Pub Date : 2025-11-11DOI: 10.1016/j.apnum.2025.11.003
Utku Erdoğan , Gabriel Lord
In this paper, we develop numerical methods for solving stochastic differential equations with solutions that evolve within a hypercube D in . Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order , and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that, in most cases, have a smaller error constant.
{"title":"Preserving invariant domains and strong approximation of stochastic differential equations","authors":"Utku Erdoğan , Gabriel Lord","doi":"10.1016/j.apnum.2025.11.003","DOIUrl":"10.1016/j.apnum.2025.11.003","url":null,"abstract":"<div><div>In this paper, we develop numerical methods for solving stochastic differential equations with solutions that evolve within a hypercube <em>D</em> in <span><math><msup><mi>R</mi><mi>d</mi></msup></math></span>. Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order <span><math><mstyle><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></math></span>, and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that, in most cases, have a smaller error constant.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 30-45"},"PeriodicalIF":2.4,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616684","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}
Pub Date : 2025-11-08DOI: 10.1016/j.apnum.2025.11.001
Changlun Ye , Hai Bi , Liangkun Xu , Xianbing Luo
In this paper, for the Cahn-Hilliard equation with dynamic boundary conditions, we establish a variational time stepping numerical scheme integrated with finite element methods. This scheme is a structure-preserving scheme, which effectively maintains the inherent physical properties of the continuous model including mass conservation and energy dissipation. We demonstrate the existence of discrete solutions without restrictions on the discretization parameters, and establish the uniqueness under mild conditions. Finally, we present ample numerical results which validate our theoretical findings and demonstrate that our numerical scheme can achieve second-order convergence in time. We also apply our scheme to the KLS (proposed by P. Knopf, K.F. Lam, and J. Stange) and KLLM (proposed by P. Knopf, K. F. Lam, C. Liu, and S. Metzger) models, two other Cahn-Hilliard models with dynamic boundaries, and verify that the solutions of KLS model converge to the solutions of KLLM model numerically.
本文针对具有动态边界条件的Cahn-Hilliard方程，建立了与有限元法相结合的变分时步数值格式。该方案是一种结构保持方案，有效地保持了连续模型固有的物理性质，包括质量守恒和能量耗散。我们证明了不受离散化参数限制的离散解的存在性，并在温和条件下证明了其唯一性。最后，我们给出了大量的数值结果来验证我们的理论发现，并证明了我们的数值格式在时间上可以达到二阶收敛。我们还将我们的方案应用于KLS （P. Knopf， K.F. Lam, and J. Stange提出）和KLLM （P. Knopf， K.F. Lam， C. Liu, and S. Metzger提出）模型以及另外两种具有动态边界的Cahn-Hilliard模型，并在数值上验证了KLS模型的解收敛于KLLM模型的解。
{"title":"A structure-preserving variational time stepping scheme for Cahn-Hilliard equation with dynamic boundary conditions","authors":"Changlun Ye , Hai Bi , Liangkun Xu , Xianbing Luo","doi":"10.1016/j.apnum.2025.11.001","DOIUrl":"10.1016/j.apnum.2025.11.001","url":null,"abstract":"<div><div>In this paper, for the Cahn-Hilliard equation with dynamic boundary conditions, we establish a variational time stepping numerical scheme integrated with finite element methods. This scheme is a structure-preserving scheme, which effectively maintains the inherent physical properties of the continuous model including mass conservation and energy dissipation. We demonstrate the existence of discrete solutions without restrictions on the discretization parameters, and establish the uniqueness under mild conditions. Finally, we present ample numerical results which validate our theoretical findings and demonstrate that our numerical scheme can achieve second-order convergence in time. We also apply our scheme to the KLS (proposed by P. Knopf, K.F. Lam, and J. Stange) and KLLM (proposed by P. Knopf, K. F. Lam, C. Liu, and S. Metzger) models, two other Cahn-Hilliard models with dynamic boundaries, and verify that the solutions of KLS model converge to the solutions of KLLM model numerically.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 346-372"},"PeriodicalIF":2.4,"publicationDate":"2025-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145576255","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}
Pub Date : 2025-11-06DOI: 10.1016/j.apnum.2025.10.017
K. Mustapha, W. Mclean, J. Dick, Q.T. Le Gia
Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter λ → ∞, or equivalently as the Poisson ratio ν → 1/2. This effect is known as locking or non-robustness. For the piecewise linear case, the error in the L2(Ω)-norm of the standard Galerkin conforming FEM is bounded by Cλh2, resulting in poor accuracy for practical values of h if λ is sufficiently large. In this paper, we show that the locking phenomenon can be reduced by replacing λ with or in the stiffness matrix, where μ is the second Lamé parameter and L is the diameter of the body Ω. We prove that with this modification, the error in the L2(Ω)-norm is bounded by Ch for a constant C that does not depend on λ. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if λ is larger than about μL/h. Our analysis also shows that the error in the H1(Ω)-norm is bounded by , which improves the Cλ1/2h estimate for the case of conforming FEM.
