• Abstract

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 $L^{\infty}$-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.

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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.

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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.

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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 $\lambda>0$, $0 < \gamma <1<p$ and $\frac{1}{p} < \alpha \leq1, $, $g \in C([0,T])$ and $f \in C^{1}([0,T]\times \mathbb{R,\mathbb{R}})$. A useful examples are presented in order to illustrate the validity of our main results.

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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.

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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.

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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 $\varepsilon$ 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.

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In this paper, two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation (of order $\alpha$, with $0<\alpha<2$) 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 $3-\alpha$. 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.

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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.

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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.

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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.

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Although the genuinely two-dimensional HLL-CPS solver holds the inherent multidimensionality property and capability of resolving contact discontinuities, the conventional low-order (second-order and below) reconstruction methods still limits its application in the two-dimensional complex flows involving shock waves and shear layers. A fifth-order reconstruction method is proposed for the genuinely two-dimensional HLL-CPS solver. The conserved variable vectors at the midpoints of interfaces are approximated by the fifth-order 1D WENO reconstruction. Meanwhile, variables at the corners are evaluated by a dimension-by-dimension reconstruction method consisting of a number of 1D fifth-order WENO sweeps. To avoid introducing spurious oscillations, each reconstruction is carried out in the corresponding local characteristic fields. Numerical results of several benchmark tests indicate  the higher-order accuracy and the multidimensionality property of the proposed scheme. Compared with the 1D HLLE, HLLC and HLL-CPS schemes, the proposed high-order genuinely two-dimensional HLL-CPS solver provides higher resolution for contact discontinuities and presents better robustness against the shock anomalies.

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In this paper, we  develop  a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity   numerical method  adopts the backward Euler scheme and  conforming simplicial finite elements  for the temporal and spatial discretizations, respectively.  Under the assumption that the time step size is sufficiently small and time steps   are not very large,  we show that the singular value distribution of the high-fidelity solution matrix $U$ is close to that of a rank one matrix.  We select the eigenfunction associated to the principal eigenvalue of the matrix $U^\top U$   as the basis of    Proper Orthogonal Decomposition (POD) method so as to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.

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We present a novel two-grid compact finite difference scheme for the viscous Burgers' equation in this paper, where the second-order Crank-Nicolson method is used to deal with the time marching, the compact finite difference formula is used to approximate the spatial second-order term, and the nonlinear convection term is discretized using the developed nonlinear fourth-order operator, providing the scheme with both high fourth-order spatial convergence and a low computational cost. The scheme is then established in three steps, with the first step being the construction of a nonlinear coarse-grid compact finite difference scheme that is solved iteratively using a fixed point iterative method, the second step being the application of the Lagrange interpolation formula to obtain a rough solution on the fine grid, and the third step being the development of the linearized fine-grid compact finite difference scheme. We also perform a convergence and stability analysis on the developed scheme, and the results show that the scheme can achieve spatial fourth-order and temporal second-order convergence. Finally, a number of numerical examples are provided to validate the theoretical predictions.

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This paper deals with an efficient two-step time split explicit/implicit scheme applied to a two-dimensional nonlinear unsteady convection-diffusion-reaction equation. The computational cost of the new algorithm at each time level is equivalent to solving a pentadiagonal matrix equation with strictly dominant diagonal elements. Such a bandwidth matrix can be easily inverted using the Gaussian Decomposition and the corresponding linear system should be solved by the back substitution method. The proposed approach is unconditionally stable, temporal second-order accuracy and fourth-order convergence in space. These results suggest that the developed technique is faster and more efficient than a large class of numerical methods studied in the literature for the considered initial-boundary value problem. Numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the performance of the constructed numerical scheme.

