Pub Date : 2026-01-08DOI: 10.1016/j.camwa.2025.12.027
Yanlai Song
In this study, we present a numerical framework founded on a specialized version of radial basis function-produced finite difference solvers, designed for application in both interpolation tasks and the computational solution of higher-dimensional PDEs arising in multi-asset options. The distinguishing feature of the proposed method lies in its utilization of integrated forms of a special power of the multiquadric kernel to construct novel derivative weights, thereby enhancing solution accuracy and robustness. The derivative approximations are obtained through the derivation of analytical expressions, which are then evaluated on stencils comprising both uniformly spaced and non-uniformly distributed nodes. These formulations are constructed to offer greater flexibility in handling a wide spectrum of discretization patterns. Numerical results are provided to support the theoretical discussions.
{"title":"On the numerical solution of high-dimensional PDEs arising in multi-asset options via a kernel-type solver","authors":"Yanlai Song","doi":"10.1016/j.camwa.2025.12.027","DOIUrl":"10.1016/j.camwa.2025.12.027","url":null,"abstract":"<div><div>In this study, we present a numerical framework founded on a specialized version of radial basis function-produced finite difference solvers, designed for application in both interpolation tasks and the computational solution of higher-dimensional PDEs arising in multi-asset options. The distinguishing feature of the proposed method lies in its utilization of integrated forms of a special power of the multiquadric kernel to construct novel derivative weights, thereby enhancing solution accuracy and robustness. The derivative approximations are obtained through the derivation of analytical expressions, which are then evaluated on stencils comprising both uniformly spaced and non-uniformly distributed nodes. These formulations are constructed to offer greater flexibility in handling a wide spectrum of discretization patterns. Numerical results are provided to support the theoretical discussions.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 97-112"},"PeriodicalIF":2.5,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940410","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 : 2026-01-08DOI: 10.1016/j.camwa.2025.12.026
Longyuan Wu , Xinlong Feng
We propose an hp-adaptive Galerkin least squares intrinsic finite element method for the stationary convection-reaction-diffusion equation on closed surfaces. The discrete variational formulation is intrinsic to the interpolated surface, incorporating the contribution of the local parametrization. This eliminates the reliance on pre-established surface parametrization and circumvents the challenge of high-order numerical integration in the space where the surface is embedded. The Galerkin least squares method, a variant of the streamline-upwind/Petrov-Galerkin method, is employed to enhance the stability of the numerical scheme. Unique solvability of the discrete variational problem is shown. To generate appropriate anisotropic mesh, an hp-adaptive refinement strategy is designed by combining the Galerkin least squares method with the quadratic intrinsic finite element method. The recovery function, based on centroid weights, yields more accurate estimates compared to area-weighted counterparts. Finally, numerical experiments show the effectiveness of the proposed method.
{"title":"An hp-adaptive Galerkin least squares intrinsic finite element method for the convection-dominated problem on surfaces","authors":"Longyuan Wu , Xinlong Feng","doi":"10.1016/j.camwa.2025.12.026","DOIUrl":"10.1016/j.camwa.2025.12.026","url":null,"abstract":"<div><div>We propose an <em>hp</em>-adaptive Galerkin least squares intrinsic finite element method for the stationary convection-reaction-diffusion equation on closed surfaces. The discrete variational formulation is intrinsic to the interpolated surface, incorporating the contribution of the local parametrization. This eliminates the reliance on pre-established surface parametrization and circumvents the challenge of high-order numerical integration in the space where the surface is embedded. The Galerkin least squares method, a variant of the streamline-upwind/Petrov-Galerkin method, is employed to enhance the stability of the numerical scheme. Unique solvability of the discrete variational problem is shown. To generate appropriate anisotropic mesh, an <em>hp</em>-adaptive refinement strategy is designed by combining the Galerkin least squares method with the quadratic intrinsic finite element method. The recovery function, based on centroid weights, yields more accurate estimates compared to area-weighted counterparts. Finally, numerical experiments show the effectiveness of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 113-132"},"PeriodicalIF":2.5,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940426","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 : 2026-01-07DOI: 10.1016/j.camwa.2025.12.023
Renhui Teng , Yonghai Li , Hongtao Yang , Qin Zhou , Xia Cui
We construct and analyze a family of upwind high-order finite volume schemes for convection-diffusion problems on rectangular meshes. The novelty is that for the upwind discretization of the convection term, we replace the trial function restricted to the dual boundary closest to the upstream element with the extension of the trial function from the upstream element. We prove coercivity and provide optimal error estimates in the H1 and L2 norms. The schemes achieve optimal convergence rates of order k in the H1 norm and in the L2 norm, whether in diffusion-dominated or convection-dominated regimes. Numerical experiments confirm the theoretical results.
