28 September 2026, Volume 48 Issue 4

    

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Youth Review
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  TIME-HARMONIC WAVE MODES FOR THE HELMHOLTZ EQUATION IN LAYERED PERIODIC STRUCTURES

  Zhang Jiayi, Hu Guanghui

  Mathematica Numerica Sinica. 2026, 48(4): 565-586. https://doi.org/10.12286/jssx.j2026-1407

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  In this paper, we review the time-harmonic wave modes of the Helmholtz equation subject to quasi-periodic boundary conditions in two-dimensional layered periodic structures. Using the spectral theory of periodic ordinary differential operators and the Bloch theory for elliptic partial differential equations, we present the wave modes for electromagnetic wave propagation under TE polarization in three different types of inhomogeneous periodic layered materials: periodic media depending only on the horizontal coordinate, periodic media depending only on the vertical coordinate, and general biperiodic media. By employing the finite energy property of wave propagation, we characterize the radiating modes of the periodic Helmholtz equation in infinite inhomogeneous media, thereby laying a mathematical foundation for model analysis and numerical computations.
Articles
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  AN ENERGY STABLE AND MAXIMUM PRINCIPLE PRESERVING SECOND-ORDER $\overline{L1}$ SCHEME FOR MULTI-TERM TIME-FRACTIONAL ALLEN-CAHN EQUATION

  Zhou Yongtao, Cai Longbo

  Mathematica Numerica Sinica. 2026, 48(4): 587-603. https://doi.org/10.12286/jssx.j2024-1246

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  In this study, we develop a $\overline{L1}$ scheme for solving multi-term time-fractional Allen-Cahn equations. To effectively manage the weak singularity of the solution at the initial time, we implement graded meshes in the temporal direction. We rigorously establish the unique solvability, discrete energy stability and discrete maximum principle of the solution associated with the proposed scheme. The numerical results corroborate the validity of the theoretical analysis and illustrate the efficacy of the proposed method in mitigating the effects of weak singularity at the initial time and accommodating long-term memory.
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  AN INERTIAL BREGMAN PROXIMAL GRADIENT ALGORITHM FOR SOLVING THE DOUBLE-LAYER OPTIMIZATION MODEL OF TENSOR COMPLETION

  Lan Ze, Yan Xihong, Li Jiaqi

  Mathematica Numerica Sinica. 2026, 48(4): 604-626. https://doi.org/10.12286/jssx.j2025-1336

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  This paper proposes an inertial Bregman proximal gradient algorithm for solving a double-layer model of tensor completion. Under the framework of the proximal gradient method, the algorithm introduces the Bregman distance into the subproblems to preserve the structural information of tensor data, thereby improving computational efficiency. The global convergence of the algorithm is established through the Kurdyka-Łojasiewicz property. Numerical experiments demonstrate the effectiveness of the proposed algorithm in terms of recovery accuracy and computational efficiency.
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  NUMERICAL SIMULATION OF CYLINDRICAL RESONANT TUNNELING DIODES

  Guo Weiyi, Jiang Haiyan, Lu Tiao, Yao Weiqi

  Mathematica Numerica Sinica. 2026, 48(4): 627-647. https://doi.org/10.12286/jssx.j2025-1347

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  This paper develops a high-accuracy numerical algorithm for quantum transport modeling in three-dimensional cylindrical resonant tunneling diodes. Utilizing the mode-space approximation, we reduce the cylindrical Schrödinger equation to coupled one-dimensional eigenvalue and transport problems. The eigenvalue problem is solved using a spectral collocation method with Chebyshev-Gauss-Lobatto points, and its rigorous convergence analysis within the Galerkin spectral framework demonstrates exponential convergence rates. Numerical experiments validate the theoretical results and show that the proposed method significantly outperforms traditional central difference schemes in computational accuracy. The transport direction is discretized using a finite difference scheme. This paper employs a high-accuracy numerical algorithm to simulate the transport characteristics of resonant tunneling diodes. For the double-barrier potential configuration, quantum resonant tunneling phenomena are clearly demonstrated through transmission coefficient calculations. The current-voltage characteristics of resonant tunneling diodes are numerically simulated, successfully reproducing the negative differential resistance behavior. Furthermore, the underlying mechanism of negative resistance is rationally explained by illustrating the electron density distribution at specific bias voltages.
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  A GLOBALLY ASYMPTOTICALLY STABLE DYNAMICAL MODEL FOR SOLVING EVLCPS

  Wei Yufei, Lin Shiping, Chen Cairong, Han Deren

  Mathematica Numerica Sinica. 2026, 48(4): 648-667. https://doi.org/10.12286/jssx.j2025-1350

