Pub Date : 2026-06-01Epub Date: 2026-01-27DOI: 10.1016/j.apnum.2026.01.017
Xiao Qi , Yubin Yan
Huang and Shen [Math. Comput.92 (2023) 2685–2713] proposed a semi-implicit tamed scheme for the numerical approximation of stochastic Allen–Cahn equations driven by multiplicative trace-class noise. They showed that the scheme is unconditionally stable on finite time intervals and can be efficiently implemented. In this paper, we investigate the long-time stability of this tamed scheme for stochastic Allen–Cahn equations driven by additive white noise. We also address the strong convergence analysis of the associated fully discrete scheme within the Galerkin finite element framework. The main contributions of this work are as follows: (i) by constructing a suitable Lyapunov functional, we establish the unconditional long-time stability of the tamed method; (ii) we rigorously derive the strong convergence rates of the fully discrete scheme obtained by coupling the tamed approach with the finite element method. Numerical experiments are provided to validate the theoretical analysis and demonstrate the effectiveness of the proposed scheme.
{"title":"Long time stability and strong convergence of an efficient tamed scheme for stochastic Allen-Cahn equation driven by additive white noise","authors":"Xiao Qi , Yubin Yan","doi":"10.1016/j.apnum.2026.01.017","DOIUrl":"10.1016/j.apnum.2026.01.017","url":null,"abstract":"<div><div>Huang and Shen [<em>Math. Comput.</em> <strong>92</strong> (2023) 2685–2713] proposed a semi-implicit tamed scheme for the numerical approximation of stochastic Allen–Cahn equations driven by multiplicative trace-class noise. They showed that the scheme is unconditionally stable on finite time intervals and can be efficiently implemented. In this paper, we investigate the long-time stability of this tamed scheme for stochastic Allen–Cahn equations driven by additive white noise. We also address the strong convergence analysis of the associated fully discrete scheme within the Galerkin finite element framework. The main contributions of this work are as follows: (i) by constructing a suitable Lyapunov functional, we establish the unconditional long-time stability of the tamed method; (ii) we rigorously derive the strong convergence rates of the fully discrete scheme obtained by coupling the tamed approach with the finite element method. Numerical experiments are provided to validate the theoretical analysis and demonstrate the effectiveness of the proposed scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"224 ","pages":"Pages 22-36"},"PeriodicalIF":2.4,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077007","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 : 2026-06-01Epub Date: 2026-01-23DOI: 10.1016/j.apnum.2026.01.013
Haibin Chen , Wenqi Zhu , Coralia Cartis
In this paper, we propose a novel tensor-based Dinkelbach–Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach–Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternating minimization method which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Łojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.
{"title":"Tensor-based Dinkelbach method for computing generalized tensor eigenvalues and its applications","authors":"Haibin Chen , Wenqi Zhu , Coralia Cartis","doi":"10.1016/j.apnum.2026.01.013","DOIUrl":"10.1016/j.apnum.2026.01.013","url":null,"abstract":"<div><div>In this paper, we propose a novel tensor-based Dinkelbach–Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach–Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternating minimization method which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Łojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"224 ","pages":"Pages 1-21"},"PeriodicalIF":2.4,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071115","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 : 2026-05-01Epub Date: 2026-01-08DOI: 10.1016/j.apnum.2026.01.002
SatpalSingh , Divyashree B K , Devendra Kumar
This article presents the development and evaluation of a collocation-based, parameter-uniform numerical method tailored for a specific class of singularly perturbed convection-diffusion problems with a boundary turning point. A set of a priori bounds is established for both the exact solution and its derivatives to enable a comprehensive error analysis. These bounds are essential for ensuring the accuracy and stability of the proposed method, as they provide necessary constraints to evaluate the solution’s behavior across various parameters. The classical Crank-Nicolson method discretizes the time direction on a uniform mesh. At the same time, a collocation approach is applied to the spatial domain using an exponentially graded mesh, which is carefully refined in the boundary layer region. The proposed method demonstrates second-order, parameter-uniform convergence, as confirmed by a thorough investigation. Extensive numerical tests support the theoretical results, showing the approach’s accuracy and efficiency.
