Pub Date : 2025-11-13DOI: 10.1016/j.camwa.2025.11.004
Zhizhong Kong , Rui Sheng , Jerry Zhijian Yang , Juntao You , Cheng Yuan
In this paper, we propose the Weaker Adversarial Network (WerAN), a novel extension of the Weak Adversarial Network (WAN) framework designed for partial differential equations with singularities. By applying integration by parts twice in the derivation of weak form, our approach relaxes the constraints on the solution space in WAN. This adjustment transitions the implementation-level requirements from to and the theoretical requirements from to . Furthermore, we provide a systematic error analysis of the proposed method. Our theoretical investigations affirm the convergence of the WerAN method and offer insights into selecting network parameters and sample sizes across the domain and its boundary. We also present several numerical examples to demonstrate the effectiveness of WerAN in addressing both smooth and singular problems.
{"title":"A weaker adversarial neural networks method for linear second order elliptic PDEs","authors":"Zhizhong Kong , Rui Sheng , Jerry Zhijian Yang , Juntao You , Cheng Yuan","doi":"10.1016/j.camwa.2025.11.004","DOIUrl":"10.1016/j.camwa.2025.11.004","url":null,"abstract":"<div><div>In this paper, we propose the Weaker Adversarial Network (WerAN), a novel extension of the Weak Adversarial Network (WAN) framework designed for partial differential equations with singularities. By applying integration by parts twice in the derivation of weak form, our approach relaxes the constraints on the solution space in WAN. This adjustment transitions the implementation-level requirements from <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the theoretical requirements from <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mo>∞</mo></mrow></msup></math></span> to <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msup></math></span>. Furthermore, we provide a systematic error analysis of the proposed method. Our theoretical investigations affirm the convergence of the WerAN method and offer insights into selecting network parameters and sample sizes across the domain and its boundary. We also present several numerical examples to demonstrate the effectiveness of WerAN in addressing both smooth and singular problems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"201 ","pages":"Pages 259-277"},"PeriodicalIF":2.5,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528581","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 : 2025-11-12DOI: 10.1016/j.camwa.2025.11.001
Junseok Kim , Zhengang Li , Xinpei Wu , Soobin Kwak
In this study, we investigate the artificial pinning phenomena that emerge in numerical simulations of high-order Allen–Cahn (AC) equations. The AC equation is widely used to describe interface motion during phase separation processes. However, discretized numerical solutions may exhibit nonphysical behavior such as interface immobilization, or “pinning,” particularly when the ratio between the interfacial thickness model parameter (ϵ) and the spatial numerical mesh size (h) is below a critical threshold. The present work systematically analyzes how this ratio, , influences interface dynamics and determines the critical values at which pinning begins to occur for various nonlinearity degrees (α) of the polynomial potential. A fully explicit finite difference method is used to simulate the AC equation in both two- and three-dimensional settings. A bisection algorithm is introduced to accurately identify the critical pinning threshold as a function of α. Numerical experiments demonstrate that as α increases, decreases, indicating that the solution becomes more sensitive to spatial discretization and requires finer grids to avoid artificial pinning. Additionally, computational results show that higher values of α yield thicker and smoother interface transitions, and the study also investigates the influence of these values on phase morphology under random initial conditions. The numerical results in this study provide quantitative guidance for selecting discretization parameters in phase-field simulations and emphasize the importance of balancing the interface resolution with model complexity. These insights contribute to the development of accurate and robust numerical schemes for simulating interfacial dynamics in complex physical systems.
