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2025年 第7卷 第4期    刊出日期：2025-08-20

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ORIGINAL PAPERS

On the Stability of IMEX Upwind gSBP Schemes for 1D Linear Advection-Diffusion Equations

Sigrun Ortleb

2025, 7(4):  1195-1224.  doi:10.1007/s42967-023-00296-4

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A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts (upwind gSBP) schemes in space and implicit-explicit Runge-Kutta (IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly, while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first- and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin (DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay (BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space, since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.

Eigenvalues and Jordan Forms of Dual Complex Matrices

Liqun Qi, Chunfeng Cui

2025, 7(4):  1225-1241.  doi:10.1007/s42967-023-00299-1

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Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an n×n dual complex matrix is diagonalizable if and only if it has exactly n eigenvalues with n appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.

A Note on Chebyshev Accelerated PMHSS Iteration Method for Block Two-by-Two Linear Systems

Zhao-Zheng Liang, Jun-Lin Tian, Hong-Yi Wan

2025, 7(4):  1242-1263.  doi:10.1007/s42967-023-00300-x

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In this paper, the efficient preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks. With the aid of the singular value decomposition technique, the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks. Moreover, the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case. On account of the favorable spectral properties of the PMHSS preconditioned matrix, a parameter free Chebyshev accelerated PMHSS (CAPMHSS) method is established to further improve its convergence rate. Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.

Unconditionally Superconvergence Error Analysis of an Energy-Stable and Linearized Galerkin Finite Element Method for Nonlinear Wave Equations

Huaijun Yang

2025, 7(4):  1264-1281.  doi:10.1007/s42967-023-00301-w

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In this paper, a linearized energy-stable scalar auxiliary variable (SAV) Galerkin scheme is investigated for a two-dimensional nonlinear wave equation and the unconditional superconvergence error estimates are obtained without any certain time-step restrictions. The key to the analysis is to derive the boundedness of the numerical solution in the H¹-norm, which is different from the temporal-spatial error splitting approach used in the previous literature. Meanwhile, numerical results are provided to confirm the theoretical findings.

Reflected Stochastic Burgers Equation with Jumps

Hongchao Qian, Jun Peng, Ruizhi Li, Yewei Gui

2025, 7(4):  1282-1307.  doi:10.1007/s42967-023-00305-6

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This paper is concerned with the reflected stochastic Burgers equation driven both by the Brownian motion and by the Poisson random measure. The existence and uniqueness of solutions are established. The penalization method plays an important role.

Modulus-Based Matrix Splitting Iteration Method for Horizontal Quasi-complementarity Problem

Lu-Xin Wang, Qin-Qin Shen, Yang Cao

2025, 7(4):  1308-1332.  doi:10.1007/s42967-023-00311-8

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In this paper, the modulus-based matrix splitting (MMS) iteration method is extended to solve the horizontal quasi-complementarity problem (HQCP), which is characterized by the presence of two system matrices and two nonlinear functions. Based on the specific matrix splitting of the system matrices, a series of MMS relaxation iteration methods are presented. Convergence analyses of the MMS iteration method are carefully studied when the system matrices are positive definite matrices and H₊-matrices, respectively. Finally, two numerical examples are given to illustrate the efficiency of the proposed MMS iteration methods.

Ramsey Numbers of Stars Versus Generalised Wheels

Yiran Zhang, Yuejian Peng

2025, 7(4):  1333-1349.  doi:10.1007/s42967-023-00316-3

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For two graphs $G$ and $H$, the Ramsey number $R(G, H)$ is the smallest integer $n$ such that for any $n$-vertex graph, either it contains $G$ or its complement contains $H$. Let $S_n$ be a star of order $n$ and $W_{s, m}$ be a generalised wheel $K_s \vee C_m$. Previous studies by Wang and Chen (Graphs Comb 35(1):189-193, 2019) and Chng et al. (Discret Math 344(8):112440, 2021) imply that a tree is $W_{s, 4^{-}}$good, $W_{s, 5^{-}}$good, $W_{s, 6}$-good, and $W_{s, 7^{-}}$good for $s \geqslant 2$. In this paper, we study the Ramsey numbers $R\left(S_n, W_{s, 8}\right)$, and our results indicate that trees are not always $W_{s, 8}$-good.