{"title":"A family of upwind high-order finite volume methods for convection-diffusion problems on rectangular meshes","authors":"Renhui Teng , Yonghai Li , Hongtao Yang , Qin Zhou , Xia Cui","doi":"10.1016/j.camwa.2025.12.023","DOIUrl":"10.1016/j.camwa.2025.12.023","url":null,"abstract":"<div><div>We construct and analyze a family of upwind high-order finite volume schemes for convection-diffusion problems on rectangular meshes. The novelty is that for the upwind discretization of the convection term, we replace the trial function restricted to the dual boundary closest to the upstream element with the extension of the trial function from the upstream element. We prove coercivity and provide optimal error estimates in the <em>H</em><sup>1</sup> and <em>L</em><sup>2</sup> norms. The schemes achieve optimal convergence rates of order <em>k</em> in the <em>H</em><sup>1</sup> norm and <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> in the <em>L</em><sup>2</sup> norm, whether in diffusion-dominated or convection-dominated regimes. Numerical experiments confirm the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 80-96"},"PeriodicalIF":2.5,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940428","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 : 2026-01-06DOI: 10.1016/j.camwa.2025.12.013
Nunzio Dimola, Nicola Rares Franco, Paolo Zunino
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the computational workload increases considerably when attempting to accurately capture the behavior of the system under significant variations or uncertainties in the low-dimensional structures such as fractures, fibers, or vascular networks, due to the inevitable necessity of running multiple simulations. In this work, we present a novel preconditioning strategy that leverages neural networks and unsupervised operator learning to design an efficient preconditioner specifically tailored to a class of 3D-1D mixed-dimensional PDEs. The proposed approach is capable of generalizing to varying shapes of the 1D manifold without retraining, making it robust to changes in the 1D graph topology. Moreover, thanks to convolutional neural networks, the neural preconditioner can adapt over a range of increasing mesh resolutions of the discrete problem, enabling us to train it on low resolution problems and deploy it on higher resolutions. Numerical experiments validate the effectiveness of the preconditioner in accelerating convergence in iterative solvers, demonstrating its appeal and limitations over traditional methods. This study lays the groundwork for applying neural network-based preconditioning techniques to a broader range of coupled multi-physics systems.
{"title":"Numerical solution of mixed-dimensional PDEs using a neural preconditioner","authors":"Nunzio Dimola, Nicola Rares Franco, Paolo Zunino","doi":"10.1016/j.camwa.2025.12.013","DOIUrl":"10.1016/j.camwa.2025.12.013","url":null,"abstract":"<div><div>Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the computational workload increases considerably when attempting to accurately capture the behavior of the system under significant variations or uncertainties in the low-dimensional structures such as fractures, fibers, or vascular networks, due to the inevitable necessity of running multiple simulations. In this work, we present a novel preconditioning strategy that leverages neural networks and unsupervised operator learning to design an efficient preconditioner specifically tailored to a class of 3D-1D mixed-dimensional PDEs. The proposed approach is capable of generalizing to varying shapes of the 1D manifold without retraining, making it robust to changes in the 1D graph topology. Moreover, thanks to convolutional neural networks, the neural preconditioner can adapt over a range of increasing mesh resolutions of the discrete problem, enabling us to train it on low resolution problems and deploy it on higher resolutions. Numerical experiments validate the effectiveness of the preconditioner in accelerating convergence in iterative solvers, demonstrating its appeal and limitations over traditional methods. This study lays the groundwork for applying neural network-based preconditioning techniques to a broader range of coupled multi-physics systems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 58-79"},"PeriodicalIF":2.5,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940425","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 : 2026-01-05DOI: 10.1016/j.camwa.2025.12.021
Ayse Nur Altunkaya, Erkan Caner Ozkat, Mete Avci
This study introduces an integrated analytical-to-AI framework for modeling and optimizing entropy generation in pulsating non-Newtonian heat and fluid flow specifically within two basic geometries: circular pipe and plane duct geometries. A semi-analytical model, based on the second law of thermodynamics, is first developed using the perturbation method to evaluate entropy generation under fully developed laminar flow and constant heat flux conditions. The model is validated against existing benchmark solutions, confirming its accuracy. Using this model, a comprehensive dataset is created by varying key dimensionless numbers: Brinkman number (Br), power-law index (n), pulsation amplitude (ε), and frequency (F). Four machine learning (ML) models are then trained to predict entropy generation, among which Gaussian Process Regression (GPR) shows the highest accuracy and is selected as the surrogate ML model for optimization. It is performed using the Grey Wolf Optimization (GWO) algorithm under different flow scenarios. The results indicate that shear-thinning fluids, especially with high amplitude (ε = 0.3) and moderate-to-high frequency pulsation (F = 60.803 for the circular pipe and F = 53.843 for the plane duct), yield the lowest entropy generation ( = 1.850 for the circular pipe and = 0.911 for the plane duct), while an increase in the power-law index leads to higher . The frequency range where entropy generation is significantly affected expands with increasing power-law index. These findings demonstrate the combined effect of fluid rheology and pulsation in reducing entropy generation. Furthermore, they emphasize that the proposed framework offers a reliable and efficient approach for analyzing and improving thermal systems using a combination of analytical modeling, machine learning, and optimization.
