中国科学院数学与系统科学研究院期刊网
JCM

15 May 2026, Volume 44 Issue 3
    

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  • INVARIANT REGION PRESERVING RECONSTRUCTION AND ENHANCED STABILITY OF THE CENTRAL SCHEME IN TWO DIMENSIONS
    Ruifang Yan, Wei Tong, Guoxian Chen
    Journal of Computational Mathematics. 2026, 44(3): 593-617. https://doi.org/10.4208/jcm.2502-m2024-0015
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    In this paper, our focus is on examining the robustness of the central scheme in two dimensions. Although stability analyses are available in the literature for the scheme's solution of scalar conservation laws, the associated Courant-Friedrichs-Lewy (CFL) number is often notably small, occasionally degenerating to zero. This challenge is traced back to the initial data reconstruction. The interface value limiter used in the reconstruction proves insufficient to maintain the invariant region of the updated solutions. To overcome this limitation, we introduce the vertex value limiter, resulting in a more suitable CFL number that is half of the one-dimensional value. We present a unified analysis of stability applicable to both types of limiters. This enhanced stability condition enables the utilization of larger time steps, offering improved resolution to the solution and ensuring faster simulations. Our analysis extends to general conservation laws, encompassing scalar problems and nonlinear systems. We support our findings with numerical examples, validating our claims and showcasing the robustness of the enhanced scheme.
  • CONVERGENCE ANALYSIS OF A CLASS OF REGULARIZATION METHODS WITH A NOVEL DISCRETE SCHEME FOR SOLVING INVERSE PROBLEMS
    M. P. Rajan
    Journal of Computational Mathematics. 2026, 44(3): 618-633. https://doi.org/10.4208/jcm.2503-m2024-0225
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    Many inverse problems that appear in applications can be modeled as an operator equation. In practice, most of these problems are ill-posed, and computing solutions to such problems in an efficient manner is challenging and has been of greatest interest among researchers in the recent past. While many approaches are developed within infinite-dimensional Hilbert space settings, practical applications often require solutions in finite-dimensional spaces, and we need to discretize the problem. In this manuscript, we study a novel discretization scheme along with a class of regularization techniques for solving linear ill-posed problems and obtain the optimal order error estimates under an a priori parameter choice strategy. We illustrate the computational efficacy of the proposed scheme through numerical examples, and the results demonstrate that the proposed scheme is more economical due to the amount of discrete information needed to solve the problem is significantly lower than the traditional finite-dimensional approach.
  • A NEW COUPLED SUBDIFFUSION MODEL AND ITS PARTITIONED TIME-STEPPING ALGORITHM
    Zheng Li, Yuting Xiang, Hui Xu, Jiaping Yu, Haibiao Zheng
    Journal of Computational Mathematics. 2026, 44(3): 634-649. https://doi.org/10.4208/jcm.2504-m2025-0005
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    This paper investigates an interface-coupled fractional subdiffusion model, featuring two subdiffusion equations in adjacent domains connected by an interface allowing bidirectional energy transfer. The fractional derivative, accounting for long-term medium effects, introduces challenges in theoretical analysis and computational efficiency. We propose a partitioned time-stepping algorithm using higher-order extrapolations on the interface term to decouple the system with improved temporal accuracy, combined with finite element spatial approximations. Rigorous theoretical analysis demonstrates unconditional stability and optimal L2 norm error estimates, supported by several numerical experiments.
  • TENSOR COMPLETION VIA MINIMUM AND MAXIMUM OPTIMIZATION WITH NOISE
    Chuanlong Wang, Rongrong Xue
    Journal of Computational Mathematics. 2026, 44(3): 650-669. https://doi.org/10.4208/jcm.2504-m2024-0005
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    In this paper, the novel optimization model for solving tensor completion with noise is proposed, its objective function is a convex combination of the minimum nuclear norm and maximum nuclear norm. The necessary condition and sufficient condition of the stationary point and optimal solution are discussed. Based on the proximal gradient algorithm and feasible direction method, we design the new algorithm for solving the proposed nonconvex and nonsmooth optimization problem and prove that the sub-sequence generated by the new algorithm converges to the stationary point. Finally, experimental results on the random sample completions and images show that the proposed optimization and algorithm are superior to the compared algorithms in CPU time or precision.