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

T. Nasiri, A. Zakeri, A. Aminataei

2025, 7(4):  1350-1363.  doi:10.1007/s42967-023-00319-0

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In this paper, the existence theorem for a quasi solution of an inverse fractional stochastic parabolic equation driven by multiplicative noise in the form ^cD^α_tu-div(g(x,▽u))=f(x,u)+σ(x,u)ẇ(t) is given. In this equation, the fractional derivative is considered in the Caputo sense. Also, the random function g is unknown and should be determined. To identify the unknown coefficient, the minimization and stochastic variational formulation methods in a fractional stochastic Sobolev space are used. Indeed, we obtain a stability estimation and then prove the continuity of the minimization functional using obtained stability estimation. These results show the existence of the quasi solution for the mentioned problem.

A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

A. M. Vargas

2025, 7(4):  1364-1377.  doi:10.1007/s42967-023-00323-4

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In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.

The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient

Qiaoqiao Dai, Dongxia Li

2025, 7(4):  1378-1397.  doi:10.1007/s42967-023-00326-1

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In this paper, an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient. Since the solution of such an equation in general has a typical weak singularity near the initial time t=0, the time-fractional derivative with order in (0, 1) is discretized by L2-1_σ formula on nonuniform meshes. For the spatial derivative, the local discontinuous Galerkin (LDG) method is employed. A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality. Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.

Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems

Divyansh Pandey, Prashant K. Pandey, Rajesh K. Pandey

2025, 7(4):  1398-1418.  doi:10.1007/s42967-023-00340-3

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This paper is devoted to studying the variational approximation for the higher order regular fractional Sturm-Liouville problems (FSLPs). Using variational principle, we demonstrate that the FSLP has a countable set of eigenvalues and corresponding unique eigenfunctions. Furthermore, we establish two results showing that the eigenfunctions corresponding to distinct eigenvalues are orthogonal, and the smallest (first) eigenvalue is the minimizer of the functional. To validate the theoretical result, we also present a numerical method using polynomials \Phi_j(t)=t^{j+1}(1-t)^2 \text { for } j=1,2,3… as a basis function. Further, the Lagrange multiplier method is used to reduce the fractional variational problem into a system of algebraic equations. In order to find the eigenvalues and eigenfunctions, we solve the algebraic system of equations. Further, the analytical convergence and the absolute error of the method are analyzed.

Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions

Tarek Aboelenen

2025, 7(4):  1419-1443.  doi:10.1007/s42967-023-00337-y

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A new set of generalized Jacobi rational functions of the first and second kinds, GJRFs-1 and GJRFs-2, which are mutually orthogonal in L²(0, ∞), are introduced and they are analytical eigensolutions to a new family of singular fractional Sturm-Liouville problems (SFSLPs) of the first and second kinds as non-polynomial functions. We establish some properties and optimal approximation results for these GJRFs-1 and GJRFs-2 in non-uniformly weighted Sobolev spaces involving fractional derivatives, which play important roles in the related spectral methods for a class of fractional differential equations. We develop Jacobi rational-Gauss quadrature type formulae andL²-orthogonal projections based on GJRFs-1 and GJRFs-2. As examples of applications, the two quadrature rules are proposed for Fermi-Dirac and Bose-Einstein integrals. Using various orthogonal properties of GJRFs-1 and GJRFs-2, the Petrov-Galerkin methods are proposed for fractional initial value problems and fractional boundary value problems. Numerical results demonstrate its efficient algorithm, and spectral accuracy for treating the above-mentioned classes of problems. The suggested numerical scheme provides an applicable substitutional to other competitive methods in the recent method-related accuracy.

An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

Xing Liu

2025, 7(4):  1444-1461.  doi:10.1007/s42967-023-00342-1

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An accelerated convergence scheme for temporal approximation of stochastic partial differential equation is presented. First, the regularity of the mild solution is provided. Combining the Itô formula and the remainder term of the exponential Euler scheme, this paper proposes a high accuracy time discretization method. Based on regularity results, a strong convergence rate for the discretization error $O\left(\tau^{\frac{3}{2}-\epsilon}\right)$ is proved for arbitrarily small $\epsilon$＞0. Here $\tau$ is the uniform time step size. Finally, the theoretical results are verified by several numerical experiments.

Numerical Stability and Convergence for Delay Space-Fractional Fisher Equations with Mixed Boundary Conditions in Two Dimensions

Jing Chen, Qi Wang

2025, 7(4):  1462-1488.  doi:10.1007/s42967-023-00346-x

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In this paper, for generalized two-dimensional delay space-fractional Fisher equations with mixed boundary conditions, we present the stability and convergence computed by a novel numerical method. The unconditional stability of analytic solutions is first derived. Next, we have established the linear θ-method with the Grünwald-Letnikov operator, which has the first-order accuracy in spatial dimensions. Moreover, approaches involved error estimations and inequality reductions are utilized to prove the stability and convergence of numerical solutions under different values of θ. Eventually, we implement a numerical experiment to validate theoretical conclusions, where the interaction impacts of fractional derivatives have been further analyzed by applying two different harmonic operators.

Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

Xuan Zhao, Zhongqin Xue

2025, 7(4):  1489-1515.  doi:10.1007/s42967-023-00350-1

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An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^-α, α∈ (0, 1). The Fourier pseudo-spectral method is applied for the spatial approximation. The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model, primarily due to the introduction of the fractional Laplacian. This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian. Subsequently, the unique solvability and the modified energy dissipation law are theoretically guaranteed. We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term. Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian. In particular, we demonstrate that the adaptive time-stepping strategy, compared with the uniform time steps, captures the multiple time scale evolutions of the solution in simulations.

On Nonlinear Analysis for Multi-term Delay Fractional Differential Equations Under Hilfer Derivative

Dildar Ahmad, Amjad Ali, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad

2025, 7(4):  1516-1539.  doi:10.1007/s42967-023-00351-0

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In this manuscript, a class of multi-term delay fractional differential equations (FDEs) under the Hilfer derivative is considered. Some newly updated results are established under boundary conditions. For the required results, we utilize the fixed point theory and tools of the nonlinear functional analysis. Further keeping in mind the importance of stability results, we develop some adequate results about the said aspect. The Hyers-Ulam (H-U)-type concept is used to derive the required stability for the solution of the considered problem. Finally, by appropriate test problems, we justify our findings.

Bifurcation Analysis of an Advertising Diffusion Model

Yong Wang, Yao Wang, Liangping Qi

2025, 7(4):  1540-1561.  doi:10.1007/s42967-023-00353-y

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This research aims to investigate the impact of diffusion on the stability and bifurcation behavior of advertising diffusion systems. The study findings suggest that in the absence of diffusion, a higher proportion of crowd contact positively contributes to the stability of the system. Specifically, the study employs the interval partitioning method to discuss the k-mode Turing bifurcation and derives a more explicit Turing bifurcation line. Moreover, the study examines the k-mode Hopf bifurcation with the proportion of crowd contact acting as the bifurcation parameter. Furthermore, the weakly nonlinear analysis method is implemented to scrutinize the pattern formation in the Turing instability region. Finally, numerical simulation is utilized to validate the analytical findings obtained in this study.

Coordinate-Adaptive Integration of PDEs on Tensor Manifolds

Alec Dektor, Daniele Venturi

2025, 7(4):  1562-1579.  doi:10.1007/s42967-023-00357-8

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We introduce a new tensor integration method for time-dependent partial differential equations (PDEs) that controls the tensor rank of the PDE solution via time-dependent smooth coordinate transformations. Such coordinate transformations are obtained by solving a sequence of convex optimization problems that minimize the component of the PDE operator responsible for increasing the tensor rank of the PDE solution. The new algorithm improves upon the non-convex algorithm we recently proposed in Dektor and Venturi (2023) which has no guarantee of producing globally optimal rank-reducing coordinate transformations. Numerical applications demonstrating the effectiveness of the new coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

On Multi-step Greedy Kaczmarz Method for Solving Large Sparse Consistent Linear Systems

Long-Ze Tan, Ming-Yu Deng, Xue-Ping Guo

2025, 7(4):  1580-1597.  doi:10.1007/s42967-023-00358-7

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Based on the greedy randomized Kaczmarz (GRK) method, we propose a multi-step greedy Kaczmarz method for solving large-scale consistent linear systems, utilizing multi-step projection techniques. Its convergence is proved when the linear system is consistent. Numerical experiments demonstrate that the proposed method is effective and more efficient than several existing classical Kaczmarz methods.

An Improved Iterative Algorithm for Identifying Strong H-Tensors

Wenbin Gong, Yan Li, Yaqiang Wang

2025, 7(4):  1598-1614.  doi:10.1007/s42967-023-00362-x

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Strong H-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor. In this paper, first, an improved iterative algorithm is proposed to determine whether a given tensor is a strong H-tensor, and the validity of the iterative algorithm is proved theoretically. Second, the iterative algorithm is employed to identify the positive definiteness of an even-order real symmetric tensor. Finally, numerical examples are presented to illustrate the advantages of the proposed algorithm.