{"title":"Analytical-to-AI pipeline: Modeling and optimization of entropy generation in pulsating non-Newtonian heat flow","authors":"Ayse Nur Altunkaya, Erkan Caner Ozkat, Mete Avci","doi":"10.1016/j.camwa.2025.12.021","DOIUrl":"10.1016/j.camwa.2025.12.021","url":null,"abstract":"<div><div>This study introduces an integrated analytical-to-AI framework for modeling and optimizing entropy generation in pulsating non-Newtonian heat and fluid flow specifically within two basic geometries: circular pipe and plane duct geometries. A semi-analytical model, based on the second law of thermodynamics, is first developed using the perturbation method to evaluate entropy generation under fully developed laminar flow and constant heat flux conditions. The model is validated against existing benchmark solutions, confirming its accuracy. Using this model, a comprehensive dataset is created by varying key dimensionless numbers: Brinkman number (<em>Br</em>), power-law index (<em>n</em>), pulsation amplitude (ε), and frequency (<em>F</em>). Four machine learning (ML) models are then trained to predict entropy generation, among which Gaussian Process Regression (GPR) shows the highest accuracy and is selected as the surrogate ML model for optimization. It is performed using the Grey Wolf Optimization (GWO) algorithm under different flow scenarios. The results indicate that shear-thinning fluids, especially with high amplitude (ε = 0.3) and moderate-to-high frequency pulsation (<em>F</em> = 60.803 for the circular pipe and <em>F</em> = 53.843 for the plane duct), yield the lowest entropy generation (<span><math><msub><mi>N</mi><msub><mi>s</mi><mrow><mi>a</mi><mi>v</mi><mi>g</mi></mrow></msub></msub></math></span> = 1.850 for the circular pipe and <span><math><msub><mi>N</mi><msub><mi>s</mi><mrow><mi>a</mi><mi>v</mi><mi>g</mi></mrow></msub></msub></math></span> = 0.911 for the plane duct), while an increase in the power-law index leads to higher <span><math><msub><mi>N</mi><msub><mi>s</mi><mrow><mi>a</mi><mi>v</mi><mi>g</mi></mrow></msub></msub></math></span>. The frequency range where entropy generation is significantly affected expands with increasing power-law index. These findings demonstrate the combined effect of fluid rheology and pulsation in reducing entropy generation. Furthermore, they emphasize that the proposed framework offers a reliable and efficient approach for analyzing and improving thermal systems using a combination of analytical modeling, machine learning, and optimization.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"205 ","pages":"Pages 195-211"},"PeriodicalIF":2.5,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145902641","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 : 2026-01-05DOI: 10.1016/j.camwa.2025.12.010
Jie Zhou , Xuelian Jiang , Ying Liu
In this paper, we propose two fully discrete convex splitting schemes to solve the Cahn-Hilliard equation based on the mixed finite element method. For these two numerical schemes, the first-order backward Euler method and the second-order backward differentiation formula (BDF2) are used for temporal discretization, and the nonlinear term is treated by the convex splitting method. In order to ensure unconditional energy stability, the second-order time scheme requires the addition of a stability term , where β ≥ 0 is a stable parameter. We strictly prove that both numerical schemes have unconditional energy stability. In particular, the second-order time scheme can be guaranteed to be unconditionally energy stable for . Additionally, we conduct rigorous error analysis on these two numerical schemes and obtain optimal error estimates in H1 norm. Lastly, we verify the effectiveness of both numerical schemes and confirm the correctness of the theoretical results.
