中国科学院数学与系统科学研究院期刊网
JCM

01 July 2026, Volume 44 Issue 5
    

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  • DYNAMIC-STABILIZATION-BASED LINEAR SCHEMES FOR THE ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
    Hongfei Fu, Dianming Hou, Zhonghua Qiao, Bingyin Zhang
    Journal of Computational Mathematics. 2026, 44(5): 1219-1247. https://doi.org/10.4208/jcm.2604-m2025-0304
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    In this paper, we investigate linear first- and second-order numerical schemes for the Allen-Cahn equation with a general (possibly degenerate) mobility, based on a novel dynamic stabilization strategy.For the first-order scheme, we prove that it unconditionally preserves both the discrete maximum bound principle (MBP) and the energy dissipation law.The second-order scheme, in contrast, preserves the discrete MBP under certain conditions on the stabilization parameters and satisfies a uniform energy bound, i.e., the discrete original energy at any time is uniformly bounded by the initial energy plus a highorder perturbation term.A key advance is that the discrete energy stability remains valid even in the presence of degenerate mobility-a property we refer to as mobility robustness.Rigorous maximum-norm error estimates are also established.In particular, for the second-order scheme, we introduce a new prediction strategy with a cut-off preprocessing procedure on the extrapolation solution, and only one linear system needs to be solved per time level.Representative numerical examples are provided to validate the theoretical findings and performance of the proposed schemes.
  • A STRUCTURE-PRESERVING NONCONFORMING FEM OF NONLINEAR KIRCHHOFF-TYPE EQUATION WITH DAMPING
    Junjun Wang, Dongyang Shi, Shuo Wang, Xuesong Che
    Journal of Computational Mathematics. 2026, 44(5): 1248-1268. https://doi.org/10.4208/jcm.2509-m2024-0004
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    Superconvergent behavior for nonlinear Kirchhoff-type with damping is researched by a structure-preserving nonconforming finite element method (FEM).A new implicit energy dissipation scheme is developed and the numerical solution is bounded in energy norm.The existence of the numerical solution is obtained with the help of the Brouwer fixed-point theorem and then the uniqueness is gained.Superconvergence characteristics is revealed by the properties of the nonconforming FE and a special splitting technique.Numerical tests confirm the correctness of the theoretical research results.
  • CONVERGENCE ANALYSIS OF AN INEXACT AUGMENTED LAGRANGIAN METHOD FOR NONCONVEX SECOND-ORDER CONE PROGRAMMING
    Benqi Liu, Liwei Zhang
    Journal of Computational Mathematics. 2026, 44(5): 1269-1296. https://doi.org/10.4208/jcm.2602-m2024-0086
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    This paper focuses on the convergence analysis of the inexact augmented Lagrangian method (iALM) for solving nonconvex second-order cone optimization problems (SOCPs).We propose an implementable termination criterion based on the iterative difference of Lagrangian multipliers.Theoretical analysis demonstrates that, under the Jacobian uniqueness conditions, the local convergence rate of the iALM is proportional to 1/r, provided the penalty parameter r exceeds a certain threshold.Furthermore, we establish the global convergence of the algorithm, proving that any accumulation point of the iterates is a stationary point.Numerical experiments validate the effectiveness of the proposed method.In particular, when applied to classical trust-region subproblems, our algorithm is compared against Mosek, SDPT3, and the classical Moré-Sorensen method.The results demonstrate that our approach exhibits superior efficiency and robustness, especially for highdimensional dense problems where it significantly outperforms factorization-based methods.