在本研究中，我们研究了高阶Allen-Cahn （AC）方程数值模拟中出现的人为钉住现象。交流方程被广泛用于描述相分离过程中的界面运动。然而，离散数值解可能表现出非物理行为，如界面固定或“钉住”，特别是当界面厚度模型参数（柱）与空间数值网格尺寸(h)之间的比率低于临界阈值时。本研究系统地分析了这个比值P= λ /h如何影响界面动力学，并确定了多项式电位的各种非线性度（α）开始发生钉住的临界值。采用全显式有限差分法模拟了二维和三维环境下的交流方程。引入了一种对分算法，以准确地识别临界钉住阈值P作为α的函数。数值实验表明，随着α的增加，P - P - P减小，表明溶液对空间离散化更加敏感，需要更细的网格来避免人为钉住。此外，计算结果表明，α值越高，界面转变越厚、越光滑，并研究了随机初始条件下α值对相形态的影响。本研究的数值结果为相场模拟中离散化参数的选择提供了定量指导，并强调了平衡界面分辨率与模型复杂性的重要性。这些见解有助于开发精确和强大的数值方案来模拟复杂物理系统中的界面动力学。
{"title":"Pinning phenomena in numerical schemes of the Allen–Cahn equation","authors":"Junseok Kim , Zhengang Li , Xinpei Wu , Soobin Kwak","doi":"10.1016/j.camwa.2025.11.001","DOIUrl":"10.1016/j.camwa.2025.11.001","url":null,"abstract":"<div><div>In this study, we investigate the artificial pinning phenomena that emerge in numerical simulations of high-order Allen–Cahn (AC) equations. The AC equation is widely used to describe interface motion during phase separation processes. However, discretized numerical solutions may exhibit nonphysical behavior such as interface immobilization, or “pinning,” particularly when the ratio between the interfacial thickness model parameter (<em>ϵ</em>) and the spatial numerical mesh size (<em>h</em>) is below a critical threshold. The present work systematically analyzes how this ratio, <span><math><mi>P</mi><mo>=</mo><mi>ϵ</mi><mo>/</mo><mi>h</mi></math></span>, influences interface dynamics and determines the critical values at which pinning begins to occur for various nonlinearity degrees (<em>α</em>) of the polynomial potential. A fully explicit finite difference method is used to simulate the AC equation in both two- and three-dimensional settings. A bisection algorithm is introduced to accurately identify the critical pinning threshold <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> as a function of <em>α</em>. Numerical experiments demonstrate that as <em>α</em> increases, <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> decreases, indicating that the solution becomes more sensitive to spatial discretization and requires finer grids to avoid artificial pinning. Additionally, computational results show that higher values of <em>α</em> yield thicker and smoother interface transitions, and the study also investigates the influence of these values on phase morphology under random initial conditions. The numerical results in this study provide quantitative guidance for selecting discretization parameters in phase-field simulations and emphasize the importance of balancing the interface resolution with model complexity. These insights contribute to the development of accurate and robust numerical schemes for simulating interfacial dynamics in complex physical systems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"201 ","pages":"Pages 248-258"},"PeriodicalIF":2.5,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145515798","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 : 2025-11-11DOI: 10.1016/j.camwa.2025.11.002
Kamana Porwal, Ritesh Singla
In this article, we propose and analyze a discontinuous Galerkin finite element method for the numerical approximation of a fourth-order parabolic variational inequality. In the fully-discrete discontinuous Galerkin scheme, we implement time discretization using the implicit backward Euler method, while spatial discretization is achieved through the utilization of a piecewise quadratic finite element space. The convergence analysis is conducted under a reasonable regularity assumption on the exact solution u, to be specific we assume , and the obstacle constraints are incorporated at the vertices of the triangulation. We derive an optimal order (with respect to the regularity) error estimate in the energy norm. Additionally, we present some numerical experiment results to illustrate the performance of the proposed method.
{"title":"Discontinuous Galerkin finite element method for a fourth order parabolic variational inequality","authors":"Kamana Porwal, Ritesh Singla","doi":"10.1016/j.camwa.2025.11.002","DOIUrl":"10.1016/j.camwa.2025.11.002","url":null,"abstract":"<div><div>In this article, we propose and analyze a discontinuous Galerkin finite element method for the numerical approximation of a fourth-order parabolic variational inequality. In the fully-discrete discontinuous Galerkin scheme, we implement time discretization using the implicit backward Euler method, while spatial discretization is achieved through the utilization of a piecewise quadratic finite element space. The convergence analysis is conducted under a reasonable regularity assumption on the exact solution <em>u</em>, to be specific we assume <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><mi>S</mi><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>)</mo></math></span>, and the obstacle constraints are incorporated at the vertices of the triangulation. We derive an optimal order (with respect to the regularity) <span><math><mi>a</mi><mspace></mspace><mi>p</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>r</mi><mi>i</mi></math></span> error estimate in the energy norm. Additionally, we present some numerical experiment results to illustrate the performance of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"201 ","pages":"Pages 233-247"},"PeriodicalIF":2.5,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145498981","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 : 2025-11-07DOI: 10.1016/j.camwa.2025.10.024
JiangShan Tong, Zhong Chen, Wei Jiang
This article presents a novel meshless approach to solve a nonlinear fourth-order reaction-diffusion equation with the time-fractional derivative of Caputo-type on arbitrary bounded domains by constructing a set of novel basis using the Sinc function as a shape function with higher accuracy and stability. The key contributions and innovations of this study are summarized as follows. Firstly, the time non-smooth problem is addressed by discretizing time term and approximating the time-fractional derivative with the piecewise fractional linear interpolation. What's more, a novel approach has been developed for the superconvergence estimation of the two-dimensional Sinc function approximation. Subsequently, the time-iterative stability analysis of the semi-analytical solution is presented, and it is demonstrated that the solution is absolutely stable on arbitrary bounded domains. We then present a detailed analysis of both local and global error estimates and prove that the space-time convergence order is with M being the number of basis and q being the length of time step, that is, the spatial convergence order is superconvergent. At last, a series of numerical examples validates the effectiveness of the proposed meshless method, and the low CPU time demonstrates its high computational efficiency.
