The Fast Multipole Method (FMM) is a highly efficient numerical algorithm for handling large-scale multi-particle systems, playing an important role in fields such as molecular dynamics, astrodynamics, acoustics, and electromagnetics. This paper first reviews the history of the Fast Multipole Method, then taking Helmholtz and Maxwell equations as examples, introduces the data structures, mathematical principles, implementation steps, and complexity analysis of the FMM based on kernel analytical expansion in two-dimensional and three-dimensional cases, and describes corresponding adaptive version of FMM. Finally, numerical experiments on multi-particle simulations in two-dimensional and three-dimensional spaces are given on the MATLAB platform.

The Bordered Block Diagonal (BBD) method is a classical approach for solving the largescale sparse linear equation systems generated in transient analysis of circuits simulations. In this paper, a new BBD method is proposed, which improves upon the traditional BBD method by addressing the issue of load imbalance through a combination of basic column decomposition and pipelined decomposition. During the matrix boundary decomposition, the introduction of pipelined decomposition overcomes the difficulty in parallelizing boundaries in traditional methods. By solving the large-scale sparse linear equations generated from 16 real-world circuits, we have verified the effectiveness of the improved BBD method. Compared to the traditional BBD method, the improved method has certain improvements in solution speed with various numbers of parallel threads.

In this paper, based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we construct a lopsided variant of the PMHSS iteration method, i.e., LVPMHSS iteration method, to solve the equivalent complex linear system. The convergence conditions of the LVPMHSS iteration method are proposed. By using a special preconditioning matrix, we not only give the detailed theoretical analysis about the spectral properties of the preconditioned matrix, but also obtain the quasi-optimal iterative parameters by minimizing the spectral radius of the iteration matrix. The results of the numerical experiments also illustrate the feasibility and efficiency of the new algorithm.

The article consider hybrid high-order methods (HHO) for the p-Laplace problem when 1$ < p < \infty$. The approximation by HHO methods utilizes a reconstruction of the gradients with piecewise Raviart-Thomas finite elements on a regular triangulation without stabilization. Using high-order gradient $\mathbf{Ru_{h}}$ for local gradient reconstruction in piecewise Raviart Thomas finite element space instead of gradient $\mathbf{Dv}$. From the perspective of energy, we perform gradient reconstruction on the minimum value of discrete energy, and determine the discrete stress in a new framework of distance. The main results are the a priori and a posteriori error estimates with global upper bound and global lower bound. Numerical benchmarks display higher convergence rates for the HHO method.

This paper considers the nonseparable nonconvex nonsmooth minimization problem, whose objective function is the sum of a proper lower semicontinuous biconvex function of the entire variables, and two nonconvex functions of their private variables without the global Lipschitz gradient continuity. This paper develops a general inertial alternating structureadapted proximal gradient descent algorithm (GIASAP for short), which not only adopts nonlinear proximal regularization and inertial strategies, but also utilizes constant and dynamical step sizes. The worst case O(1/k) nonasymptotic convergence rate of GIASAP algorithm is established. Furthermore, the bounded sequence generated by GIASAP globally converges to a critical point under the condition that the objective function possesses the Kurdyka-Łojasiewicz property. In addition, numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

The Two-Step Backward Difference Formula (BDF2) of variable-step in time has exceptional stability, making it an excellent choice for handling stiff problems and multi-scale dynamics issues. However, there is limited research on the optimal control of Partial Differential Equations. This paper introduces a variable-step method to solve the optimal control problem of source term control for a class of reaction-diffusion equations. Specifically, the BDF2 scheme is employed in time, while in space, we utilize the center-difference method for variable-step difference scheme in the L² norm, provided that the ratio of adjacent time-steps falls within the range of $\frac{1}{4.8645}$ to 4.8645. Furthermore, it achieves second-order convergence accuracy in both time and space. Finally, two numerical examples are provided to validate the feasibility and effectiveness of the proposed scheme.

Immersed finite element methods are a group of effective numerical methods for solving interface problems using unfitted meshes. Currently, there are many works on immersed finite element methods for solving interface problems with traditional interface jump conditions. However, there is limited research on interface problems with Robin type jump conditions. In this paper, an immersed finite element method is proposed for solving one-dimensional interface problems with Robin-type jump conditions. The optimal approximation properties and the optimal convergence of the proposed immersed finite element method are proved rigorously. Some numerical examples are provided to validate the theoretical results.