  • A RESTARTED AND RANDOMIZED ALGORITHM FOR EVALUATING ENERGY OF EXTREMELY LARGE-SCALE GRAPH
    Gang Wu, Ke Li, Jianjian Wang
    Journal of Computational Mathematics. 2026, 44(5): 1297-1329. https://doi.org/10.4208/jcm.2506-m2024-0192
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    Graph energy is a spectrum-based graph invariant that has been studied extensively in network sciences.However, as far as we are aware, most of the existing works try to establish theoretical bounds, and there are few efficient algorithms for computing energy of extremely large-scale graph.To fill-in this gap, we first propose a randomized algorithm for evaluating energy of large-scale graphs, under the assumption that the adjacency matrix is approximately low (numerical) rank.However, the number of sampling used in this algorithm is difficult to determine in advance, and the graph energy is often underestimated.In order to improve the quality of the evaluation, we then propose a non-restarted randomized algorithm that updates the columns of the search basis incrementally.The error analysis and the convergence of the algorithm are established.However, the nonrestarted algorithm may suffer from heavy overhead as the iteration proceeds.So as to release the overhead of the non-restarted algorithm, we finally propose a restarted randomized algorithm for evaluating energy of extremely large-scale graphs.The rationality of the restarted algorithm is given.Extensive numerical experiments are performed on extremely large-scale real-world and synthetic graphs, to show the effectiveness of our strategies and efficiency of the proposed algorithms.
  • ON THE ADAPTIVE DETERMINISTIC BLOCK COORDINATE DESCENT METHOD WITH MOMENTUM FOR SOLVING LARGE LINEAR LEAST-SQUARES PROBLEMS
    Longze Tan, Mingyu Deng, Jiali Qiu, Xueping Guo
    Journal of Computational Mathematics. 2026, 44(5): 1330-1353. https://doi.org/10.4208/jcm.2506-m2025-0047
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    Inspired by Polyak’s heavy-ball method, this paper proposes an adaptive deterministic block coordinate descent method with momentum (mADBCD) for efficiently solving large-scale linear least-squares problems.The proposed method introduces a novel column selection criterion based on the Euclidean norm of the residual vector of the normal equation.In contrast to classical block coordinate descent methods, mADBCD does not require a fixed pre-partitioning of the column indices of the coefficient matrix and avoids the expensive computation of Moore-Penrose pseudoinverses of submatrices at each iteration.The method adaptively updates the block index set at each step, thereby improving both flexibility and scalability.When the coefficient matrix is of full column rank, we prove that mADBCD converges linearly to the unique solution of the least-squares problem.Numerical experiments are conducted to show that mADBCD outperforms several recent block coordinate descent methods in terms of iteration count and CPU time.In particular, when solving extremely sparse least-squares problems, mADBCD is the first block coordinate descent method reported to achieve CPU time nearly comparable to that of the classical least squares QR (LSQR) method [Paige and Saunders, ACM Trans.Math.Softw., 8 (1982)].
  • A NOVEL TGMFECN DIMENSION REDUCTION METHOD BASED ON POD FOR THE IMPROVED NONLINEAR FOURTH-ORDER CAHN-HILLIARD EQUATIONS
    Huanrong Li, Yuejie Li, Liang He, Zhendong Luo
    Journal of Computational Mathematics. 2026, 44(5): 1354-1386. https://doi.org/10.4208/jcm.2506-m2024-0245
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    An improved nonlinear fourth-order Cahn-Hilliard (INFOCH) equation is first developed to ensure that its numerical model is symmetric, positive definite, and solvable.Then, by introducing an auxiliary function, the INFOCH equation is decomposed into the nonlinear system of equations with second-order derivatives of spatial variables.Subsequently, by using the Crank-Nicolson (CN) technique to discretize the time derivative, a new time semi-discretized mixed CN (TSDMCN) scheme with second-order accuracy is constructed, and the existence, stability, and error estimates of TSDMCN solutions are analyzed.Thenceforth, a new two-grid mixed finite element (MFE) CN (TGMFECN) method is created by using two-grid MFE method to discretize the TSDMCN scheme, and the existence, stability, and error estimates of TGMFECN solutions are discussed.Next, it is most important that by using proper orthogonal decomposition to reduce the dimension of unknown coefficient vectors of TGMFECN solutions and keep the MFE basis functions unchanged, a new TGMFECN dimensionality reduction (TGMFECNDR) method with very few unknowns, unconditional stability, and second-order time precision is created, and the existence, stability, and error estimates of TGMFECNDR solutions are proved.Finally, the superiority of TGMFECNDR method and the correctness of the obtained theoretical results are showed by two sets of numerical experiments.
