Applied Numerical Mathematics最新文献

{"title":"Global superlinear linearization schemes based on adaptive strategies for solving Richards’ equation","authors":"Guillermo Albuja ,&nbsp;Andrés I. Ávila ,&nbsp;Miguel Murillo","doi":"10.1016/j.apnum.2025.10.011","DOIUrl":"10.1016/j.apnum.2025.10.011","url":null,"abstract":"<div><div>The Richards’ equation is a nonlinear degenerate parabolic differential equation, whose numerical solutions depend on the linearization methods used to deal with the degeneracy. Those methods have two main properties: convergence (global v.s. local) and order (linear v.s. quadratic). Among the main methods, Newton’s Method, the modified Picard method, and the L-scheme have one good property but not the other. Mixed schemes get the best of both properties, starting with a global linear method and following with a quadratic local scheme without a clear rule to switch from a global method to a local method.</div><div>In this work, we use two different approaches to define new global superlinear and quadratic schemes. First, we use an error-correction convex combination of classical linearization methods, a global linear method, and a quadratic local method by selecting the parameter <span><math><msubsup><mi>λ</mi><mi>k</mi><mi>n</mi></msubsup></math></span> via an error-correction approach to get fixed-point convergent sequences. We built an error-correction type-Secant scheme (ECtS) without derivatives to get a superlinear global scheme. Next, we build the convex combination of the L-scheme with three global schemes: the type-Secant scheme (ECLtS), the modified Picard scheme (ECLP), and Newton’s scheme (ECLN) to obtain global superlinear convergent schemes. Second, we use a parameter <span><math><mi>τ</mi></math></span> to adapt the time step in the general Newton-Raphson method, applying to three classical linearizations and the new three error-correction linearizations. For the new schemes, we first apply the <span><math><mi>τ</mi></math></span>-adaptation to the classical methods (<span><math><mi>τ</mi></math></span>-Newton’s, <span><math><mi>τ</mi></math></span>-L-scheme, and <span><math><mi>τ</mi></math></span>-modified Picard). Next, we apply to the error-correction schemes (<span><math><mi>τ</mi></math></span>-AtS, <span><math><mi>τ</mi></math></span>-ALtS, <span><math><mi>τ</mi></math></span>-ALP, <span><math><mi>τ</mi></math></span>-ALN). Finally, we consider a combination of the L-scheme and the <span><math><mi>τ</mi></math></span>-adaptive Newton’s Method, mixing both methods (<span><math><mi>τ</mi></math></span>-LAN).</div><div>We test the twelve new schemes with five examples given in the literature, showing that they are robust and fast, including cases when Newton’s scheme does not converge. Moreover, we include an example which uses the Gardner exponential nonlinearities, showing that L- and L2-schemes are as slow as linearization techniques. Some new schemes show high performance in different examples. The <span><math><mi>τ</mi></math></span>-LAN scheme has advantages, using fewer iterations in most examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 189-215"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145413755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A feasible interior-point method with full-Newton step for P*(κ)-weighted linear complementarity problem via the algebraically equivalent transformation","authors":"Xiaoni Chi ,&nbsp;Lin Gan ,&nbsp;Zhuoran Gao ,&nbsp;Jein-Shan Chen","doi":"10.1016/j.apnum.2025.10.003","DOIUrl":"10.1016/j.apnum.2025.10.003","url":null,"abstract":"<div><div>This paper investigates feasible interior-point method (IPM) with full-Newton step for <span><math><mrow><msub><mi>P</mi><mo>*</mo></msub><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow></mrow></math></span>-weighted linear complementarity problem (WLCP). In particular, by applying the algebraically equivalent transformation (AET) for linear optimization, we obtain the new search directions by solving the perturbed Newton system. The AET of the Newton system is based on the kernel function <span><math><mrow><mi>φ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>t</mi><mo>−</mo><msqrt><mi>t</mi></msqrt></mrow></math></span>, which is used for solving WLCP for the first time. At each iteration, our algorithm takes only full-Newton steps. Therefore, no line-searches are needed to update the iterates. We show the strict feasibility of the full-Newton step and the polynomial iteration complexity of our algorithm under suitable assumptions. Some numerical experiments demonstrate the effectiveness of the proposed algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 144-166"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal error estimate of a conservative, efficient and accurate finite difference scheme for the nonlinear Schrodinger equation with Dirac delta potentials","authors":"Jianfeng Liu ,&nbsp;Tingchun Wang ,&nbsp;Jingjun Zhang ,&nbsp;Xuanxuan Zhou","doi":"10.1016/j.apnum.2025.10.013","DOIUrl":"10.1016/j.apnum.2025.10.013","url":null,"abstract":"<div><div>This paper is concerned with constructing and analyzing an efficient and accurate finite difference scheme for the nonlinear Schrödinger equation with Dirac delta potentials. The proposed scheme exhibits several notable features: (1) It is derived from an accurate approximation of the internal interface matching conditions, enabling the use of varying mesh sizes in subintervals divided by the singular points <span><math><mrow><msub><mi>ξ</mi><mi>z</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>. This ensures