In the last two decades, many edge detection methods have been developed and widely used in image processing for edge detection and the hybrid compact-WENO finite difference (hybrid) schemes for solving the system of hyperbolic conservation laws with solutions containing both discontinuous and complex fine-scale structures. However, many edge detection methods include the problem-dependent parameters such as the high order multi-resolution (MR) analysis (Harten, JCP, 49 (1983)). Therefore, we combined the Tukey’s boxplot method with MR analysis (Gao et al., JSC, 73 (2017)) to overcome this problem in a sense. But the Tukey’s boxplot method needs to sort the data at the beginning of Runge-Kutta time integration method, which is relatively time-consuming and inefficient. In this study, we employ the Pauta criterion and remove the problem-dependent parameters in the MR analysis. Furthermore, two new edge detection approaches, which are based on second-order central difference scheme and Ren’s idea (Ren et al., JCP, 192 (2003)), are also proposed. The accuracy, efficiency and robustness of the hybrid scheme with the new edge detectors are verified by numerous classical one- and two-dimensional examples in the image processing and compressible Euler equations with discontinuous solutions.

In this paper, a quadratic finite volume method (FVM) for parabolic problems is studied. We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme. We then employ the backward Euler method and the Crank-Nicolson method respectively to further disctetize the time vatiable so as to derive two full-discrete schemes. The existence and uniqueness of the semi-discrete and full-discrete FVM solutions are established and their optimal error estimates are derived. Finally, we give numerical examples to illustrate the theoretical results.

This article aims to identify the partial topological structures of delayed complex network. Based on the drive-response concept, a more universal model, which includes nonlinear couplings, stochastic perturbations and multi-weights, is considered into drive-response networks. Different from previous methods, we obtain identification criteria by combining graph-theoretic method and adaptive synchronization. After that, the partial topological structures of stochastic multi-weighted complex networks with or without time delays can be identified successfully. Moreover, response network can reach synchronization with drive network. Ultimately, the effectiveness of the proposed theoretical results is validated through numerical simulations.

Gas distribution function plays a crucial role in the description of gas flows at the mesoscopic scale. In the presence of non-equilibrium flow, the distribution function loses its rotational symmetricity, making the mathematical derivation difficult. From both the Chapman-Enskog expansion and the Hermite polynomial expansion (Grad’s method), we observe that the non-equilibrium effect is closely related to the peculiar velocity space $(C).$ Based on this recognition, we propose a new methodology to construct the non-equilibrium distribution function from the perspective of polynomial expansion in the peculiar velocity space of molecules. The coefficients involved in the non-equilibrium distribution function can be exactly determined by the compatibility conditions and the moment relationships. This new framework allows constructing non-equilibrium distribution functions at any order of truncation, and the ones at the third and the fourth order have been presented in this paper for illustration purposes. Numerical validations demonstrate that the new method is more accurate than the Grad’s method at the same truncation error for describing non-equilibrium effects. Two-dimensional benchmark tests are performed to shed light on the applicability of the new method to practical engineering problems.

A generalized homotopy-based Coiflet-type wavelet method for solving strongly nonlinear PDEs with nonhomogeneous edges is proposed. Based on the improvement of boundary difference order by Taylor expansion, the accuracy in wavelet approximation is largely improved and the accumulated error on boundary is successfully suppressed in application. A unified high-precision wavelet approximation scheme is formulated for inhomogeneous boundaries involved in generalized Neumann, Robin and Cauchy types, which overcomes the shortcomings of accuracy loss in homogenizing process by variable substitution. Large deflection bending analysis of orthotropic plate with forced boundary moments and rotations on nonlinear foundation is used as an example to illustrate the wavelet approach, while the obtained solutions for lateral deflection at both smally and largely deformed stage have been validated compared to the published results in good accuracy. Compared to the other homotopy-based approach, the wavelet scheme possesses good efficiency in transforming the differential operations into algebraic ones by converting the differential operators into iterative matrices, while nonhomogeneous boundary is directly approached dispensing with homogenization. The auxiliary linear operator determined by linear component of original governing equation demonstrates excellent approaching precision and the convergence can be ensured by iterative approach.

In this article, we develop a new well-balanced finite volume central weighted essentially non-oscillatory (CWENO) scheme for one- and two-dimensional shallow water equations over uneven bottom. The well-balanced property is of paramount importance in practical applications, where many studied phenomena can be regarded as small perturbations to the steady state. To achieve the well-balanced property, we construct numerical fluxes by means of a decomposition algorithm based on a novel equilibrium preserving reconstruction procedure and we avoid applying the traditional hydrostatic reconstruction technique accordingly. This decomposition algorithm also helps us realize a simple source term discretization. Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed scheme maintains the well-balanced property exactly. Furthermore, extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions at the same time.