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  The extended vertical linear complementarity problem (EVLCP) is first transformed into a kind of new generalized absolute value equation. On this basis, a dynamical system is developed to solve the EVLCP, and the global asymptotic stability of the proposed model is established under certain conditions. In contrast to existing approaches, the proposed model eliminates the need for smoothing the EVLCP, and numerical experiments demonstrate its superior performance compared with existing continuous dynamical systems.
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  A CONTINUOUS-DOMAIN DYNAMIC PROGRAMMING METHOD FOR ONE-DIMENSIONAL CHAIN L₁-TV MODELS

  Yan Kai, Fu Meiyan, Yao Chengbao

  Mathematica Numerica Sinica. 2026, 48(4): 668-687. https://doi.org/10.12286/jssx.j2025-1353

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  This paper studies a class of one-dimensional chain-structured convex optimization problems with total variation (TV) regularization, in which each position contains two $L_1$ data terms and the TV weights are allowed to vary across positions. We propose an exact method based on continuous-domain dynamic programming (CD-DP). In the forward stage, the algorithm maintains the piecewise-linear convex structure of the value function using a breakpoint stack and a lower convex hull representation, enforces the dual constraints via bounded-slope projection, and preserves convexity through breakpoint refinement and stack-based lower hull maintenance. Subgradient jumps are introduced at prescribed positions to handle nonsmooth penalties. In the backward stage, back substitution recovers the primal variables while satisfying the complementary slackness conditions, thereby yielding an exact solution that satisfies the first-order optimality conditions in finitely many steps. We prove that at most a constant number of new breakpoints are introduced at each layer of the value function, implying that the total number of segments in its piecewise-linear representation grows only linearly. When the size of the candidate breakpoint set $S$ is fixed, the overall time complexity is $O(n|S|)$ and the additional memory requirement is $O(|S|)$. Numerical experiments show that, compared with the alternating direction method of multipliers (ADMM) and linear programming (LP), the proposed method achieves orders-of-magnitude speedups on large-scale instances. For the typical parameter ranges considered in this paper, the algorithm still exhibits near-linear scaling while maintaining stable and reproducible results. The method is well suited to long-chain structures and batch optimization settings, and offers a new perspective on solving $L_1$-TV convex optimization problems.
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  ENERGY-STABLE HIGH-ORDER IMPLICIT LAGRANGIAN FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL INCOMPRESSIBLE VARIABLE-DENSITY FLOWS

  Dong Xiaoyue, Li Lingxiao, Zhang Jie

  Mathematica Numerica Sinica. 2026, 48(4): 688-706. https://doi.org/10.12286/jssx.j2025-1354

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  An energy stable high-order implicit Lagrangian finite element method is proposed for three-dimensional incompressible variable density flows, based on moving curvilinear tetrahedral meshes. The spatial discretization of velocity and pressure is carried out using high-order curvilinear finite element methods that satisfy the inf-sup condition, while the implicit Euler method is chosen for time discretization. The resulting fully discrete scheme requires the solving of saddle point problems at each time step, and an augmented Lagrangian preconditioner is proposed to accelerate the convergence of the Krylov subspace methods. The Lagrangian method can accurately track the material interfaces, and thereby one can convert the calculation of surface tension into the calculation of surface divergence of the test functions on the interface. Numerical examples are conducted to verify the high-order accuracy, energy stability, effectiveness and robustness of the method.
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  ENERGY-PRESERVING EXPONENTIAL WAVE INTEGRATOR METHOD FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATIONS AND ITS ERROR ANALYSIS

  Tang Lin, Ran Maohua, Luo Jialing

  Mathematica Numerica Sinica. 2026, 48(4): 707-731. https://doi.org/10.12286/jssx.j2025-1355

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  Aiming at the computational challenges in long-term simulations of the generalized nonlinear Schrödinger equation with high-order dispersion terms, this paper proposes an energy-preserving exponential wave integrator Fourier pseudospectral method (EP-EWIFP). The study primarily investigates the impact of high-order dispersion terms on the construction of numerical methods and improves the limitations of existing methods in preserving physical properties. The proposed scheme possesses time symmetry and stability, and can strictly preserve the energy conservation law in a discrete sense. Theoretical analysis confirms that over long time scales up to \( T = \mathcal{O}(\varepsilon^{-\alpha}) \) (\( 0 \leq \alpha \leq 2 \), where \( \varepsilon \) is the nonlinear strength parameter), the method achieves uniformly bounded numerical errors of order \( \mathcal{O}(h^{m_0} + \varepsilon^{2-\alpha}\tau^2) \), where \( h \) is the spatial step size, \( \tau \) is the temporal step size, and \( m_0 \) depends on the spatial regularity of the solution. Numerical experiments verify the second-order temporal convergence, spatial spectral accuracy, and excellent energy conservation properties of the method, providing an efficient simulation tool for long-term dynamic modeling in fields such as nonlinear optics and plasma physics.
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  THEORETICAL AND NUMERICAL ANALYSIS OF THE OPTIMAL CONTROL PROBLEM FOR THE AGE-STRUCTURED SEIR MODEL