  • A MATRIX OPTIMIZATION METHOD FOR BLIND EXTRACTION OF EXTERNAL EQUITABLE PARTITIONS FROM LOW PASS GRAPH SIGNALS
    Wenshun Teng, Qingna Li
    Journal of Computational Mathematics. 2026, 44(5): 1387-1405. https://doi.org/10.4208/jcm.2506-m2024-0128
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    Seeking the external equitable partitions (EEPs) of networks under unknown structures is an emerging problem in network analysis.The special structure of EEPs has found widespread applications in many fields such as cluster synchronization and consensus dynamics.While most literature focuses on utilizing the special structural properties of EEPs for network studies, there has been little work on the extraction of EEPs or their connection with graph signals.In this paper, we address the interesting connection between low pass graph signals and EEPs, which, as far as we know, is the first time.We provide a method BE-EEPs for extracting EEPs from low pass graph signals and propose an optimization model, which is essentially a problem involving nonnegative orthogonality matrix decomposition.We derive theoretical error bounds for the performance of our proposed method under certain assumptions and apply three algorithms to solve the resulting model, including the K-means algorithm, the practical exact penalty method and the iterative Lagrangian approach.Numerical experiments verify the effectiveness of the proposed method.Under strong low pass graph signals, the iterative Lagrangian and Kmeans perform equally well, outperforming the exact penalty method.However, under complex weak low pass signals, all three perform equally well.
  • A LOW ORDER GLOBALLY DIVERGENCE-FREE WG FINITE ELEMENT METHOD FOR STEADY THERMALLY COUPLED INCOMPRESSIBLE MHD FLOW
    Min Zhang, Qijia Zhai, Xiaoping Xie
    Journal of Computational Mathematics. 2026, 44(5): 1406-1437. https://doi.org/10.4208/jcm.2506-m2025-0061
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    This paper develops a low order weak Galerkin (WG) finite element method for the steady thermally coupled incompressible magnetohydrodynamics flow.In the interior of elements, the WG scheme uses piecewise linear polynomials for the approximations of the velocity, the magnetic field and the temperature, and piecewise constants for the approximations of the pressure and the magnetic pseudo-pressure; and on the interfaces of elements, the scheme uses piecewise constants for the numerical traces of velocity and the temperature, and piecewise linear polynomials for the numerical traces of the magnetic fields, the pressure and the magnetic pseudo-pressure.This WG method is shown to yield globally divergence-free approximations of the velocity and magnetic fields.Existence and uniqueness results as well as optimal a priori error estimates for the discrete scheme are obtained.A convergent linearized iterative algorithm is presented.Numerical experiments are provided to verify the theoretical analysis.
  • STOCHASTIC VARIANCE REDUCED GRADIENT FOR TENSOR RECOVERY
    Li Li, Chen Xu, Jian Lu, Ningning Han, Lixin Shen
    Journal of Computational Mathematics. 2026, 44(5): 1438-1457. https://doi.org/10.4208/jcm.2508-m2024-0288
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    Low-rank tensor recovery is pivotal in numerous applications, including image and video processing, machine learning, and data analysis.A common approach to this problem involves convex relaxation, where the tensor rank function is minimized by using the tensor nuclear norm.However, this method can be significantly suboptimal.In addition, the stochastic variance reduced gradient (SVRG) method, a variant of stochastic gradient descent, has been applied to matrix recovery problems.
    In this paper, we extend the SVRG method to the tensor framework, introducing the tensor stochastic variance reduced gradient (TSVRG) algorithm for tensor recovery with CP or Tucker rank constraints.TSVRG is designed to achieve higher precision solutions by escaping local minima and identifying superior global optima.Moreover, TSVRG offers reduced computational complexity compared to traditional gradient descent methods.We establish a convergence theorem for TSVRG under the tensor restricted isometry condition when the measurements are linear.Finally, we present numerical results using both synthetic and real data, demonstrating the competitive performance of TSVRG compared to other advanced algorithms.