{"title":"Two unconditionally energy stable schemes for the Cahn-Hilliard equation","authors":"Jie Zhou , Xuelian Jiang , Ying Liu","doi":"10.1016/j.camwa.2025.12.010","DOIUrl":"10.1016/j.camwa.2025.12.010","url":null,"abstract":"<div><div>In this paper, we propose two fully discrete convex splitting schemes to solve the Cahn-Hilliard equation based on the mixed finite element method. For these two numerical schemes, the first-order backward Euler method and the second-order backward differentiation formula (BDF2) are used for temporal discretization, and the nonlinear term is treated by the convex splitting method. In order to ensure unconditional energy stability, the second-order time scheme requires the addition of a stability term <span><math><mrow><mo>−</mo><mi>β</mi><mi>τ</mi><mstyle><mi>Δ</mi></mstyle><mo>(</mo><msubsup><mi>φ</mi><mi>h</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>−</mo><msubsup><mi>φ</mi><mi>h</mi><mi>m</mi></msubsup><mo>)</mo></mrow></math></span>, where <em>β</em> ≥ 0 is a stable parameter. We strictly prove that both numerical schemes have unconditional energy stability. In particular, the second-order time scheme can be guaranteed to be unconditionally energy stable for <span><math><mrow><mi>β</mi><mo>≥</mo><mfrac><mn>1</mn><mn>16</mn></mfrac></mrow></math></span>. Additionally, we conduct rigorous error analysis on these two numerical schemes and obtain optimal error estimates in <em>H</em><sub>1</sub> norm. Lastly, we verify the effectiveness of both numerical schemes and confirm the correctness of the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 29-57"},"PeriodicalIF":2.5,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145902644","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 : 2026-01-02DOI: 10.1016/j.camwa.2025.12.025
Wenquan Liang , Yanfei Wang
The Lax-Wendroff high-order time discretization method is well known for its larger stability range and its ability to reduce temporal dispersion introduced by large time steps. However, the Lax-Wendroff high-order time discretization method substantially increases the computation burden since it requires computing many more spatial derivatives. In this paper, we propose two novel finite-difference schemes for the high-order Lax-Wendroff time discretization. With comparable accuracy, the proposed finite-difference schemes reduce numerical simulation time by approximately 50 % and 63 %, respectively, compared with a conventional finite-difference implementation of the Lax-Wendroff time discretization method. We then present dispersion error analyses and derive the stability conditions. Finally, numerical experiments with progressively larger time steps validate the accuracy and efficiency of the proposed finite-difference schemes.
{"title":"Accelerating the Lax-Wendroff time discretization method for the first-order acoustic wave equation simulation by designing proper SGFD schemes","authors":"Wenquan Liang , Yanfei Wang","doi":"10.1016/j.camwa.2025.12.025","DOIUrl":"10.1016/j.camwa.2025.12.025","url":null,"abstract":"<div><div>The Lax-Wendroff high-order time discretization method is well known for its larger stability range and its ability to reduce temporal dispersion introduced by large time steps. However, the Lax-Wendroff high-order time discretization method substantially increases the computation burden since it requires computing many more spatial derivatives. In this paper, we propose two novel finite-difference schemes for the high-order Lax-Wendroff time discretization. With comparable accuracy, the proposed finite-difference schemes reduce numerical simulation time by approximately 50 % and 63 %, respectively, compared with a conventional finite-difference implementation of the Lax-Wendroff time discretization method. We then present dispersion error analyses and derive the stability conditions. Finally, numerical experiments with progressively larger time steps validate the accuracy and efficiency of the proposed finite-difference schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"205 ","pages":"Pages 178-194"},"PeriodicalIF":2.5,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885774","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 : 2026-01-02DOI: 10.1016/j.camwa.2025.12.028
Hu Li , Jin Huang , Yanying Ma
This paper study the numerical solutions for three-dimensional axisymmetric boundary integral equations. First, a quadrature method achieving O(h3) accuracy with low computational complexity is developed. Convergence is proven using the compact operator theory. An error analysis yields a single-parameter asymptotic expansion with odd powers, enabling the creation of extrapolation algorithms to enhance accuracy. After one extrapolation, accuracy improves to O(h5), and further improvements are possible with using extrapolation again. Three numerical examples illustrate the algorithm’s efficiency.