{"title":"A higher order collocation method for a singularly perturbed system having boundary turning points","authors":"SatpalSingh , Divyashree B K , Devendra Kumar","doi":"10.1016/j.apnum.2026.01.002","DOIUrl":"10.1016/j.apnum.2026.01.002","url":null,"abstract":"<div><div>This article presents the development and evaluation of a collocation-based, parameter-uniform numerical method tailored for a specific class of singularly perturbed convection-diffusion problems with a boundary turning point. A set of a priori bounds is established for both the exact solution and its derivatives to enable a comprehensive error analysis. These bounds are essential for ensuring the accuracy and stability of the proposed method, as they provide necessary constraints to evaluate the solution’s behavior across various parameters. The classical Crank-Nicolson method discretizes the time direction on a uniform mesh. At the same time, a collocation approach is applied to the spatial domain using an exponentially graded mesh, which is carefully refined in the boundary layer region. The proposed method demonstrates second-order, parameter-uniform convergence, as confirmed by a thorough investigation. Extensive numerical tests support the theoretical results, showing the approach’s accuracy and efficiency.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 76-100"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145975292","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 : 2026-05-01Epub Date: 2026-01-04DOI: 10.1016/j.apnum.2026.01.001
Hao Han, Zengqiang Tan
The Gray-Scott reaction-diffusion (GS-RD) model plays an important role to study the pattern formation dynamics in nature. This paper studies linear and high-order accurate schemes for numerically solving the GS-RD model, where the schemes are built on the one-parameter methods in time and the compact difference scheme in space. The error estimates and stability of the proposed schemes are analysed and it is proved that the fully discrete schemes have second-order accuracy in temporal direction and four-order accuracy in spatial direction. Specifically, for two- and three-dimensional cases, the alternating direction implicit (ADI) technique is adopted to reduce computational cost of the schemes. Several numerical experiments are conducted to validate the theoretical results as well as the computational effectiveness and accuracy of the derived schemes.
{"title":"Linearized and high-order accurate one-parameter compact difference schemes for solving the Gray-Scott reaction-diffusion model","authors":"Hao Han, Zengqiang Tan","doi":"10.1016/j.apnum.2026.01.001","DOIUrl":"10.1016/j.apnum.2026.01.001","url":null,"abstract":"<div><div>The Gray-Scott reaction-diffusion (GS-RD) model plays an important role to study the pattern formation dynamics in nature. This paper studies linear and high-order accurate schemes for numerically solving the GS-RD model, where the schemes are built on the one-parameter methods in time and the compact difference scheme in space. The error estimates and stability of the proposed schemes are analysed and it is proved that the fully discrete schemes have second-order accuracy in temporal direction and four-order accuracy in spatial direction. Specifically, for two- and three-dimensional cases, the alternating direction implicit (ADI) technique is adopted to reduce computational cost of the schemes. Several numerical experiments are conducted to validate the theoretical results as well as the computational effectiveness and accuracy of the derived schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 16-44"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145915176","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 : 2026-05-01Epub Date: 2026-01-07DOI: 10.1016/j.apnum.2026.01.005
Zhili Chen, Yong Wang, Zhixian Gui
The Generalized Finite Difference Method (GFDM) is a promising meshless approach for seismic wave simulation, offering superior flexibility in modeling complex geological structures. However, its critical weakness lies in the sensitivity of numerical stability to the node distribution within the computational stencil. Asymmetric or irregular point clouds often lead to ill-posed systems and unstable simulations, especially when using the theoretical minimum number of nodes required for a given order of accuracy. This instability severely limits GFDM's practical application in high precision seismic modeling. To address this challenge, we propose a novel Mixed-Order Compact GFDM (MOCGFDM) inspired by the principles of the Compact Finite Difference Method (CFDM). Our key innovation is the incorporation of derivative information from neighboring stencil points directly into the objective function used to compute the difference coefficients. This introduces a compact constraint that significantly enhances the robustness of the system. Crucially, the proposed method modifies only the pre-computation of the difference coefficients, leaving the core wave propagation algorithm unchanged and thus preserving computational efficiency. Numerical experiments across various models demonstrate that MOCGFDM achieves markedly higher stability in high order simulations compared to the conventional GFDM. It effectively enables stable computations with fewer nodes in irregular point clouds and allows for larger critical time steps. Consequently, this method not only improves reliability but also indirectly boosts simulation efficiency, providing a robust and practical meshless solution for high fidelity seismic wavefield simulation.