that all singular points <span><math><mrow><msub><mi>ξ</mi><mi>z</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> align with the grid nodes; (2) The scheme is proved to preserve the total mass and energy in the discrete sense; (3) The nonlinear term is discretized in a way that facilitates temporal linearization, and the spatial grid stencil comprises only three nodes. This translates to solving a tridiagonal system of linear algebraic equations efficiently using the Thomas algorithm at each time step; (4) The convergence order of the proposed scheme is proved to be <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mi>τ</mi><mn>2</mn></msup><mo>)</mo></mrow></math></span> in the maximum norm, with no restrictions on the grid ratio, where, <span><math><mi>h</mi></math></span> represents the mesh size and <span><math><mi>τ</mi></math></span> denotes the time step. We then derive two other efficient and accurate finite difference schemes by enhancing the accuracy of the approximation of the internal interface matching conditions, one still preserves the mass and energy in the discrete sense but needs uniform grid, the other one is nonconservative but allows different mesh sizes in different subintervals. Numerical results are carried out to validate our theoretical conclusions and simulate several dynamical behaviors of the nonlinear Schrödinger equation with Dirac delta potentials.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 246-264"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145463524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical analysis of a space-time finite volume element method for the nonlinear Schrödinger equation","authors":"Zhihui Zhao,&nbsp;Hong Li","doi":"10.1016/j.apnum.2025.10.014","DOIUrl":"10.1016/j.apnum.2025.10.014","url":null,"abstract":"<div><div>In this paper, for the first time, we propose a space-time finite volume element (STFVE) method for the cubic nonlinear Schrödinger (NLS) equation. In contrast to the space-time finite element (STFE) method, this method not only is easy to achieve high accuracy in both space and time directions, but also the method itself can maintain the conservation laws of physical quantities, and thus it is well suited to solve the conservation laws equations. For the constructed STFVE scheme, the rigorous theoretical analyses are given including the proof of the existence of the resulting approximations and the optimal <span><math><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><msup><mi>L</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><msup><mi>H</mi><mn>1</mn></msup><mo>)</mo></mrow></mrow></math></span> norms estimates are obtained in the case that the spatial mesh parameter is not related to the time step size. Finally, some numerical tests are shown to confirm the theoretical findings, unconditional stability and the conservation properties of the STFVE method. Also, the numerical tests show that the STFVE method simulates the NLS equation well.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 277-293"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145463526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On implicit second derivative two-step peer methods with RK stability for ODEs","authors":"M. Sharifi ,&nbsp;A. Abdi ,&nbsp;M. Braś ,&nbsp;G. Hojjati","doi":"10.1016/j.apnum.2025.10.019","DOIUrl":"10.1016/j.apnum.2025.10.019","url":null,"abstract":"<div><div>Two-step peer methods (TSPMs) for solving ODEs have been extended to incorporate both first and second derivatives of the solution, leading to the introduction of second derivative TSPMs and referred to as STSPMs. In this paper, we introduce second derivative diagonally implicit two-step peer methods as a subclass of STSPMs. This class of the methods is categorized into four different types based on their specific applications (for non-stiff or stiff ODEs) as well as their architectures (parallel or sequential). We investigate the derivation of these methods equipped with the Runge–Kutta stability property with <em>A</em>–stability for implicit ones. Furthermore, we derive examples of such methods up to order four. Finally, the proposed methods are examined through numerical experiments on some well-known stiff problems, demonstrating their effectiveness in terms of both accuracy and efficiency.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 329-345"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145517224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Structure-preserving numerical methods for phase-field surfactant models based on the supplemental variable method (SVM)","authors":"Mengchun Yuan,&nbsp;Qi Li","doi":"10.1016/j.apnum.2025.10.007","DOIUrl":"10.1016/j.apnum.2025.10.007","url":null,"abstract":"<div><div>Numerous efficient numerical algorithms have been developed for phase-field surfactant models, including the convex splitting method, the scalar auxiliary variable (SAV) approach, the invariant energy quadratization (IEQ) approach, and the new Lagrange multiplier method. In this paper, we introduce novel numerical schemes based on the supplementary variable method for solving and simulating the binary fluid phase-field surfactant model of the Cahn-Hilliard type. The key innovation of our method is the introduction of an auxiliary variable, which reformulates the original problem into a constrained optimization framework. This reformulation offers several significant advantages over existing approaches. Firstly, our schemes require solving only a few Poisson-type systems with constant coefficient matrices, significantly reducing