In this paper, a monotone coordinate descent algorithm for solving absolute value equations is presented, and the global convergence of the algorithm is analyzed under appropriate conditions. The feasibility and effectiveness of the proposed algorithm are verified by numerical experiments. Another purpose of this paper is to point out a mistake in the paper by Noor et al. [Optim. Lett., 6:1027-1033, 2012], which is caused by misuse of the second-order Taylor expansion in constructing the descending direction of the objective function.

Floquet transform is a mathematical tool for studying operators with periodic translation invariance. This paper discusses the basic mathematical properties of quantum eigenvalue problems of periodic systems from this perspective. The Bloch function is obtained by the Floquet transformation, and the Wannier function is defined by the inverse of Floquet transformation. In this process, the operator H(k) is proved for square-integrable functions of periodic units. The self-adjoint and resolvable set compactness, Wannier function as $L^2\left(\mathbb{R}^d\right)$ Orthogonality and completeness of the basis. The continuous differentiability of the isolated energy band with respect to k is also proved. The smoothing of non-isolated band groups is introduced. Finally, based on the Floquet transform of Wannier function, the interpolation calculation of energy band is introduced.

Under the condition that the objective function satisfies the Lojasiewicz property, the rates of convergence of the proximal point algorithm on general manifolds are established. The results are new on Riemannian manifolds and improve the corresponding ones in Euclidean space settings.

In this paper, we introduce a line search criterion and propose a Bregman proximal gradient algorithm with a inertial term to solve a class of nonconvex composite optimization problem, where the objective functions are the sum of a relatively smooth loss function and a nonsmooth regular function. Under the assumption of the generalized concave Kurdyka-Łojasiewicz (KL) property, the global convergence of the algorithm is proved. The numerical results on image restoration and nonconvex sparse approximation with l_1/2 regularization are reported to demonstrate the effectiveness and superiority of the inertial Bregman proximal gradient algorithm.

Radial basis function neural network (RBFNN) is a method applied to interpolation and classification prediction. In this article, we propose an improved algorithm for the RBFNN, based on the singular value decomposition (SVD) technique, in order to greatly simplify the network structure. In particular, the proposed algorithm is able to automatically choose core neurons in the hidden layer, while deleting redundant ones, which can therefore save the CPU memory and computational cost. Meanwhile, we propose to use the $K$-fold cross validation method to determine the radial parameter $\varepsilon$ in RBF, to keep the algorithm accuracy. More importantly, there is no need to load all the sample data into the CPU memory. Instead, we propose to load and deal with the sample data row by row, based on the approximate SVD algorithm proposed by Halko in [2]. All numerical experiments show that, our proposed algorithm greatly improve the computational efficiency and simplify the RBFNN structure, compared to the traditional RBFNN, while not losing the computational accuracy.

In this paper, for a class of linear equations with block 2×2 structure, the preconditioning techniques of two kinds of Schur complement matrices and their relations are discussed. We also get a new structure-constrained preconditioner in the derivation process, which possesses both theoretical advantages and computing advantages. By minimizing the spectral clustering of the preconditioned matrices, we obtained two kinds of effective parameter selection strategies and exact eigenvalue distribution of the preconditioned matrices. Also, we proved that under certain special conditions, the preconditioned technologies based on Schur’s complement approximation can be further improved and optimized. At the same time, the effects of these two kinds of Schur approximate matrices and their applications are compared. Finally, a general, reliable and effective preprocessing technique is summarized, which is applied to the three most effective preconditioners at present. Several numerical examples show that the theoretical analysis is convincing, and the effectiveness of the optimized preconditioners are also verified.

Multidimensional scaling (MDS) is a statistical method used to analyze and visualize the similarity or distance relationships between data points. It represents the relative distances or similarities between data points by mapping them to coordinates in a low-dimensional space. The classical solution to the multidimensional scaling problem involves a double centering process on the squared (non-Euclidean) distance matrix, followed by truncating the eigenvalue decomposition to seek a low-dimensional approximate configuration of points. In this paper, we directly fit the squared distance matrix and reformulate the reconstruction problem as a constrained matrix optimization model in the product manifold composed of zero column and column orthogonal matrices and diagonal matrices. By leveraging the geometric properties of the product manifold and incorporating an extended Riemannian gradient descent algorithm based on Zhang-Hager technique, we design a class of adaptive problem models. Numerical experiments demonstrate that direct fitting yields a smaller error in fitting the Euclidean distance matrix. Moreover, the proposed algorithm exhibits certain advantages in terms of iterative efficiency compared to existing projection gradient flow algorithms and first-order and second-order Riemannian algorithms in the Riemannian optimization toolbox.