  • ROTHE METHOD AND NUMERICAL ANALYSIS FOR A SUB-DIFFUSION EQUATION WITH CLARKE SUBDIFFERENTIAL
    Yujie Li and Chuanju Xu
    Journal of Computational Mathematics. 2026, 44(3): 670-699. https://doi.org/10.4208/jcm.2504-m2024-0235
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    This paper is devoted to the study of a sub-diffusion equation involving a Clarke subdifferential boundary condition. It describes transport of particles governed by the anomalous diffusion in media with boundary semipermeability. The weak formulation of the model problem results in a time fractional parabolic hemivariational inequality. We first construct an abstract hemivariational evolutionary inclusion and prove its unique solvability using a time-discretization approximation, known as the Rothe method. In addition, a numerical approach based on a finite difference scheme in time and finite dimensional approximation in space is proposed and analyzed for the abstract problem. These results are then applied to establish the convergence of the numerical solution of the model problem. Under appropriate regularity assumptions, an optimal order error estimate for the linear finite element method is derived. Some numerical examples are provided to support the theoretical results.
  • STABILITY ANALYSIS FOR MAXIMUM PRINCIPLE PRESERVING EXPLICIT ISOTROPIC SCHEMES OF THE ALLEN-CAHN EQUATION
    Jyoti, Seokjun Ham, Soobin Kwak, Youngjin Hwang, Seungyoon Kang, Junseok Kim
    Journal of Computational Mathematics. 2026, 44(3): 700-721. https://doi.org/10.4208/jcm.2504-m2024-0106
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    In practical applications, the Allen-Cahn (AC) equation is commonly used to model microstructure evolutions, including alloy solidification, crystal growth, fingerprint image restoration, and image segmentation. However, when we discretize the AC equation with a conventional finite difference scheme, the directional bias in error terms introduces anisotropy into the numerical results, affecting interface dynamics. To address this issue, we use two- and three-dimensional isotropic finite difference schemes to solve the AC equation. Stability of the proposed algorithm is verified by deriving the time step constraints in both 2D and 3D domains. To demonstrate the sharp estimation of the stability constraints, we conducted several numerical experiments and found the maximum principle is guaranteed under the analyzed time-step constraint.
  • NEWTON ITERATIVE INVERSION METHOD FOR INVERSE OBSTACLE SCATTERING IN A LAYERED MEDIUM
    Hao Wu, Jiaqing Yang
    Journal of Computational Mathematics. 2026, 44(3): 722-746. https://doi.org/10.4208/jcm.2504-m2024-0272
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    Consider the inverse acoustic scattering of time-harmonic point sources by a locally perturbed interface with bounded obstacles embedded in the lower half-space. A Newton-type iterative method is proposed to simultaneously reconstruct the locally rough interface and embedded obstacles by taking partial near-field measurements in the upper half-space. The method relies on a differentiability analysis of the scattering problem with respect to the locally rough interface and the embedded obstacle, which is established by introducing a kind of new shape derivatives and reducing the original model to an equivalent system of integral equations defined in a bounded domain. With a slight modification, the inversion method can be easily generalized to reconstruct local perturbations of a global rough interface. Finally, numerical results are presented to illustrate the effectiveness of the inversion method with the multi-frequency data.
  • SPECTRAL SOLUTIONS OF A FRACTIONAL-ORDER MATHEMATICAL MODEL FOR LUNG CANCER, SENSITIVITY ANALYSIS, AND FEEDBACK CONTROL
    Khadijeh Sadri, David Amilo
    Journal of Computational Mathematics. 2026, 44(3): 747-778. https://doi.org/10.4208/jcm.2504-m2024-0211
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    A fractional-order mathematical model of lung cancer is used to describe the dynamics of tumor growth and the interactions between cancer cells and immune cells. To obtain approximate solutions and better understand the behavior of the state functions, a pseudoo-perational collocation scheme employing shifted Jacobi polynomials as basis functions is introduced. Initially, the existence and uniqueness of solutions to the model are established using the Leray-Schauder fixed-point theorem. Error bounds for the residual functions are estimated within a Jacobi-weighted L2-space. To enhance the accuracy and reliability of the results, two distinct strategies are implemented: sensitivity analysis and feedback control. The feedback control of the proposed pseudo-operational spectral method is performed using the method of Lagrange multipliers, marking its first application in this context. Spectral solutions are derived by applying the pseudo-operational scheme to both the original model and the model with control functions. Improved performance and outputs are anticipated following the application of the feedback control strategy. Finally, comprehensive biological interpretations of the results are provided, offering insights into the practical implications of the model.