  • MULTI-LEVEL MIXED FINITE ELEMENT METHODS BASED ON DIFFERENT ITERATIONS FOR THE STEADY BOUSSINESQ PROBLEM
    Liwei Liu, Tong Zhang, Chuanjun Chen
    Journal of Computational Mathematics. 2026, 44(5): 1458-1491. https://doi.org/10.4208/jcm.2508-m2024-0257
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    Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper.The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size h0 is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes hj, j = 1, 2, ..., J.The computational scales are reduced and the computational costs are saved.Furthermore, the uniform stability and convergence results in both L2- and H1-norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems.Theoretical results show that the multi-level methods have the same order of numerical solutions in the H1-norm as the one level method with the mesh sizes hj = h2j-1, j = 1, 2, ..., J.Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.
  • SERENDIPITY VIRTUAL ELEMENT METHOD FOR THE SECOND ORDER ELLIPTIC EIGENVALUE PROBLEM IN TWO AND THREE DIMENSIONS
    Haokun Li, Yulin Liu, Jian Meng, Xu Qian
    Journal of Computational Mathematics. 2026, 44(5): 1492-1525. https://doi.org/10.4208/jcm.2508-m2024-0265
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    In this paper, we analyze the virtual element method for the symmetric second order elliptic eigenvalue problem with variable coefficients in two and three dimensions, which reduces the number of degrees of freedom of the standard virtual element method.We attempt to prove the interpolation theory and stability analysis for the serendipity nodal virtual element space, which provides new stabilization terms for the virtual element schemes.Then we prove the spectral approximation and the optimal a priori error estimates.Moreover, we construct a fully computable residual-type a posteriori error estimator applied to the adaptive serendipity virtual element method and prove its upper and lower bounds with respect to the approximation error.Finally, we show numerical examples to verify the theoretical results and show the comparison between standard and serendipity virtual element methods.
  • REGULARIZATION METHOD WITH TWO DIFFERENTIAL OPERATORS FOR SIMULTANEOUS INVERSION OF SOURCE TERM AND INITIAL VALUE IN A TIME-FRACTIONAL BLACK-SCHOLES EQUATION
    Shuang Yu, Hongqi Yang
    Journal of Computational Mathematics. 2026, 44(5): 1526-1553. https://doi.org/10.4208/jcm.2508-m2025-0012
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    This paper is devoted to identifying the source term and initial value simultaneously in a time-fractional Black-Scholes equation, which is an ill-posed problem.The inverse problem is transformed into a system of operator equations, and under certain source conditions, conditional stability is established.We propose a regularization method with two differential operators to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived, respectively.Numerical experiments are presented to validate the effectiveness of the proposed regularization method.
  • COMPACT FINITE DIFFERENCE SCHEME FOR SOME SOBOLEV TYPE EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS
    Lavanya V Salian, Rathan Samala, Rakesh Kumar
    Journal of Computational Mathematics. 2026, 44(5): 1554-1582. https://doi.org/10.4208/jcm.2509-m2024-0296
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    This study aims to construct a stable, high-order compact finite difference method for solving Sobolev-type equations with Dirichlet boundary conditions.Approximation of higher-order mixed derivatives in some specific Sobolev-type equations requires a bigger stencil information.One can approximate such derivatives on compact stencils, which are higher-order accurate and take less stencil information but are implicit and sparse.Spatial derivatives in this work are approximated using the sixth-order compact finite difference method, while temporal derivatives are handled with the explicit forward Euler difference scheme.We examine the accuracy and convergence behavior of the proposed scheme.Using the von Neumann stability analysis, we establish L2-stability theory for the linear case.We derive conditions under which fully discrete schemes are stable.Also, the amplification factor C(θ) is analyzed to ensure the decay property over time.Real parts of C(θ) lying on the negative real axis confirm the exponential decay of the solution.A series of numerical experiments were performed to verify the effectiveness of the proposed scheme.These tests include both one-dimensional and two-dimensional cases of advection-free and advectiondiffusion flows.They also cover applications to the equal width equation, such as the propagation of a single solitary wave, interactions between two and three solitary waves, undular bore formation, and the Benjamin-Bona-Mahony-Burgers equation.