In this paper, combined with the relaxation algorithm, we present new tensor splitting methods for solving multilinear PageRank problem. The convergence analysis of the proposed algorithms is also shown. It is shown that the proposed algorithms perform well from some numerical experiments when relaxation parameters are properly selected.

In this paper, we consider a numerical approximation for phase field model of nematic liquid crystal and viscous fluids. An equivalent model of the phase field model of nematic liquid crystal and viscous fluid is obtained based on the convex splitting strategy of the Ginzburg-Landau functional. In the numerical scheme, the backward Euler method is used for temporal discretization, and the hybrid finite element method is used for spacial discretization. Here the pressure correction method is used to decouple the computation of the pressure from that of the velocity. Hence, a new first-order scheme is proposed. This proposed scheme is unconditionally stable, as rigorously proven by theoretical analysis. In addition, numerical simulations are given on the temporal convergence rates with different parameters, the spacial convergence rates with different parameters, the evolution of energies, and the annihilation of singularities for the variables d, u, φ. Ample numerical simulations are performed to validate the accuracy and efficiency of the proposed scheme.

This paper considers a special nonconvex optimization problem, namely DC optimization problem, whose objective function can be written as the sum of a smooth convex function, a proper closed convex function and a continuous possibly nonsmooth concave function. This paper develops a general inertial proximal DC algorithm (GIPDCA), which adopts three different extrapolation points for the inertial direction and the gradient center and the proximal center in solving subproblems based on the classical proximal DC algorithm. The proposed GIPDCA can include some classical algorithms as special cases. Under the assumption that the objective function satisfies the Kurdyka-Łojasiewicz property and some suitable conditions on the parameters, we prove that each bounded sequence generated by GIPDCA globally converges to a critical point. In addition, numerical simulations demonstrate the feasibility and effectiveness of the proposed approach.

The numerical methods of highly nonlinear stochastic differential equations can be divided into two types: explicit methods and implicit methods. In general, the explicit method has cheap computational cost but the stable property is bad, in contrast, the implicit method has good stable property but computational cost is expensive. In this paper, we present the implicit partially truncated Euler-Maruyama method and prove that it is strongly convergent and stable in mean-square sense. In addition, the obtained results show that the presented method has approximate computational cost and better stable property compared with the explicit partially truncated Euler-Maruyama method for the case that the drift coefficient contains a linear function, that is, it posses concurrently the merit of explicit and implicit methods.

The Multiple-sets Split Feasibility Problem (MSSFP) is an extension of the Split Feasibility Problem and found applications in many practical problems, such as, image reconstruction and phase recovery. Based on selection techniques, Yao et al. [Optimization,2020,69(2): 269-281] proposed two projection algorithms (SPA) for solving MSSFP in Hilbert space. In this paper, we modify the step-size parameter of SPA and present two modified inertial projection algorithms (MISPA) for solving MSSFP. The weak and strong convergence of MISPA are established, respectively, whenever the solution set of MSSFP is nonempty. Numerical experiments are used to show the feasibility of MISPA. Moreover, inertial technique can be used to accelerate SPA.

In this manuscript we proposed using the implicit-explicit splitting method to solve the linear complementarity problem satisfied by American options in financial option pricing problems. Although implicit-explicit methods have been widely used in jump-diffusion models, they are mostly applied in European options, and there is little stability analysis in numerical solutions for American options. In this paper, we proposed that in terms of time, we adopted three discretization methods: the implicit-explicit Backward differential formula of order two (BDF2), the implicit-explicit Crank-Nikolson Leap-Frog(CNLF), and the implicit-explicit Crank-Nikolson AdamBashforth(CNAB), and proved their stability. In space, finite difference discretization is presented, and due to the nonsmoothness of the initial value function, a local mesh refinement strategy is considered near the strike price to improve accuracy. To verify the theoretical results, numerical results for pricing American options under Merton type and Kou type jump-diffusion models were presented. The numerical experimental results show that our proposed method is stable and effective.