{"title":"A mixed-order compact generalized finite difference method for stable seismic wavefield simulation","authors":"Zhili Chen, Yong Wang, Zhixian Gui","doi":"10.1016/j.apnum.2026.01.005","DOIUrl":"10.1016/j.apnum.2026.01.005","url":null,"abstract":"<div><div>The Generalized Finite Difference Method (GFDM) is a promising meshless approach for seismic wave simulation, offering superior flexibility in modeling complex geological structures. However, its critical weakness lies in the sensitivity of numerical stability to the node distribution within the computational stencil. Asymmetric or irregular point clouds often lead to ill-posed systems and unstable simulations, especially when using the theoretical minimum number of nodes required for a given order of accuracy. This instability severely limits GFDM's practical application in high precision seismic modeling. To address this challenge, we propose a novel Mixed-Order Compact GFDM (MO<img>CGFDM) inspired by the principles of the Compact Finite Difference Method (CFDM). Our key innovation is the incorporation of derivative information from neighboring stencil points directly into the objective function used to compute the difference coefficients. This introduces a compact constraint that significantly enhances the robustness of the system. Crucially, the proposed method modifies only the pre-computation of the difference coefficients, leaving the core wave propagation algorithm unchanged and thus preserving computational efficiency. Numerical experiments across various models demonstrate that MO<img>CGFDM achieves markedly higher stability in high order simulations compared to the conventional GFDM. It effectively enables stable computations with fewer nodes in irregular point clouds and allows for larger critical time steps. Consequently, this method not only improves reliability but also indirectly boosts simulation efficiency, providing a robust and practical meshless solution for high fidelity seismic wavefield simulation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 181-195"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024186","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 : 2026-05-01Epub Date: 2026-01-12DOI: 10.1016/j.apnum.2026.01.004
Keyang Zhang
The earlier convergence analysis of boundary integral method (Ambrose et. al.) for viscous Stokes flow was discrete in space and continuous in time. The main result derived showed that the numerical method with filtering converges to the exact solution with spectral accuracy. In this paper, a harder problem with nonuniform spatial viscosity contrast is proposed. Also, time discretization methods of such problem are given, and analyses of such methods are presented. The numerical results are reported at the end of the manuscript.
早期边界积分法（Ambrose et al.）对粘性Stokes流的收敛性分析在空间上是离散的，在时间上是连续的。推导出的主要结果表明，带滤波的数值方法收敛于具有谱精度的精确解。本文提出了一个具有非均匀空间粘度对比的难题。同时给出了该问题的时间离散化方法，并对这些方法进行了分析。数值结果报告在文章的最后。
{"title":"Analyses and simulation of boundary integral methods of viscous Stokes flow","authors":"Keyang Zhang","doi":"10.1016/j.apnum.2026.01.004","DOIUrl":"10.1016/j.apnum.2026.01.004","url":null,"abstract":"<div><div>The earlier convergence analysis of boundary integral method (Ambrose et. al.) for viscous Stokes flow was discrete in space and continuous in time. The main result derived showed that the numerical method with filtering converges to the exact solution with spectral accuracy. In this paper, a harder problem with nonuniform spatial viscosity contrast is proposed. Also, time discretization methods of such problem are given, and analyses of such methods are presented. The numerical results are reported at the end of the manuscript.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 121-137"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145975290","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 : 2026-05-01Epub Date: 2026-01-16DOI: 10.1016/j.apnum.2026.01.007
Yonghui Bo , Yushun Wang
In this paper, we develop an exponential invariant energy quadratization (EIEQ) approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. This novel method, which builds upon an exponential reformulation of the nonlinear term in the Hamiltonian, provides improved efficiency and broader applicability compared to the conventional IEQ method, which is a widely used technique for constructing linear energy-preserving schemes. The introduced auxiliary variable eliminates the requirement for the nonlinear term to be bounded from below and allows for fully explicit treatment, which not only simplifies the numerical implementation but also leads to a simple framework for deriving high-order linear schemes. Moreover, the EIEQ method preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the IEQ method. Rigorous proofs regarding the energy conservation and the numerical accuracy are provided for all the schemes. Notably, unlike the IEQ-based schemes, the EIEQ schemes decouple the solution variable from the auxiliary variable, leading to improved computational efficiency. A comparative analysis between the IEQ and EIEQ methods is presented, along with numerical results that demonstrate the efficiency, accuracy and structure-preserving properties of the EIEQ schemes.