{"title":"Meshless approach for solving nonlinear time-fractional fourth-order reaction–diffusion equation with convergence order analysis and stability analysis","authors":"JiangShan Tong, Zhong Chen, Wei Jiang","doi":"10.1016/j.camwa.2025.10.024","DOIUrl":"10.1016/j.camwa.2025.10.024","url":null,"abstract":"<div><div>This article presents a novel meshless approach to solve a nonlinear fourth-order reaction-diffusion equation with the time-fractional derivative of Caputo-type on arbitrary bounded domains by constructing a set of novel basis using the Sinc function as a shape function with higher accuracy and stability. The key contributions and innovations of this study are summarized as follows. Firstly, the time non-smooth problem is addressed by discretizing time term and approximating the time-fractional derivative with the piecewise fractional linear interpolation. What's more, a novel approach has been developed for the superconvergence estimation of the two-dimensional Sinc function approximation. Subsequently, the time-iterative stability analysis of the semi-analytical solution is presented, and it is demonstrated that the solution is absolutely stable on arbitrary bounded domains. We then present a detailed analysis of both local and global error estimates and prove that the space-time convergence order is <span><math><mi>O</mi><mo>(</mo><mfrac><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>4</mn><mo>−</mo><mi>m</mi></mrow></msup></mrow><mrow><mi>q</mi></mrow></mfrac><mo>+</mo><mi>q</mi><mo>)</mo></math></span> with <em>M</em> being the number of basis and <em>q</em> being the length of time step, that is, the spatial convergence order is superconvergent. At last, a series of numerical examples validates the effectiveness of the proposed meshless method, and the low CPU time demonstrates its high computational efficiency.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"201 ","pages":"Pages 214-232"},"PeriodicalIF":2.5,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145461923","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}
In this study, we propose and examine a local discontinuous Galerkin finite element approach for solving the Kelvin-Voigt viscoelastic fluid flow equations, incorporating a forcing term within the space for . The method employs an upwind scheme to efficiently manage the nonlinear convective term. We establish new a priori bounds for the semidiscrete local discontinuous Galerkin approximations. Furthermore, we derive optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in -norm and the pressure approximation in -norm for . Assuming the smallness of the data, we also prove uniform error estimates in time. Additionally, we consider the first- and second-order backward difference schemes for full discretization and derive the corresponding fully discrete error estimates. Finally, numerical experiments are presented to support the theoretical findings.
