerkin discretization can be recovered by corrected schemes on perturbed grids. In addition, shock waves and discontinuities can also be well captured by corrected schemes with the help of gradient limiter function.

Using Nehari manifold method combined with fibring maps, we show the existence of nontrivial, weak, positive solutions of the nonlinear $\psi$-Riemann-Liouville fractional boundary value problem involving the $p$-Laplacian operator, given by 

where $λ>0, 0<\gamma<1< p$ and $\frac{1}{p}<\alpha≤1,$ $g∈C([0,T])$ and $f ∈C^1 ([0,T]×\mathbb{R},\mathbb{R}).$ A useful examples are presented in order to illustrate the validity of our main results.

In this paper, we investigate the numerical stability for solving the three dimensional (3D) poroelastic wave equations with the high-order staggered-grid method. First by introducing some proper difference operators, we construct the arbitrary high-order staggered-grid schemes for 3D poroelastic wave equations with spatially varying media parameters. Then the stability condition of the schemes is derived firstly. The result is an explicit restriction of time step, which only depends on the difference coefficients and the spatially varying media parameters. The condition is sufficient and can be computed prior to numerical computations, which is very helpful for us to find suitable time step and spatial grid size in numerical computations efficiently. For numerical computations, absorbing boundary conditions with perfectly matched layer (PML) based on real prolongation variables are derived. Numerical computations of 3D poroelastic wave propagation with PML are completed. The results show the effectiveness of our developed method.

In this paper we consider the finite element approximation to an elliptic optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma).$ This problem is motivated by certain applications in optimal control of semiconductor devices where the discrepancy of the current density on the boundary to the target one is the objective to be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization, whereas the state and adjoint state variables are approximated by piecewise linear and continuous finite elements. Second order convergence rate for the optimal control is theoretically proved and verified by numerical results.

In this paper, we propose a newly designed fifth-order finite difference well-balanced mapped unequal-sized weighted essentially non-oscillatory (WBMUS-WENO) scheme for simulating the shallow water systems on multi-dimensional structured meshes. We design new non-linear weights and a new mapping function, so that the WBMUS-WENO scheme can maintain fifth-order accuracy with a small $ε$ even nearby the extreme points in smooth regions. The truncation errors of the scheme is smaller and it has better convergence in simulating some steady-state problems. Unlike the traditional well-balanced WENO-XS scheme [29], this new WBMUS-WENO scheme uses three unequal-sized stencils, denotes the linear weights to be any positive numbers on condition that their summation is one. By incorporating a quartic polynomial on the whole big stencil into WENO reconstruction, the WBMUS-WENO scheme is simple and efficient. Extensive examples are performed to testify the exact C-property, absolute convergence property, and good representations of this new WBMUS-WENO scheme.

In this paper, two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation (of order $α,$ with $0<α<2$) are developed to solve the initial value problem. Firstly, we study a direct numerical scheme that uses quadratic Charles Hermite and Newton (H2N2) interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction. We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order $3−α.$ In order to improve computational efficiency, a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed. Numerical examples confirm the error estimation results and the validity of the fast scheme.

We present a numerical study that investigates the collision between a falling droplet and a sessile droplet on a superhydrophobic surface with various impact velocities, size ratios, and eccentric angles. After impact, the two droplets may merge into a bigger droplet. We use the OpenFOAM volume of the fluid method to monitor the droplet geometry, analyze the dynamic characteristics of the impact, determine the contact time of the merged droplet on the surface, and locate the contact line of the merged droplet. When a large droplet impacts a small droplet, increasing the eccentric angle decreases both the maximum spreading distance and the contact time for all impact velocities. Additionally, it reduces the number of broken small droplets that splash following fragmentation at high impact velocity. Droplets of equal size produce impacts that are nearly the same as those of a large droplet impacting a small droplet. The only difference occurs at the moment of droplet fragmentation. When a small droplet impacts a large droplet, increasing the eccentric angle by less than $10^{\circ}$ shortens the maximum spreading distance while only slightly increasing the contact time. In addition, droplet fragmentation is significantly reduced. The results provide insight into potential applications of droplet-solid interactions, which are valuable for engineering applications such as anti-icing and self-cleaning.

Strong field concentration may occur between two nearly-touching high-contrast material inclusions due to an incident field. The degree of concentration is characterised by the blowup rate of the underlying gradient field. This phenomenon has received considerable attention in the literature due to its practical implications in the theory of composite materials. However, most of the existing studies are concerned with the static cases. In this paper, we present a comprehensive numerical investigation of this intriguing phenomenon associated to the Helmholtz system in the quasi-static frequency regime. On the one hand, we present extensive numerical results to corroborate the theoretical findings in [16], and on the other hand, we derive new findings that cannot be handled by the theoretical analysis yet. Our focus is on the static effect, frequency effect, the geometric (curvature) effect to the gradient estimates.

In this study, an efficient regularized lattice Boltzmann model aimed at solving the consistent and conservative phase-field model is developed. This model is composed of the conservative Allen-Cahn equation, the momentum equation featuring a modified mass flux, and the associated consistency conditions. Consequently, two distribution functions are introduced within the framework of the regularized lattice Boltzmann model: one dedicated to the conservative Allen-Cahn equation, and the other designed for addressing the fluid dynamics equations. In order to accurately recover the momentum equation and ensure the consistency of mass and momentum transport, a simple force distribution function with a auxiliary term is incorporated into the regularized lattice Boltzmann model. To assess the capabilities of the current regularized lattice Boltzmann model, simulations of various two-phase flow problems with substantial density ratios have been conducted, including layered Poiseuille flow and spinodal decomposition. These simulations demonstrate excellent agreement with previously published numerical results. Additionally, numerical investigations into Rayleigh-Taylor instability indicate that the present regularized lattice Boltzmann model can accurately and stably track interfaces with high precision.