{"title":"High-order energy-preserving schemes for general Hamiltonian PDEs and their explicit computation","authors":"Yonghui Bo , Yushun Wang","doi":"10.1016/j.apnum.2026.01.007","DOIUrl":"10.1016/j.apnum.2026.01.007","url":null,"abstract":"<div><div>In this paper, we develop an exponential invariant energy quadratization (EIEQ) approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. This novel method, which builds upon an exponential reformulation of the nonlinear term in the Hamiltonian, provides improved efficiency and broader applicability compared to the conventional IEQ method, which is a widely used technique for constructing linear energy-preserving schemes. The introduced auxiliary variable eliminates the requirement for the nonlinear term to be bounded from below and allows for fully explicit treatment, which not only simplifies the numerical implementation but also leads to a simple framework for deriving high-order linear schemes. Moreover, the EIEQ method preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the IEQ method. Rigorous proofs regarding the energy conservation and the numerical accuracy are provided for all the schemes. Notably, unlike the IEQ-based schemes, the EIEQ schemes decouple the solution variable from the auxiliary variable, leading to improved computational efficiency. A comparative analysis between the IEQ and EIEQ methods is presented, along with numerical results that demonstrate the efficiency, accuracy and structure-preserving properties of the EIEQ schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 138-152"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024277","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 : 2026-05-01Epub Date: 2026-01-14DOI: 10.1016/j.apnum.2026.01.006
Linghua Kong , Fuguang Zhou , Lihai Ji
A TS-LRBFCM structure-preserving scheme is proposed for the stochastic coupled nonlinear Schrödinger equations. One of the starting point in designing this scheme is efficient and structure-preserving. To the goal, the splitting method is used to decouple the nonlinear algebraic system, and the midpoint rule is employed to temporal derivatives to make the scheme symplectic-preserving and mass-preserving. It demonstrates that the scheme preserves the discrete stochastic symplectic conservation law and discrete mass conservation law almost surely. Numerical experiments corroborate the theoretical results well. Furthermore, numerical facts also indicate that noise accelerates the oscillation of the wave. The solitary wave will be completely destroyed if the noise is relatively strong. Additionally, one can observe that the phase shift is significantly influenced by the noise.
{"title":"A TS-LRBFCM structure-preserving scheme for stochastic coupled nonlinear Schrödinger equations","authors":"Linghua Kong , Fuguang Zhou , Lihai Ji","doi":"10.1016/j.apnum.2026.01.006","DOIUrl":"10.1016/j.apnum.2026.01.006","url":null,"abstract":"<div><div>A TS-LRBFCM structure-preserving scheme is proposed for the stochastic coupled nonlinear Schrödinger equations. One of the starting point in designing this scheme is efficient and structure-preserving. To the goal, the splitting method is used to decouple the nonlinear algebraic system, and the midpoint rule is employed to temporal derivatives to make the scheme symplectic-preserving and mass-preserving. It demonstrates that the scheme preserves the discrete stochastic symplectic conservation law and discrete mass conservation law almost surely. Numerical experiments corroborate the theoretical results well. Furthermore, numerical facts also indicate that noise accelerates the oscillation of the wave. The solitary wave will be completely destroyed if the noise is relatively strong. Additionally, one can observe that the phase shift is significantly influenced by the noise.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 166-180"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024185","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 : 2026-05-01Epub Date: 2026-01-16DOI: 10.1016/j.apnum.2026.01.008
Youngjin Hwang , Yunjae Nam , Junseok Kim
This paper presents an efficient hybrid numerical algorithm to solve the high-order Allen–Cahn (AC) equation, which uses a polynomial free energy potential with a high-order exponent. The high-order AC equation preserves fine structures, shows traveling wave phenomena, suppresses noise for larger polynomial orders, and accurately captures interface motion driven by mean curvature. The proposed scheme combines an operator splitting method, a finite difference discretization, and an interpolation technique to overcome the challenge of solving nonlinear implicit equations that arise from the high-order polynomial potential. A theoretical condition on the time step is derived to guarantee monotonicity and ensure that the computational solution obtained by the proposed method satisfies the discrete maximum principle. Compared to fully explicit methods, the proposed approach allows a significantly larger time step size and thus results in improved numerical efficiency. Computational tests are performed to evaluate the accuracy, stability, and ability of the proposed algorithm to capture key physical behaviors of the phase-field model, such as curvature-driven interface motion and the suppression of high-frequency noise. The computational results demonstrate that higher polynomial orders lead to cleaner interface evolution by eliminating spurious oscillations and preserving non-random features. Furthermore, the method reliably captures the shrinking of interfaces even with relatively low numerical resolution. The proposed hybrid algorithm thus provides a robust and practical numerical method for simulating the complex dynamics of high-order AC equations in multi-phase systems and has potential applications in materials science, image processing, and biological modeling.