{"title":"Local discontinuous Galerkin method for Kelvin-Voigt viscoelastic fluid flow model","authors":"Debendra Kumar Swain , Saumya Bajpai , Deepjyoti Goswami","doi":"10.1016/j.camwa.2025.10.025","DOIUrl":"10.1016/j.camwa.2025.10.025","url":null,"abstract":"<div><div>In this study, we propose and examine a local discontinuous Galerkin finite element approach for solving the Kelvin-Voigt viscoelastic fluid flow equations, incorporating a forcing term within the <span><math><msup><mrow><mtext>L</mtext></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> space for <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>. The method employs an upwind scheme to efficiently manage the nonlinear convective term. We establish new <em>a priori</em> bounds for the semidiscrete local discontinuous Galerkin approximations. Furthermore, we derive optimal <em>a priori</em> error estimates for the semidiscrete discontinuous Galerkin velocity approximation in <span><math><msup><mrow><mtext>L</mtext></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and the pressure approximation in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm for <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>. Assuming the smallness of the data, we also prove uniform error estimates in time. Additionally, we consider the first- and second-order backward difference schemes for full discretization and derive the corresponding fully discrete error estimates. Finally, numerical experiments are presented to support the theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"200 ","pages":"Pages 467-499"},"PeriodicalIF":2.5,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145434663","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.camwa.2025.10.026
Xiang Li , Du Zhou , Zihan Liu , Chao Xu , Likuan Chen , Bingliang Yan , Chuanjiang Shen , Zhixiong Wang , Henghu Yang , Yongzhi Zhao
Transient simulations offer the advantage of capturing time-dependent flow behavior, making them more suitable than steady simulations for modeling complex phenomena such as turbulence, vibration, cavitation, and noise. While traditional CFD methods are more suitable for handling steady simulations, they are less effective for transient simulations due to limited parallel processing capabilities, leading to high computational costs. As a result, the lattice Boltzmann method (LBM) is employed in this study, which is a more efficient approach for transient simulation owing to its efficient handling of complex geometries, programming simplicity, and strong parallel scalability. In order to enhance the stability of LBM in the numerical simulation of high Reynolds number flow fields, the multiple relaxation time (MRT) collision model and the Smagorinsky-Lilly large eddy simulation (LES) turbulence model are utilized. To address the high dissipation near the wall in the Lilly model, the van Driest damping function is incorporated, improving the accuracy of the LES model in boundary regions. Additionally, to minimize memory consumption and reduce computation time without sacrificing accuracy, wall functions and local grid refinement techniques are applied, reducing the overall number of computational grids required. An experimental platform was established to measure the flow characteristics of control valves, and the accuracy of the proposed method was validated by comparing the flow coefficient Cv at various valve openings. Finally, the effects of local grid refinement and wall functions on simulation accuracy were compared, demonstrating that these techniques significantly improve the precision of transient simulations.
{"title":"Transient numerical simulation of control valve flow characteristics using a wall function and local grid refinement in LES-LBM","authors":"Xiang Li , Du Zhou , Zihan Liu , Chao Xu , Likuan Chen , Bingliang Yan , Chuanjiang Shen , Zhixiong Wang , Henghu Yang , Yongzhi Zhao","doi":"10.1016/j.camwa.2025.10.026","DOIUrl":"10.1016/j.camwa.2025.10.026","url":null,"abstract":"<div><div>Transient simulations offer the advantage of capturing time-dependent flow behavior, making them more suitable than steady simulations for modeling complex phenomena such as turbulence, vibration, cavitation, and noise. While traditional CFD methods are more suitable for handling steady simulations, they are less effective for transient simulations due to limited parallel processing capabilities, leading to high computational costs. As a result, the lattice Boltzmann method (LBM) is employed in this study, which is a more efficient approach for transient simulation owing to its efficient handling of complex geometries, programming simplicity, and strong parallel scalability. In order to enhance the stability of LBM in the numerical simulation of high Reynolds number flow fields, the multiple relaxation time (MRT) collision model and the Smagorinsky-Lilly large eddy simulation (LES) turbulence model are utilized. To address the high dissipation near the wall in the Lilly model, the van Driest damping function is incorporated, improving the accuracy of the LES model in boundary regions. Additionally, to minimize memory consumption and reduce computation time without sacrificing accuracy, wall functions and local grid refinement techniques are applied, reducing the overall number of computational grids required. An experimental platform was established to measure the flow characteristics of control valves, and the accuracy of the proposed method was validated by comparing the flow coefficient <em>C<sub>v</sub></em> at various valve openings. Finally, the effects of local grid refinement and wall functions on simulation accuracy were compared, demonstrating that these techniques significantly improve the precision of transient simulations.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"201 ","pages":"Pages 195-213"},"PeriodicalIF":2.5,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145434664","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.camwa.2025.10.019
Patrick Bammer, Lothar Banz, Andreas Schröder
In this paper, a reliable a posteriori error estimator for a model problem of elastoplasticity with linearly kinematic hardening is derived, which satisfies some (local) efficiency estimates. It is applicable to any discretization that is conforming with respect to the displacement field and the plastic strain. Furthermore, the paper presents hp-finite element discretizations relying on a variational inequality as well as on a mixed variational formulation and discusses their equivalence by using biorthogonal basis functions. Numerical experiments demonstrate the applicability of the theoretical findings and underline the potential of h- and hp-adaptive finite element discretizations for problems of elastoplasticity.
