Pub Date : 2026-04-15Epub Date: 2026-02-11DOI: 10.1016/j.camwa.2026.01.041
Muhammad Mohebujjaman , Mengying Xiao , Cheng Zhang
Newton’s method for solving stationary Navier-Stokes equations (NSE) is known for its fast convergence; however, it may fail when provided with a poor initial guess. This work presents a simple-to-implement nonlinear preconditioning technique for Newton’s iteration that retains quadratic convergence and expands the domain of convergence. The proposed AAPicard-Newton method adds an Anderson accelerated Picard step at each iteration of Newton’s method for solving NSE. This approach has been shown to be globally stable with a relaxation parameter in the Anderson acceleration optimization step, converges quadratically, and achieves faster convergence with a small convergence rate for large Reynolds numbers. Several benchmark numerical tests have been carried out and are well aligned with the theoretical results.
{"title":"A simple-to-implement nonlinear preconditioning of Newton’s method for solving the steady Navier-Stokes equations","authors":"Muhammad Mohebujjaman , Mengying Xiao , Cheng Zhang","doi":"10.1016/j.camwa.2026.01.041","DOIUrl":"10.1016/j.camwa.2026.01.041","url":null,"abstract":"<div><div>Newton’s method for solving stationary Navier-Stokes equations (NSE) is known for its fast convergence; however, it may fail when provided with a poor initial guess. This work presents a simple-to-implement nonlinear preconditioning technique for Newton’s iteration that retains quadratic convergence and expands the domain of convergence. The proposed AAPicard-Newton method adds an Anderson accelerated Picard step at each iteration of Newton’s method for solving NSE. This approach has been shown to be globally stable with a relaxation parameter <span><math><mrow><msub><mi>β</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≡</mo><mn>1</mn></mrow></math></span> in the Anderson acceleration optimization step, converges quadratically, and achieves faster convergence with a small convergence rate for large Reynolds numbers. Several benchmark numerical tests have been carried out and are well aligned with the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 127-146"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146160313","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 : 2026-04-01Epub Date: 2026-02-09DOI: 10.1016/j.camwa.2026.02.001
Heyan Xu, Haichao Li, Fuzhen Pang, Tianyi Hang, Chuanshuai Yi
For the optimization of vibration analysis methods for cylindrical shells, a numerical method based on Walsh series is proposed here for analyzing the free vibration of laminated cylindrical shells under general boundary conditions. The theoretical model is established using first-order shear deformation theory, with consideration of the effects of rotational inertia. The Walsh series method is applied in the axial direction, and a Fourier series is used in the circumferential direction. The resulting system of algebraic equations contains unknown Walsh series coefficients. By solving this system, the eigenfrequencies and related parameters of the laminated cylindrical shell are calculated. The convergence and accuracy of the proposed method are evaluated by comparing the results with those from existing literature and finite element analysis.
{"title":"Free vibration analysis of laminated cylindrical shells based on the Walsh series method","authors":"Heyan Xu, Haichao Li, Fuzhen Pang, Tianyi Hang, Chuanshuai Yi","doi":"10.1016/j.camwa.2026.02.001","DOIUrl":"10.1016/j.camwa.2026.02.001","url":null,"abstract":"<div><div>For the optimization of vibration analysis methods for cylindrical shells, a numerical method based on Walsh series is proposed here for analyzing the free vibration of laminated cylindrical shells under general boundary conditions. The theoretical model is established using first-order shear deformation theory, with consideration of the effects of rotational inertia. The Walsh series method is applied in the axial direction, and a Fourier series is used in the circumferential direction. The resulting system of algebraic equations contains unknown Walsh series coefficients. By solving this system, the eigenfrequencies and related parameters of the laminated cylindrical shell are calculated. The convergence and accuracy of the proposed method are evaluated by comparing the results with those from existing literature and finite element analysis.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 204-221"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146659","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 : 2026-04-01Epub Date: 2026-01-29DOI: 10.1016/j.camwa.2026.01.023
Clément Lasuen
In this paper, we propose a finite volume scheme for the linear transport equation in two space dimensions. This scheme is based on a second order upwind flux where the velocity is modified so as to recover the correct diffusion limit. A partially implicit time discretization is used. This allows to have good properties while keeping the computational cost per iteration very low. The resulting scheme is asymptotic preserving, positive under a classical CFL condition, conservative and second order consistent in all the regimes. These properties are valid on general unstructured meshes and the computational cost is similar to an explicit scheme. Eventually, the extension of this scheme to 3D unstructured meshes is straightforward and its properties remain valid.