  Sun Xinyao, Gao Jianfang, Yang Zhanwen

  Mathematica Numerica Sinica. 2026, 48(4): 732-748. https://doi.org/10.12286/jssx.j2025-1357

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  This paper investigates the optimal control and numerical stability of the Susceptible-Exposed-Infected-Recovered (SEIR) model. By introducing vaccination and treatment as control variables, we derive the adjoint equations of the original problem and establish the necessary conditions for optimal control. Subsequently, the linear Implicit-Explicit (IMEX) Euler method is employed to discretize the model. It is proven that this numerical method unconditionally preserves the positivity of solutions and ensures first-order convergence between the numerical solution and the exact solution. Furthermore, a numerical optimization scheme is proposed for the optimal control problem. Through theoretical analysis and numerical simulations, the optimal control strategy is formulated, which can provide a reference for infectious disease prevention and control.
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  A FINITE DIFFERENCE-SPECTRAL COLLOCATION METHOD FOR PRICING EUROPEAN CALL OPTIONS UNDER THE FINITE MOMENT LOG STABLE MODEL

  Jiang Xiaoying, Wang Tianyu, Wei Yujie

  Mathematica Numerica Sinica. 2026, 48(4): 749-759. https://doi.org/10.12286/jssx.j2025-1359

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  This paper investigates an efficient algorithm for pricing European call options under the finite moment log stable (FMLS) model, which stands out in capturing the large movements of option prices during small time steps. The FMLS model can be characterized by the space-fractional Black-Scholes (SFBS) equation, which is more challenging to solve than the classical Black-Scholes model. To address this, we propose a new finite difference-spectral collocation scheme based on the Legendre spectral collocation method to approximate the modified SFBS model. The error analysis for the proposed approach is rigourously established. Additionally, Numerical results demonstrate the effectiveness of the proposed algorithm and are consistent with the theoretical analysis.
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  THE C³ QUASI CATMULL-ROM SPLINE FUNCTION WITH LOCAL SHAPE ADJUSTABILITY

  Li Juncheng, Liu Chengzhi

  Mathematica Numerica Sinica. 2026, 48(4): 760-774. https://doi.org/10.12286/jssx.j2025-1360

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  To overcome the limitation of the $C^{3}$ quasi Catmull-Rom spline function in shape adjustment, a set of quasi Catmull-Rom spline basis functions with local free parameters are constructed, and then the corresponding quasi Catmull-Rom spline function with local shape adjustability, named improved quasi Catmull-Rom spline function, is defined. The interpolation error of the proposed spline function is analyzed, and the method for constructing the optimal interpolation is provided. Numerical examples show that the improved quasi Catmull-Rom spline function can flexibly achieve local shape adjustment by adjusting the values of free parameters, and its optimal interpolation has significantly smaller overall errors than the optimal interpolation of the previous quasi Catmull-Rom spline function.
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  A VARIANT OF GLOBAL GMRES BY INTEGRATING SEARCH SPACE WITH HARMONIC F-RITZ VECTORS

  Zhong Yuhao, Wang Jiahui, Jing Yanfei

  Mathematica Numerica Sinica. 2026, 48(4): 775-796. https://doi.org/10.12286/jssx.j2026-1373

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  Based on the idea of Robbé and Sadkane's inexact breakdowns handling technology in IB-BGMRES [Linear Algebra Appl., 419 (2006), pp. 265-285], from the point of view of search subspace partition, a new version of the global GMRES method (Gl-GMRES), named as GLGMRES-S, is presented for solving nonsymmetric linear systems with multiple right-hand sides, which can be regarded as a global extension of IB-BGMRES. In Gl-GMRES, the block Krylov basis matrices generated by the global Arnoldi process must retain the same dimension, and therefore the two criteria used in IB-BGMRES to detect inexact breakdowns are no longer applicable. Instead of detecting inexact breakdowns, a subspace-splitting technique is introduced into the global Arnoldi process to implicitly realize subspace partitioning within the global GMRES framework, and the associated Arnoldi-like relations are theoretically established. Numerical experiments demonstrate that the practical implementation of our proposed GLGMRES-S can improve the convergence of Gl-GMRES. Furthermore, GLGMRES-S is competitive with the deflated global GMRES method GLGMRES-DR [Numer. Algorithms, 82 (2019), pp. 155-181], which augments the search space produced by the global Arnoldi process by adding harmonic $F$-Ritz vectors in an outer manner.