{"title":"A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity","authors":"K. Mustapha, W. Mclean, J. Dick, Q.T. Le Gia","doi":"10.1016/j.apnum.2025.10.017","DOIUrl":"10.1016/j.apnum.2025.10.017","url":null,"abstract":"<div><div>Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter <em>λ</em> → ∞, or equivalently as the Poisson ratio <em>ν</em> → 1/2. This effect is known as <em>locking</em> or <em>non-robustness</em>. For the piecewise linear case, the error in the <strong>L</strong><sup>2</sup>(Ω)-norm of the standard Galerkin conforming FEM is bounded by <em>Cλh</em><sup>2</sup>, resulting in poor accuracy for practical values of <em>h</em> if <em>λ</em> is sufficiently large. In this paper, we show that the locking phenomenon can be reduced by replacing <em>λ</em> with <span><math><mrow><msub><mi>λ</mi><mi>h</mi></msub><mo>=</mo><mi>λ</mi><mi>μ</mi><mo>/</mo><msqrt><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>λ</mi><mi>h</mi><mo>/</mo><mi>L</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo><</mo><mi>λ</mi></mrow></math></span> or <span><math><mrow><msub><mi>λ</mi><mi>h</mi></msub><mo>=</mo><mi>λ</mi><mi>μ</mi><mo>/</mo><mrow><mo>(</mo><mi>μ</mi><mo>+</mo><mi>λ</mi><mi>h</mi><mo>/</mo><mi>L</mi><mo>)</mo></mrow><mo><</mo><mi>λ</mi></mrow></math></span> in the stiffness matrix, where <em>μ</em> is the second Lamé parameter and <em>L</em> is the diameter of the body Ω. We prove that with this modification, the error in the <strong>L</strong><sup>2</sup>(Ω)-norm is bounded by <em>Ch</em> for a constant <em>C</em> that does not depend on <em>λ</em>. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if <em>λ</em> is larger than about <em>μL</em>/<em>h</em>. Our analysis also shows that the error in the <strong>H</strong><sup>1</sup>(Ω)-norm is bounded by <span><math><mrow><mi>C</mi><msubsup><mi>λ</mi><mi>h</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mspace></mspace><mi>h</mi></mrow></math></span>, which improves the <em>Cλ</em><sup>1/2</sup> <em>h</em> estimate for the case of conforming FEM.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 18-29"},"PeriodicalIF":2.4,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571207","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}
Pub Date : 2025-11-06DOI: 10.1016/j.apnum.2025.10.019
M. Sharifi , A. Abdi , M. Braś , G. Hojjati
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 A–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.
{"title":"On implicit second derivative two-step peer methods with RK stability for ODEs","authors":"M. Sharifi , A. Abdi , M. Braś , 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":"2025-11-06","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}
Pub Date : 2025-11-03DOI: 10.1016/j.apnum.2025.10.018
Stefan R. Panic
A novel quasi-Newton method for solving systems of nonlinear equations by leveraging directional approximations of both the gradient and curvature via Legendre-Gauss quadrature has been proposed. The method reformulates the root-finding problem as the minimization of a scalar merit function , and approximates second-order information using three-node orthogonal polynomial integration along search directions. A rank-1 approximation of the Jacobian action is constructed without requiring explicit derivative information. The resulting scheme features a scalar curvature parameter that dynamically controls the step size, enabling stable updates through an inexact Armijo-type line search. The method remains numerically stable across problems without requiring explicit Jacobian evaluations or storage. We establish global convergence under mild assumptions and explore quasi-Newton properties under additional curvature conditions. Extensive numerical experiments demonstrate competitive accuracy, robustness, and reduced iteration counts compared to existing diagonal quasi-Newton methods.