{"title":"A posteriori error estimates for hp-FE discretizations in elastoplasticity","authors":"Patrick Bammer, Lothar Banz, Andreas Schröder","doi":"10.1016/j.camwa.2025.10.019","DOIUrl":"10.1016/j.camwa.2025.10.019","url":null,"abstract":"<div><div>In this paper, a reliable a posteriori error estimator for a model problem of elastoplasticity with linearly kinematic hardening is derived, which satisfies some (local) efficiency estimates. It is applicable to any discretization that is conforming with respect to the displacement field and the plastic strain. Furthermore, the paper presents <em>hp</em>-finite element discretizations relying on a variational inequality as well as on a mixed variational formulation and discusses their equivalence by using biorthogonal basis functions. Numerical experiments demonstrate the applicability of the theoretical findings and underline the potential of <em>h</em>- and <em>hp</em>-adaptive finite element discretizations for problems of elastoplasticity.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"200 ","pages":"Pages 448-466"},"PeriodicalIF":2.5,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145404566","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.camwa.2025.10.017
Yuhao Zhang , Haifei Liu , Xiaoqiang Li , Yingjie Zhao , Jianxin Liu
This paper proposes the modified radial point interpolation method (M-RPIM) for solving the partial differential equations governing steady point current source fields. Unlike conventional RPIM, M-RPIM constructs support domains based on a cell, only one matrix inversion operation is needed when computing the shape functions in a integration cell, thereby improving computational efficiency. It determines the support domain by searching for neighboring cells layer by layer from the integration domain outward and controls the support domain range by adjusting the search depth. A systematic investigation of search depth effects on M-RPIM shape functions is conducted in this work. Numerical analysis demonstrates that the computational time of M-RPIM increases exponentially as the search depth increases, whereas the solution accuracy does not necessarily enhance. This work proves that M-RPIM with a search depth of 2 or 3 (employing approximately 15 support points) provides effective solutions for the partial differential equations governing steady point current source fields. When selecting a proper search depth, M-RPIM exhibits superior stability and accuracy compared to linear FEM, especially when dealing with complex geological models.
{"title":"Study on solving partial differential equations governing steady point current source field using M-RPIM","authors":"Yuhao Zhang , Haifei Liu , Xiaoqiang Li , Yingjie Zhao , Jianxin Liu","doi":"10.1016/j.camwa.2025.10.017","DOIUrl":"10.1016/j.camwa.2025.10.017","url":null,"abstract":"<div><div>This paper proposes the modified radial point interpolation method (M-RPIM) for solving the partial differential equations governing steady point current source fields. Unlike conventional RPIM, M-RPIM constructs support domains based on a cell, only one matrix inversion operation is needed when computing the shape functions in a integration cell, thereby improving computational efficiency. It determines the support domain by searching for neighboring cells layer by layer from the integration domain outward and controls the support domain range by adjusting the search depth. A systematic investigation of search depth effects on M-RPIM shape functions is conducted in this work. Numerical analysis demonstrates that the computational time of M-RPIM increases exponentially as the search depth increases, whereas the solution accuracy does not necessarily enhance. This work proves that M-RPIM with a search depth of 2 or 3 (employing approximately 15 support points) provides effective solutions for the partial differential equations governing steady point current source fields. When selecting a proper search depth, M-RPIM exhibits superior stability and accuracy compared to linear FEM, especially when dealing with complex geological models.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"200 ","pages":"Pages 431-447"},"PeriodicalIF":2.5,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145404567","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 : 2025-10-28DOI: 10.1016/j.camwa.2025.10.013
Carlos Friedrich Loeffler , Jose Ronaldo Soares Ramos , Luciano de Oliveira Castro Lara , Thiago Galdino Balista , Julio Tomás Aquije Chacaltana
The Direct Interpolation Boundary Element Method (DIBEM) has proven to be a versatile, precise, and robust tool for transforming domain integrals into boundary integrals in the most diverse applications of the scalar field equation, such as the cases governed by the Equation of Stationary Poisson, Helmholtz, and Diffusion-Advection. DIBEM achieves this objective using radial basis functions to approximate the whole kernel of domain integrals composed of non-self-adjoint operators, frequently occurring in the mathematical modeling of complex problems. Using a simplified fundamental solution also allows for a more immediate and faster numerical solution without significantly losing precision in the results. In this work, DIBEM is used to solve transient heat transmission problems, which are governed by time-dependent partial differential equations and consist of one of the most important mathematical models for application in engineering, describing heat dissipation and absorption in equipment, machines, buildings, and metallurgical industrial processes, among others. Numerical tests evaluate the DIBEM model in two dimensions, in which different thermal loading variations over time, severe initial conditions, and accentuated variations in the diffusivity value are tested, aiming to assess the stability and consistency of the method.