Based on a subspace method and a linear approximation method, a convex algorithm is designed to solve a kind of non-convex PDE constrained fractional optimization problem in this paper. This PDE constrained problem is an infinite-dimensional Hermitian eigenvalue optimization problem with non-convex and low regularity. Usually, such a continuous optimization problem can be transformed into a large-scale discrete optimization problem by using the finite element methods. We use a subspace technique to reduce the scale of discrete problem, which is really effective to deal with the large-scale problem. To overcome the difficulties caused by the low regularity and non-convexity, we creatively introduce several new artificial variables to transform the non-convex problem into a convex linear semidefinite programming. By introducing linear approximation vectors, this linear semidefinite programming can be approximated by a very simple linear relaxation problem. Moreover, we theoretically prove this approximation. Our proposed algorithm is used to optimize the photonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals as an application. The results of numerical examples show the effectiveness of our proposed algorithm, while they also provide several optimized photonic crystal structures with a desired wide-band-gap. In addition, our proposed algorithm provides a technical way for solving a kind of PDE constrained fractional optimization problems with a generalized eigenvalue constraint.

In this paper, a node-based smoothed finite element method (NS-FEM) with linear gradient fields (NS-FEM-L) is presented to solve elastic wave scattering by a rigid obstacle. By using Helmholtz decomposition, the problem is transformed into a boundary value problem with coupled boundary conditions. In numerical analysis, the perfectly matched layer (PML) and transparent boundary condition (TBC) are introduced to truncate the unbounded domain. Then, a linear gradient is constructed in a node-based smoothing domain (N-SD) by using a complete order of polynomial. The unknown coefficients of the smoothed linear gradient function can be solved by three linearly independent weight functions. Further, based on the weakened weak formulation, a system of linear equation with the smoothed gradient is established for NS-FEM-L with PML or TBC. Some numerical examples also demonstrate that the presented method possesses more stability and high accuracy. It turns out that the modified gradient makes the NS-FEM-L-PML and NS-FEM-L-TBC possess an ideal stiffness matrix, which effectively overcomes the instability of original NS-FEM. Moreover, the convergence rates of $L^2$ and $H^1$ semi-norm errors for the two NS-FEM-L models are also higher.

This paper proposes a high order deep domain decomposition method (HOrderDeepDDM) for solving high-frequency interface problems, which combines high order deep neural network (HOrderDNN) with domain decomposition method (DDM). The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM, and apply HOrderDNNs to solve the high-frequency problem on each sub-domain. Besides, we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains, on the interface and the boundary during the optimization process. The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems. The results indicate that: HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems. In detail, HOrderDeepDDMs $(p>1)$ could capture the high-frequency information very well. When compared to the deep domain decomposition method (DeepDDM), HOrderDeepDDMs $(p >1)$ converge faster and achieve much smaller relative errors with the same number of trainable parameters. For example, when solving the high-frequency interface elliptic problems in Section 3.3.1, the minimum relative errors obtained by HOrderDeepDDMs $(p =9)$ are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same, as shown in Fig. 4.

In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated. To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized. The first order optimality condition of the extended optimal control problem is derived. A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed. A priori error estimates for the spectral Galerkin discrete scheme is proved. Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.

Recently, the use of porous materials has grown widely in many structures, such as beams, plates, and shells. The characteristics of porous materials change in the thickness direction by different functions. This study has investigated the free vibration analysis of a sandwich porous elliptical micro-shell with a magneto-rheological fluid (MRF) core for the first time. Initially, we examined the displacement of the middle layer’s macro- and micro-components, using Love’s shell theory. Next, we used the modified couple stress theory (MCST) to obtain the strain and symmetrical curvature tensors for the three layers. The Hamilton’s principle was implemented to derive the equations of motion. We also used the Galerkin’s method to solve the equations of motion, resulting in a system of equations in the form of a linear eigenvalue problem. By solving the governing equations, we obtained the various natural frequencies and loss factors of the elliptical micro-shell, and compared them with the results in earlier studies. Lastly, we investigated the effects of thickness, porosity distribution pattern, aspect ratio, length scale parameter, and magnetic field intensity on the natural frequency and loss factor of the micro-shell. The data accuracy was validated by comparing them with those of reputable previous articles.