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Although there are many improvements to WENO3-Z with the target to achieve the optimal order in the occurrence of the first-order critical point ($CP_1$), they mainly address resolution performance, while the robustness of schemes is of less concern and lacks understanding accordingly. In light of our analysis considering the occurrence of critical points within grid intervals, we theoretically prove that it is impossible for a scale-independent scheme that has the stencil of WENO3-Z to fulfill the above order achievement, and current scale-dependent improvements barely fulfill the job when $CP_1$ occurs at the middle of the grid cell. In order to achieve scale-independent improvements, we devise new smoothness indicators that increase the error order from 2 to 4 when $CP_1$ occurs and perform more stably. Meanwhile, we construct a new global smoothness indicator that increases the error order from 4 to 5 similarly, through which new nonlinear weights with regard to WENO3-Z are derived and new scale-independent improvements, namely WENO-Z$_\text{{ES2}}$ and -Z$_\text{{ES3}}$, are acquired. Through 1D scalar and Euler tests, as well as 2D computations, in comparison with typical scale-dependent improvements, the following performances of the proposed schemes are demonstrated: the schemes can achieve third-order accuracy at $CP_1$ no matter its location in the stencil, indicate high resolution in resolving flow subtleties, and manifest strong robustness in hypersonic simulations (e.g., the accomplishment of computations on hypersonic half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well as essentially non-oscillatory solutions of inviscid sharp double cone flow at $M=9.59$), which contrasts the comparative WENO3-Z improvements.

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In this paper, we consider an implicit method for solving the nonlinear time-space fractional Ginzburg-Landau equation. The scheme is based on the $L2$-$1_\sigma$ formula to approximate the Caputo fractional derivative and the weighted and shifted Gr{\"u}nwald difference method to approximate the Riesz space fractional derivative. In order to overcome the non-local property of Riesz space fractional derivatives and the historical dependence brought by Caputo time fractional derivatives, this paper introduces the fractional Sobolev norm and the fractional Sobolev inequality. It is proved in detail that the difference scheme is stable and uniquely solvable by the discrete energy method. In particular, the difference scheme is unconditionally stable when $\gamma\leq 0$, where $\gamma$ is a coefficient of the equation.  Moreover, the scheme is shown to be convergent in $l_h^2$ norm at the optimal order of $\mathcal{O}(\tau^2+h^2)$ with time step $\tau$ and mesh size $h$. Finally, we provide a linearized iterative algorithm, and the numerical results are present to verify the accuracy and efficiency of the proposed scheme.

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A quadratic spline collocation method combined with the Crank-Nicolson time discretization of the space fractional diffusion equations gives discrete linear systems, whose coefficient matrix is the sum of a tridiagonal matrix and two diagonal-multiply-Toeplitz-like matrices. By exploiting the Toeplitz-like structure, we split the Toeplitz-like matrix as the sum of a Toeplitz matrix and a rank-2 matrix and Strang's circulant preconditioner is constructed to accelerate the convergence of Krylov subspace method like generalized minimal residual method for solving the discrete linear systems. In theory, both the invertibility of the proposed preconditioner and the clustering spectrum of the corresponding preconditioned matrix are discussed in detail. Finally, numerical results are given to demonstrate that the performance of the proposed preconditioner is better than that of the generalized T. Chan's circulant preconditioner proposed recently by Liu et al. (J. Comput. Appl. Math., 360 (2019),  pp. 138--156) for solving the discrete linear systems of one-dimensional and two-dimensional space fractional diffusion equations.

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In this paper, the superconvergence error estimate of a low-order conforming mixed finite element scheme, which is called bilinear-constant scheme, for the Stokes equations with damping is established. In terms of the integral identity technique and dealing with the damping term carefully, the superclose estimates between the interpolation of the exact solution and the finite element solution for the velocity in $H^1$-norm and the pressure in $L^2$-norm are first derived. Then, the global superconvergence results for the velocity in $H^1$-norm and the pressure in $L^2$-norm are derived by a simple postprocessing technique with an economical workload. Finally, some numerical results are presented to demonstrate the correctness of the theoretical analysis.

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This paper investigates the reduction of backscatter radar cross

section (RCS) for cavities embedded in the ground plane. It is established by placing a multilayered radar absorbing material (RAM) at the bottom of the cavities. The minimization of the backscatter RCS is formulated as a bound constrained optimization problem. A new vertical mode expansion method is developed to solve the scattering problem, and it is integrated into the gradient projection method to obtain the optimal composite coating. Numerical examples show that the method we developed is efficient and the backscatter RCS can be significantly reduced.

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