This study focuses on the numerical solutions of the delayed Fisher’s equations by a class of non-negativity-preserving finite difference methods (FDMs) and a kind of maximumprinciple-satisfying FDMs. At first, by using a class of weighted difference formulas and explicit Euler method to discrete the diffusion term and first-order temporal derivative, respectively, a class of non-negativity-preserving FDMs are established for the delayed Fisher’s equations. Secondly, by applying cut-off technique to adjust the numerical solutions obtained by non-negativity-preserving FDMs, a kind of maximum-principle-satisfying FDMs are developed for the delayed Fisher’s equations. Thirdly, by using the non-negativity and boundedness of numerical and exact solutions, the maximum norm error estimations and stabilities for them are given, rigorously. Numerical results confirm the correctness of theoretical findings and the efficiency of the current methods.

With the rise of the AI technology revolution represented by ChatGPT, data-center AI research is rapidly emerging. Data analysis techniques including linear separability have received increasing attention from researchers. Linear separability is a fundamental mathematical problem in data analysis, but in the current big data era, an efficient method for testing linear separability is still an unsatisfied demand. This paper proposes and proves a sufficient and necessary condition for the linear separability between a point and a set based on the sphere model; and based on this necessary and sufficient condition, a parallel rapid preliminary screening method for determining the linear separability between two sets is further proposed and demonstrated. The advantages of the method proposed in this paper are: (1) its inherent parallelization properties enable low time complexity in implementation and more efficiency compared to the existing methods; and (2) the universality of the parallel framework. Any method for determining linear separability can be accelerated using the parallel framework described in this paper. The verification experiments based on benchmark data sets and artificial data sets in this paper also fully demonstrate the accuracy of the method of this paper and the efficiency in implementation.

In this paper, we established a new non-convex optimization based on L_*-L_F for low Tucker rank tensor completion problem. Three algorithms for solving the new optimization are designed based on the augmented Lagrange multiplier methods. In theory, we analyze the global convergence of the algorithms. In numerical experiment, the simulation data and real image inpainting for the new non-convex optimization and the traditional convex optimization based on nuclear norm are carried out. Experiments results show the new model outperform the nuclear norm model in CPU times under the same precision.

he purpose of this paper is to reconstruct the diffusive viscous wave equation with timevarying sources. The source can be divided into an unknown temporal part and a known spatial part. The unknown part is determined by additional detection values within a nonglobal scope. We propose a source reconstruction method based on the additional detection values and prove the existence and uniqueness of the weak solution. Finally, the theoretical results are verified through numerical examples.

In this paper, we consider the generalized double dimensional inverse eigenvalue problem for a kind of pseudo-Jacobi matrices, which is reconstructed from the eigenvalues of these matrices and their $r$×$r$ leading principle submatrices. The eigenvalue distribution of this kind of matrices is related to the size relationship between the eigenvalues of two complementary principle submatrices. When the size relationship is different, the eigenvalue distribution of this kind of matrices will change greatly. Therefore, the eigenvalue distribution of these matrices is discussed according to the distribution of the root of the secular equation, and the necessary and sufficient conditions for the problem to have a solution are given. Then the problem is solved by equivalently converting such a problem into the $k$ problem proposed by Erxiong Jiang. Finally, two numerical examples are given to verify the effectiveness and feasibility of the proposed algorithm.

A new high-order well-balanced finite difference scheme based on weighted compact nonlinear scheme (WCNS) is proposed for the Euler equation with gravitational source on the generalized coordinate system. The basic idea is to reconstruct the gravitational source term using steady-state solution, so that it can correspond to the pressure gradient at the lefthand-side of the equations in an equilibrium state. To ensure that the reconstructed value of the conserved variables is exactly equal to the reconstructed value of the steady-state solution in an equilibrium state, a nonlinear interpolation with scale invariance property is used in the reconstruction procedure. Since the same central difference scheme can be used for both flux derivatives and grid derivatives, the proposed scheme satisfy geometric conservation laws on curvilinear grids. By theoretical analysis and experimental results, it is indicated that the proposed WCNS scheme can preserve the general steady state which include both ispthermal and polytropic equilibria, and geometric conservation laws. Moreover, it can achieve fifth-order accuracy and capture exactly small perturbations near steady-state solutions on curvilinear grids.