{"title":"Directional gradient and curvature approximation via Legendre quadrature in unconstrained optimization","authors":"Stefan R. Panic","doi":"10.1016/j.apnum.2025.10.018","DOIUrl":"10.1016/j.apnum.2025.10.018","url":null,"abstract":"<div><div>A novel quasi-Newton method for solving systems of nonlinear equations by leveraging directional approximations of both the gradient and curvature via Legendre-Gauss quadrature has been proposed. The method reformulates the root-finding problem <span><math><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span> as the minimization of a scalar merit function <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mrow><mo>∥</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∥</mo></mrow><mn>2</mn></msup></mrow></math></span>, and approximates second-order information using three-node orthogonal polynomial integration along search directions. A rank-1 approximation of the Jacobian action is constructed without requiring explicit derivative information. The resulting scheme features a scalar curvature parameter <span><math><msub><mi>γ</mi><mi>k</mi></msub></math></span> that dynamically controls the step size, enabling stable updates through an inexact Armijo-type line search. The method remains numerically stable across problems without requiring explicit Jacobian evaluations or storage. We establish global convergence under mild assumptions and explore quasi-Newton properties under additional curvature conditions. Extensive numerical experiments demonstrate competitive accuracy, robustness, and reduced iteration counts compared to existing diagonal quasi-Newton methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 294-309"},"PeriodicalIF":2.4,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145463216","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}
Pub Date : 2025-10-30DOI: 10.1016/j.apnum.2025.10.014
Zhihui Zhao, Hong Li
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 and 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.
{"title":"Numerical analysis of a space-time finite volume element method for the nonlinear Schrödinger equation","authors":"Zhihui Zhao, 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":"2025-10-30","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}
Pub Date : 2025-10-25DOI: 10.1016/j.apnum.2025.10.015
Yanping Lin , Shangyou Zhang
Stabilizer-free virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous 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.
{"title":"Stabilizer-free polygonal and polyhedral virtual elements","authors":"Yanping Lin , 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":"2025-10-25","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}
Pub Date : 2025-10-24DOI: 10.1016/j.apnum.2025.10.013
Jianfeng Liu , Tingchun Wang , Jingjun Zhang , Xuanxuan Zhou
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 . This ensures that all singular points 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 in the maximum norm, with no restrictions on the grid ratio, where, represents the mesh size and 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.
{"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 , Tingchun Wang , Jingjun Zhang , 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":"2025-10-24","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}
Pub Date : 2025-10-22DOI: 10.1016/j.apnum.2025.10.012
Akbar Shirilord, Mehdi Dehghan
In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images.
{"title":"A unified preconditioned minimal residual (PMR) algorithm for matrix problems: Linear systems, multiple right-hand sides linear systems, least squares problems, inversion and pseudo-inversion with application to color image encryption","authors":"Akbar Shirilord, Mehdi Dehghan","doi":"10.1016/j.apnum.2025.10.012","DOIUrl":"10.1016/j.apnum.2025.10.012","url":null,"abstract":"<div><div>In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 216-245"},"PeriodicalIF":2.4,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145413757","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}
Pub Date : 2025-10-22DOI: 10.1016/j.apnum.2025.10.011
Guillermo Albuja , Andrés I. Ávila , Miguel Murillo
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.
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 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 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 -adaptation to the classical methods (-Newton’s, -L-scheme, and -modified Picard). Next, we apply to the error-correction schemes (-AtS, -ALtS, -ALP, -ALN). Finally, we consider a combination of the L-scheme and the -adaptive Newton’s Method, mixing both methods (-LAN).
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 -LAN scheme has advantages, using fewer iterations in most examples.
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