computational costs. Secondly, the proposed schemes preserve mass conservation and adhere to the original energy dissipation law at the discrete level, in contrast to the SAV and IEQ approaches, which adhere to modified dissipation laws. Additionally, we rigorously establish the energy stability of the schemes. Extensive 2D and 3D numerical experiments confirm the accuracy and efficiency of the proposed schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 123-143"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Schur method for robust eigenvalue assignment of second-order singular systems","authors":"Kang Hu,&nbsp;Huiqing Xie","doi":"10.1016/j.apnum.2025.09.002","DOIUrl":"10.1016/j.apnum.2025.09.002","url":null,"abstract":"<div><div>A new method for eigenvalue assignment problem of a second-order singular system is proposed via displacement-velocity-acceleration feedback. The real Schur form of a regular quadratic pencil and a robustness measurement for the closed-loop second-order system are introduced. On these grounds, a Schur method for robust eigenvalue assignment of a second-order singular system is proposed. The presented method also can be seen as an extension of the Schur method for eigenvalue assignment of a first-order system to a second-order system. However, our method directly deals with a second-order system and avoids to transform a second-order system into a first-order system. There are two open problems in the Schur method for eigenvalue assignment problem. One is how to determine the first Schur vector and another is how to order the eigenvalues in the Schur form. We provide the strategies for these two open problems, which are our main contributions. The efficiency of the proposed method is illustrated by some numerical examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 44-65"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Error estimate for the Cahn-Hilliard equation by the SAV-Runge-Kutta scheme","authors":"Lizhen Chen ,&nbsp;Xiaozhuang Ma ,&nbsp;Guohui Zhang ,&nbsp;Jing Zhang","doi":"10.1016/j.apnum.2025.10.001","DOIUrl":"10.1016/j.apnum.2025.10.001","url":null,"abstract":"<div><div>In this paper, we introduce a Scalar Auxiliary Variable (SAV) approach combined with the Runge-Kutta method for solving the Cahn-Hilliard equation. Within the framework of the SAV method, this Runge-Kutta method is particularly well-suited for obtaining numerical solutions that preserve key structural properties of the system, including energy conservation and momentum conservation. In the SAV-Runge-Kutta method, through a comparison of approximate solutions derived at various time points, the error for each step can be rigorously estimated. This approach thereby guarantees the stability and accuracy of the entire solution process. Finally, to illustrate the effectiveness and precision of our proposed method, we present several numerical examples. These examples demonstrate the capability of the SAV-Runge-Kutta method to accurately capture the intricate dynamics of the Cahn-Hilliard equation while maintaining energy conservation and momentum conservation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 66-83"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A semi-implicit finite difference approach for the swift hohenberg equation: Stability, convergence, and pattern formation","authors":"Adérito Araújo,&nbsp;Diogo Cotrim","doi":"10.1016/j.apnum.2025.11.002","DOIUrl":"10.1016/j.apnum.2025.11.002","url":null,"abstract":"<div><div>The Swift-Hohenberg equation (SH-PDE) is a fundamental model for pattern formation in nonlinear systems with symmetry breaking instabilities. This work presents a semi-implicit finite difference scheme for solving the SH-PDE, taking advantage of a linear/nonlinear decomposition to optimise stability and computational efficiency. We rigorously establish the theoretical properties of the method, including bounds and error estimates, proving stability and convergence under appropriate conditions. Numerical experiments confirm these conclusions, demonstrating second-order spatial accuracy and first-order temporal accuracy. The method is tested under various initial conditions and nonlinearities, capturing characteristic patterns such as stripes, rolls and dots, in line with the expected behaviour of SH-PDE. These results emphasise the robustness and efficiency of the proposed approach, positioning it as a powerful tool for studying pattern formation in nonlinear systems.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 373-383"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145576256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stabilizer-free polygonal and polyhedral virtual elements","authors":"Yanping Lin ,&nbsp;Shangyou Zhang","doi":"10.1016/j.apnum.2025.10.015","DOIUrl":"10.1016/j.apnum.2025.10.015","url":null,"abstract":"<div><div>Stabilizer-free <span><math><msub><mi>P</mi><mi>k</mi></msub></math></span> virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous <span><math><msub><mi>P</mi><mi>k</mi></msub></math></span> polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision of each polyhedron. With such an accurate and proper interpolation, the stabilizer of the virtual elements is eliminated while the system is kept positive-definite. We show that the stabilizer-free virtual elements converge at the optimal order in 2D and 3D. Numerical examples are computed, validating the theory.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 265-276"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145463525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}