Multidimensional scaling (MDS) is a data analysis technology that displays and analyzes the corresponding multidimensional data structure in the low-dimensional space. The Individual Difference Scaling (INDSCAL) is a specific model for simultaneous metric multidimensional scaling (MDS) of several data matrices, which not only analyzes the structure of the analysis object, but also takes into account the difference in scales between subjects. In the present work the orthogonal INDSCAL(O-INDSCAL) problem is considered and the problem of fitting the O-INDSCAL model is constructed as a matrix optimization model constrained by Stiefel manifold and linear manifolds. By leveraging the geometric properties of the product manifold, basing on the strong Wolfe line search, we design an adaptive extended hybrid Riemannian conjugate gradient algorithm for the underlying problem and its global convergence is further discussed. Numerical experiments demonstrate that the hybrid method is feasible and effective for the model. Moreover, the proposed algorithm exhibits certain advantages in terms of iterative efficiency compared to the algorithms in the Riemannian optimization toolbox and other Riemannian first-order algorithms.

Vertical nonlinear complementarity problems have wide applications. The design of numerical methods for solving vertical nonlinear complementarity problems has been a hot topic among researchers in recent years. In this paper, a modulus-based synchronous multisplitting iteration method is established by matrix multisplitting and equivalent modulus equation for vertical nonlinear complementarity problem. Some convergence conditions of the proposed method are presented under H-matrix assumption. The convergence domain of the relaxation parameters of the accelerated overrelaxation iteration is obtained. By OpenACC framework, numerical tests are given to show the high parallel computational efficiency of the proposed method.

This paper provides a brief exploration of the basic principles and applications of mathematical formalization, with a focus on the formal language Lean and its application in mathematical optimization. We first review the development background of mathematical formalization, explain the construction principles of the Lean language and its correctness guarantee mechanisms, and introduce the role of the theorem library Mathlib4 in Lean. By comparing natural language with formalized expressions, we illustrate the advantages of using formalization to verify mathematics, emphasizing its important role in the accurate verification of mathematical theories. In the field of mathematical optimization, this paper discusses the current progress of formalizing mathematical optimization theory, using formalized examples of classic theorems such as the quadratic upper bound lemma, and further highlights the characteristics and advantages of formalized mathematics. Additionally, we explore the formalization goals in operations research and investigate the technologies of automated formalization and automated theorem proving, analyzing the potential and challenges of automation tools in the mathematical formalization process. Finally, we summarize the current state of research in mathematical formalization, give suggestions for further advancing the field, and discuss the significant role of formalization in the development of applied mathematical theory.

Single-phase multi-component flow problems are widely existed in oil and gas reservoirs, groundwater pollution, etc., and numerical simulation is of great significance. In this paper, for the traditional IMPEC (Implicit Pressure-Explicit Concentration) method, which does not follow the problem of mass conservation for all components, a physically preserving IMPEC method with mass conservation for each component is used for the time discretization, and the upwind block-centered finite-difference method is used for the spatial discretization of the pressure equation, Darcy’s equation, and the component equations. In this paper, the physics-preserving mass conservation properties of all the constructed components as well as the molar concentrations are rigorously proved under reasonable condition, respectively. Finally, numerical simulations are carried out to demonstrate the validity of the proposed algorithm by means of a single-phase multi-component flow problem.

In this paper, the Hermite polynomials are used as the hidden layer of the neural network, the initial weights of the Hermite neural network are optimized using genetic algorithm, and the inverse of the error function between the actual output and the desired output of the Hermite neural network optimized by the genetic algorithm is also chosen as the fitness function of the genetic algorithm, to construct a new genetic algorithm-optimized Hermite neural network to solve the Caputo fractal-fractional order Bagley-Torvik differential equation numerical method. The general form of the numerical solution of the Caputo fractal-fractional order Bagley-Torvik differential equation is given in conjunction with the Taylor’s formula at multiple points, and the absolute error and convergence of the algorithm are theoretically investigated. Comparison with existing numerical methods is made and the results show the effectiveness and feasibility of the method in this paper.