{"title":"Modified quadrature method for solving three-dimensional axisymmetric boundary integral equations based on two extrapolations","authors":"Hu Li , Jin Huang , Yanying Ma","doi":"10.1016/j.camwa.2025.12.028","DOIUrl":"10.1016/j.camwa.2025.12.028","url":null,"abstract":"<div><div>This paper study the numerical solutions for three-dimensional axisymmetric boundary integral equations. First, a quadrature method achieving <em>O</em>(<em>h</em><sup>3</sup>) accuracy with low computational complexity is developed. Convergence is proven using the compact operator theory. An error analysis yields a single-parameter asymptotic expansion with odd powers, enabling the creation of extrapolation algorithms to enhance accuracy. After one extrapolation, accuracy improves to <em>O</em>(<em>h</em><sup>5</sup>), and further improvements are possible with using extrapolation again. Three numerical examples illustrate the algorithm’s efficiency.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 16-28"},"PeriodicalIF":2.5,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886222","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 : 2026-01-02DOI: 10.1016/j.camwa.2025.12.018
Gyeonggyu Lee , Seunggyu Lee
The Cahn–Hilliard equation describes the phase separation phenomena at the microscale, such as those observed for diblock copolymers. However, its standard form is limited to capturing diverse real-world behaviors. To address this issue, we propose a structure-preserving Cahn–Hilliard equation with a generalized source term. Based on the total energy of the suggested total energy functional, the schemes were constructed using a linearly stabilized splitting method and a fast Fourier transform. A second-order extension was achieved using the implicit-explicit Runge–Kutta method. We prove the unique solvability, mass conservation, and energy gradient stability of both first- and second-order schemes. Temporal accuracy was validated through convergence tests. Numerical experiments further illustrate the phase behaviors under varying source term orders.
{"title":"First- and second-order accurate, unconditionally energy gradient stable, uniquely solvable, and mass-preserving linear numerical schemes for Cahn–Hilliard equation with source term","authors":"Gyeonggyu Lee , Seunggyu Lee","doi":"10.1016/j.camwa.2025.12.018","DOIUrl":"10.1016/j.camwa.2025.12.018","url":null,"abstract":"<div><div>The Cahn–Hilliard equation describes the phase separation phenomena at the microscale, such as those observed for diblock copolymers. However, its standard form is limited to capturing diverse real-world behaviors. To address this issue, we propose a structure-preserving Cahn–Hilliard equation with a generalized source term. Based on the total energy of the suggested total energy functional, the schemes were constructed using a linearly stabilized splitting method and a fast Fourier transform. A second-order extension was achieved using the implicit-explicit Runge–Kutta method. We prove the unique solvability, mass conservation, and energy gradient stability of both first- and second-order schemes. Temporal accuracy was validated through convergence tests. Numerical experiments further illustrate the phase behaviors under varying source term orders.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 1-15"},"PeriodicalIF":2.5,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886221","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 : 2026-01-01DOI: 10.1016/j.camwa.2025.12.022
Xue Wang , Hongxing Rui , Xiaozhe Hu
In this work, we develop a nonconforming finite element method for the three-field Biot model, where the variables are displacement, Darcy velocity, and pore pressure. The discretization employs the lowest-order Crouzeix-Raviart (CR) element for both displacement and Darcy velocity, and piecewise constant element for pressure. We establish the well-posedness of the discrete problem with respect to a carefully chosen weighted norm, ensuring robustness with respect to both discretization and physical parameters. Furthermore, we prove optimal convergence of the proposed scheme. To improve computational efficiency, we introduce a reduced-order CR method based on the proper orthogonal decomposition (POD) technique. Numerical experiments are provided to verify the theoretical convergence rates and to demonstrate the effectiveness of the reduced-order approach.
{"title":"Nonconforming finite element method and reduced order algorithm for poroelasticity problem","authors":"Xue Wang , Hongxing Rui , Xiaozhe Hu","doi":"10.1016/j.camwa.2025.12.022","DOIUrl":"10.1016/j.camwa.2025.12.022","url":null,"abstract":"<div><div>In this work, we develop a nonconforming finite element method for the three-field Biot model, where the variables are displacement, Darcy velocity, and pore pressure. The discretization employs the lowest-order Crouzeix-Raviart (CR) element for both displacement and Darcy velocity, and piecewise constant element for pressure. We establish the well-posedness of the discrete problem with respect to a carefully chosen weighted norm, ensuring robustness with respect to both discretization and physical parameters. Furthermore, we prove optimal convergence of the proposed scheme. To improve computational efficiency, we introduce a reduced-order CR method based on the proper orthogonal decomposition (POD) technique. Numerical experiments are provided to verify the theoretical convergence rates and to demonstrate the effectiveness of the reduced-order approach.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"204 ","pages":"Pages 216-230"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884879","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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