  • AN ADAPTIVE ALGORITHM FOR L1-FIDELITY COLOR IMAGE RESTORATION
    Wei Wang, Chengyun Yang, Qifan Song
    Journal of Computational Mathematics. 2026, 44(3): 779-793. https://doi.org/10.4208/jcm.2503-m2024-0010
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    In this paper, we propose an adaptive algorithm for L1-fidelity color image restoration by using saturation-value total variation. The main contribution of this paper is to employ the generalized cross validation method efficiently and automatically to estimate the regularization parameter in a saturation-value total variation plus L1-fidelity color image restoration model. We consider Poisson noise and mixed noise in this paper, and the experimental results show that the visual quality and the SSIM/PSNR/SAM values of the restored images by using the proposed algorithm are competitive with other tested existing methods, which makes the proposed algorithm to be comparable both quantitatively and qualitatively.
  • AN IMPROVED ADAPTIVE ORTHOGONAL BASIS DEFLATION METHOD FOR MULTIPLE SOLUTIONS WITH APPLICATIONS TO NONLINEAR ELLIPTIC EQUATIONS IN VARYING DOMAINS
    Yangyi Ye, Lin Li, Pengcheng Xie, Haijun Yu
    Journal of Computational Mathematics. 2026, 44(3): 794-818. https://doi.org/10.4208/jcm.2505-m2024-0276
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    Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in the existing literature are verified. For the nonlinear sine-Gordon equation with parameter λ, nontrivial solutions are found for λ > λ2, here λ2 is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically verified, suggesting that their convergent limit is a delta function when the perturbation parameter goes to zero.
  • SUPERLINEARLY CONVERGENT ALGORITHMS FOR STOCHASTIC TIME-FRACTIONAL EQUATIONS DRIVEN BY WHITE NOISE
    Zhen Song, Minghua Chen, Jiankang Shi
    Journal of Computational Mathematics. 2026, 44(3): 819-842. https://doi.org/10.4208/jcm.2505-m2025-0062
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    The numerical analysis of stochastic time-fractional equations exhibits a significantly low-order convergence rate since the limited regularity of model caused by the nonlocal operator and the presence of noise. In this work, we consider stochastic time-fractional equations driven by integrated white noise, where ${ }^C D_t^\alpha \psi(x, t), 0<\alpha<2$ and $I_t^\gamma \dot{W}(x, t)$,, 0 < γ < 1. We first establish the regularity of the mild solution. Then superlinear convergence rate $\left(\mathbb{E}\left\|\psi\left(\cdot, t_n\right)-\psi^n\right\|^2\right)^{\frac{1}{2}}=O\left(\tau^{\alpha+\gamma-\frac{\alpha d}{4}-\frac{1}{2}-\varepsilon}\right)$ with sufficiently small ε term in the exponent is established based on the modified twostep backward difference formula methods. Here d represents the spatial dimension, ψn denotes the approximate solution at the n-th time step, and $\mathbb{E}$ is the expectation operator. Numerical experiments are performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the superlinear convergence analysis for the stochastic time-fractional equations with integrated white noise.
  • ANALYSIS OF ARBITRARY HIGH ORDER SPECTRAL VOLUME METHOD FOR HYPERBOLIC CONSERVATION LAWS OVER RECTANGULAR MESHES
    Waixiang Cao, Zhimin Zhang, Qingsong Zou
    Journal of Computational Mathematics. 2026, 44(3): 843-870. https://doi.org/10.4208/jcm.2504-m2024-0201
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    This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the L2 norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.
  • AN ANALYSIS OF THE SATURATION ASSUMPTION FOR POLYNOMIAL INTERPOLATION WITH APPLICATION TO ADAPTIVE FINITE ELEMENT METHODS
    Randolph E. Bank, Jinchao Xu, Harry Yserentant
    Journal of Computational Mathematics. 2026, 44(3): 871-890. https://doi.org/10.4208/jcm.2510-m2024-0087
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    The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods. In this work we provide an analysis of this assumption in the simple setting of interpolation. We have proved elsewhere [Bank and Yserentant, Numer. Math., 131:1 (2015)] that interpolation error is both reliable and efficient as an a posteriori error estimate. Thus behavior of interpolation error is indicative of the behavior of the error in the exact finite element solution of a PDE as well as any practical a posteriori error estimate that is also reliable and efficient.