A compact difference scheme for the two-dimensional Sobolev equation with fourth-order accuracy in space is derived, and the convergence of the compact difference scheme is proved. The difference scheme is rewritten in vector form, a reduced-order high-order compact difference scheme is constructed using the Proper Orthogonal Decomposition (POD) method, and error estimate for the approximate solution is also provided. Numerical example is presented to calculate the numerical error, spatial convergence order and temporal convergence order of both the compact difference scheme and the reduced-order compact difference scheme, verifying that the experimental results are consistent with the theoretical analysis. Furthermore, by comparing the CPU computation time before and after dimension reduction, the superiority of applying the POD method for dimension reduction in compact schemes is demonstrated.

This thesis proposes a class of space-time mixed finite element method for time fractional diffusion equations involving Riemann-Liouville derivatives of order α ∈ (0, 1). The spatial discretization employs m(m ≥ 0) order Raviart-Thomas (RT) finite elements, while the temporal discretization uses piecewise r(r ≥ 0) degree discontinuous Galerkin (DG) finite elements. To address the solution singularity near t = 0, graded meshes are used in the time direction. The well-posedness of the fully discrete scheme is analyzed. Error estimates are derived in two cases: r = 0, i.e. the temporal discretization uses the piecewise constant DG scheme, and r = 1, i.e. the temporal discretization uses the piecewise linear DG scheme. Numerical experiments are provided.

In this paper, we consider a special kind of matrix combined by a skew-symmetric tridiagonal matrix and a periodic skew-symmetric tridiagonal matrix. It is referred to as a generalized periodic skew-symmetric tridiagonal matrix. An inverse eigenvalue problem for constructing thus a matrix is studied, that is, the matrix is reconstructed from three prescribed balanced sets and a positive number. Firstly, the matrix can be transformed into a generalized periodic symmetric tridiagonal matrix by unitary similarity, and then the relationships between the eigenvalues of this matrix and those of its leading and trailing principal submatrices are analyzed. Two aspects are respectively discussed about whether the spectra of these two principal submatrices have a disjoint set and the parity of the order of the above trailing principal submatrix. Then, the necessary and sufficient conditions for the existence of solutions to the inverse eigenvalue problems in various cases are provided, and the largest number of solutions and reconstruction algorithms are determined. Finally, the effectiveness of the algorithm is validated by two numerical examples.

In this paper, we propose an adaptive step-size rule three-operator splitting algorithm with an inertia term to solve the nonsmooth DC programming problem. Under suitable assumptions, we prove that the sequence generated by the algorithm converges to a critical point of the problem. Finally, we apply the algorithm for solving the sparse recovery problem, and numerical experiments show the effectiveness and superiority of the new algorithm.

Multidimensional scaling (MDS) is a technique used in multidimensional data analysis that depicts the similarities or relationships between observed objects as distances between points in a lower-dimensional space. By representing high-dimensional data within a lowdimensional framework, MDS preserves the relative distances between data points. This study focuses on developing an efficient numerical algorithm for a specific type of individual differences scaling model, known as O-INDSCAL, within symmetric multidimensional scaling, which accounts for individual differences among observed objects. Initially, leveraging the concept of the alternating least squares algorithm, the multivariable constrained matrix optimization model associated with the O-INDSCAL model is transformed into a fixed-point iteration problem. By thoroughly examining the acceleration principles and implementation processes of various polynomial extrapolation and Anderson acceleration methods in vector sequence acceleration, we have designed a matrix sequence acceleration algorithm tailored to this problem model. Numerical experiments indicate that the proposed acceleration algorithms significantly enhance the convergence speed of sequences generated by fixed-point iterations. Furthermore, when compared with existing continuous-time projection gradient flow algorithms and the first-order and second-order Riemannian algorithms available in the Manopt toolbox for manifold optimization, the proposed algorithms demonstrate notable improvements in iterative efficiency.

This paper mainly discusses the oscillation of numerical solutions for a class of neutral delay differential equations with multiple delays. The original equation is discretized by using the linear θ-method to obtain the corresponding difference equation. By discussing the properties of the solutions of the difference equation, the oscillation properties of the numerical solutions for the original equation are transformed into that of a non-neutral difference equation. According to the relationship between the oscillation of the difference equation and the characteristic roots of the characteristic equation, the oscillation of the numerical solutions is discussed for $0\leq \theta \leq \frac{1}{2}$ and $\frac{1}{2}<\theta \leq1$， respectively. Meanwhile, the properties of non-oscillatory numerical solutions are also investigated. Finally, numerical examples are given to illustrate the conclusions.