{"title":"An efficient hybrid numerical method for the high-order Allen–Cahn equation","authors":"Youngjin Hwang , Yunjae Nam , Junseok Kim","doi":"10.1016/j.apnum.2026.01.008","DOIUrl":"10.1016/j.apnum.2026.01.008","url":null,"abstract":"<div><div>This paper presents an efficient hybrid numerical algorithm to solve the high-order Allen–Cahn (AC) equation, which uses a polynomial free energy potential with a high-order exponent. The high-order AC equation preserves fine structures, shows traveling wave phenomena, suppresses noise for larger polynomial orders, and accurately captures interface motion driven by mean curvature. The proposed scheme combines an operator splitting method, a finite difference discretization, and an interpolation technique to overcome the challenge of solving nonlinear implicit equations that arise from the high-order polynomial potential. A theoretical condition on the time step is derived to guarantee monotonicity and ensure that the computational solution obtained by the proposed method satisfies the discrete maximum principle. Compared to fully explicit methods, the proposed approach allows a significantly larger time step size and thus results in improved numerical efficiency. Computational tests are performed to evaluate the accuracy, stability, and ability of the proposed algorithm to capture key physical behaviors of the phase-field model, such as curvature-driven interface motion and the suppression of high-frequency noise. The computational results demonstrate that higher polynomial orders lead to cleaner interface evolution by eliminating spurious oscillations and preserving non-random features. Furthermore, the method reliably captures the shrinking of interfaces even with relatively low numerical resolution. The proposed hybrid algorithm thus provides a robust and practical numerical method for simulating the complex dynamics of high-order AC equations in multi-phase systems and has potential applications in materials science, image processing, and biological modeling.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 153-165"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024184","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 : 2026-05-01Epub Date: 2026-01-24DOI: 10.1016/j.apnum.2026.01.014
Xiongbo Zheng, He Liu, Xiaole Li, Mingze Ji
This paper, based on block-centered grids and employing the Integral Method with Variational Limit, develops a class of high-order numerical schemes with two parameters for one- and two-dimensional parabolic equations with variable coefficients. Through parameter adjustment, the proposed scheme can degenerate into the classical fourth-order compact scheme, and can further give rise to a single-parameter controlled sixth-order scheme and a novel eighth-order compact scheme. Theoretical analysis demonstrates that this class of numerical schemes guarantees stability, convergence, and mass conservation. Numerical experiments systematically analyze the effect of parameters on the results, revealing the intrinsic relationship between the parameters and the fourth-, sixth-, and eighth-order schemes. The results show that the schemes preserve mass conservation, and the convergence order of both the solution and flux agree with theoretical analysis.
{"title":"High-order schemes for variable-coefficient parabolic equations via integral method with variational limit","authors":"Xiongbo Zheng, He Liu, Xiaole Li, Mingze Ji","doi":"10.1016/j.apnum.2026.01.014","DOIUrl":"10.1016/j.apnum.2026.01.014","url":null,"abstract":"<div><div>This paper, based on block-centered grids and employing the Integral Method with Variational Limit, develops a class of high-order numerical schemes with two parameters for one- and two-dimensional parabolic equations with variable coefficients. Through parameter adjustment, the proposed scheme can degenerate into the classical fourth-order compact scheme, and can further give rise to a single-parameter controlled sixth-order scheme and a novel eighth-order compact scheme. Theoretical analysis demonstrates that this class of numerical schemes guarantees stability, convergence, and mass conservation. Numerical experiments systematically analyze the effect of parameters on the results, revealing the intrinsic relationship between the parameters and the fourth-, sixth-, and eighth-order schemes. The results show that the schemes preserve mass conservation, and the convergence order of both the solution and flux agree with theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"223 ","pages":"Pages 211-234"},"PeriodicalIF":2.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074822","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}
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