{"title":"Solving transient heat conduction problems through the direct interpolation technique","authors":"Carlos Friedrich Loeffler , Jose Ronaldo Soares Ramos , Luciano de Oliveira Castro Lara , Thiago Galdino Balista , Julio Tomás Aquije Chacaltana","doi":"10.1016/j.camwa.2025.10.013","DOIUrl":"10.1016/j.camwa.2025.10.013","url":null,"abstract":"<div><div>The Direct Interpolation Boundary Element Method (DIBEM) has proven to be a versatile, precise, and robust tool for transforming domain integrals into boundary integrals in the most diverse applications of the scalar field equation, such as the cases governed by the Equation of Stationary Poisson, Helmholtz, and Diffusion-Advection. DIBEM achieves this objective using radial basis functions to approximate the whole kernel of domain integrals composed of non-self-adjoint operators, frequently occurring in the mathematical modeling of complex problems. Using a simplified fundamental solution also allows for a more immediate and faster numerical solution without significantly losing precision in the results. In this work, DIBEM is used to solve transient heat transmission problems, which are governed by time-dependent partial differential equations and consist of one of the most important mathematical models for application in engineering, describing heat dissipation and absorption in equipment, machines, buildings, and metallurgical industrial processes, among others. Numerical tests evaluate the DIBEM model in two dimensions, in which different thermal loading variations over time, severe initial conditions, and accentuated variations in the diffusivity value are tested, aiming to assess the stability and consistency of the method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"200 ","pages":"Pages 409-430"},"PeriodicalIF":2.5,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145382501","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 : 2025-10-27DOI: 10.1016/j.camwa.2025.10.016
Xiangran Zheng , Qiang Wang , Wenxiang Sun , Yan Gu , Wenzhen Qu
This study presents a numerical framework with arbitrary-order accuracy for dynamic electroelastic analysis of two-dimensional (2D) and three-dimensional (3D) piezoelectric structures. The methodology achieves arbitrary-order convergence through a unified spatiotemporal coupling strategy. Temporal discretization is handled via the Krylov deferred correction (KDC) technique, which ensures asymptotically exact integration in time. For spatial approximation, the generalized finite difference method (GFDM) is employed, utilizing adaptive Taylor series expansions with matched orders to maintain consistency with temporal accuracy. An enhanced scheme is introduced for the enforcement of coupled electromechanical boundary conditions, improving numerical stability in cases where these conditions are not prescribed as explicit functions of time. To investigate the performance and reliability of the KDC-GFDM, four representative numerical examples are considered, covering both 2D and 3D cases with various geometric features and initial/boundary conditions. Numerically calculated results get systematically evaluated in comparison with available analytical solutions, when accessible, and with high-resolution reference solutions obtained from COMSOL Multiphysics.
{"title":"The general finite difference method with the Krylov deferred correction technique for dynamic 2D and 3D piezoelectric analysis","authors":"Xiangran Zheng , Qiang Wang , Wenxiang Sun , Yan Gu , Wenzhen Qu","doi":"10.1016/j.camwa.2025.10.016","DOIUrl":"10.1016/j.camwa.2025.10.016","url":null,"abstract":"<div><div>This study presents a numerical framework with arbitrary-order accuracy for dynamic electroelastic analysis of two-dimensional (2D) and three-dimensional (3D) piezoelectric structures. The methodology achieves arbitrary-order convergence through a unified spatiotemporal coupling strategy. Temporal discretization is handled via the Krylov deferred correction (KDC) technique, which ensures asymptotically exact integration in time. For spatial approximation, the generalized finite difference method (GFDM) is employed, utilizing adaptive Taylor series expansions with matched orders to maintain consistency with temporal accuracy. An enhanced scheme is introduced for the enforcement of coupled electromechanical boundary conditions, improving numerical stability in cases where these conditions are not prescribed as explicit functions of time. To investigate the performance and reliability of the KDC-GFDM, four representative numerical examples are considered, covering both 2D and 3D cases with various geometric features and initial/boundary conditions. Numerically calculated results get systematically evaluated in comparison with available analytical solutions, when accessible, and with high-resolution reference solutions obtained from COMSOL Multiphysics.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"200 ","pages":"Pages 382-408"},"PeriodicalIF":2.5,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145382514","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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