{"title":"A positive and asymptotic preserving scheme for the linear transport equation on 2D unstructured meshes","authors":"Clément Lasuen","doi":"10.1016/j.camwa.2026.01.023","DOIUrl":"10.1016/j.camwa.2026.01.023","url":null,"abstract":"<div><div>In this paper, we propose a finite volume scheme for the linear transport equation in two space dimensions. This scheme is based on a second order upwind flux where the velocity is modified so as to recover the correct diffusion limit. A partially implicit time discretization is used. This allows to have good properties while keeping the computational cost per iteration very low. The resulting scheme is <em>asymptotic preserving</em>, positive under a classical <em>CFL</em> condition, conservative and second order consistent in all the regimes. These properties are valid on general unstructured meshes and the computational cost is similar to an explicit scheme. Eventually, the extension of this scheme to 3<em>D</em> unstructured meshes is straightforward and its properties remain valid.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 137-151"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072041","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 : 2026-04-01Epub Date: 2026-01-19DOI: 10.1016/j.camwa.2026.01.014
Chaobao Huang , Yujie Yu , Na An , Hu Chen
This paper considers the subdiffusion equation with a weakly singular solution. To achieve the optimal accuracy with a smaller grading parameter r than that of the standard Alikhanov scheme, it’s essential to correct the Alikhanov scheme and the approximation (0 < θ < 1) by investigating the corrected terms and , respectively. After that, the truncation error of the corrected Alikhanov scheme and the corrected approximation of are given. By adopting the corrected Alikhanov scheme for the Caputo derivative and the corrected approximation for , we construct a fully discrete scheme for the subdiffusion equation, employing a standard finite element method in space. Furthermore, the stability analysis and the optimal convergent analysis for the proposed scheme are investigated. Finally, numerical experiments are conducted to verify the theoretical results.
{"title":"A corrected Alikhanov scheme for a subdiffusion equation","authors":"Chaobao Huang , Yujie Yu , Na An , Hu Chen","doi":"10.1016/j.camwa.2026.01.014","DOIUrl":"10.1016/j.camwa.2026.01.014","url":null,"abstract":"<div><div>This paper considers the subdiffusion equation with a weakly singular solution. To achieve the optimal accuracy with a smaller grading parameter <em>r</em> than that of the standard Alikhanov scheme, it’s essential to correct the Alikhanov scheme and the approximation <span><math><mrow><msup><mi>v</mi><mrow><mi>n</mi><mo>−</mo><mi>θ</mi></mrow></msup><mo>≈</mo><mi>θ</mi><msup><mi>v</mi><mi>n</mi></msup><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>θ</mi><mo>)</mo></mrow><msup><mi>v</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> (0 < <em>θ</em> < 1) by investigating the corrected terms <span><math><mrow><msubsup><mi>β</mi><mrow><mi>n</mi></mrow><mi>σ</mi></msubsup><mrow><mo>(</mo><msup><mi>v</mi><mn>1</mn></msup><mo>−</mo><msup><mi>v</mi><mn>0</mn></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msubsup><mi>μ</mi><mi>n</mi><mi>σ</mi></msubsup><mrow><mo>(</mo><msup><mi>v</mi><mn>1</mn></msup><mo>−</mo><msup><mi>v</mi><mn>0</mn></msup><mo>)</mo></mrow></mrow></math></span>, respectively. After that, the truncation error of the corrected Alikhanov scheme and the corrected approximation of <span><math><msup><mi>v</mi><mrow><mi>n</mi><mo>−</mo><mi>θ</mi></mrow></msup></math></span> are given. By adopting the corrected Alikhanov scheme for the Caputo derivative and the corrected approximation for <span><math><msup><mi>v</mi><mrow><mi>n</mi><mo>−</mo><mi>θ</mi></mrow></msup></math></span>, we construct a fully discrete scheme for the subdiffusion equation, employing a standard finite element method in space. Furthermore, the stability analysis and the optimal convergent analysis for the proposed scheme are investigated. Finally, numerical experiments are conducted to verify the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 15-26"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145996271","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 : 2026-04-01Epub Date: 2026-02-02DOI: 10.1016/j.camwa.2026.01.025
F.S. Oğlakkaya , C. Bozkaya
This study examines unsteady thermal convection of an Al2O3-water nanofluid in a differentially heated, wavy-walled inclined enclosure under a partially applied magnetic field. Utilizing a two-level time integration scheme combined with the dual reciprocity boundary element method (DRBEM) in space, the research investigates the impact of key parameters, including a wide range of Rayleigh and Hartmann numbers, magnetic field width, cavity inclination angle, number of undulations of wavy walls, and nanofluid solid volume fraction, on the flow dynamics and heat transfer. DRBEM approach, which focuses only on the boundary discretization, enables efficient numerical analysis while reducing computational load. Results presented through streamlines, isotherms, and average Nusselt number, reveal that increasing Hartmann number suppresses the convective motion, leading to a reduction of average Nusselt number, while increasing the Rayleigh number or nanoparticle concentration intensifies the heat transfer rate in enclosures with both flat and wavy-walls. The highest thermal performance is obtained when the enclosure with flat walls is tilted by a right angle under the presence of partially applied magnetic field for various combinations of the governing parameters. This research provides a comprehensive understanding of how multi-physical parameters and a partially applied magnetic field influence thermal convection, particularly within complex geometries, thereby contributing to advancements in the design and analysis of thermal systems.
{"title":"Impact of partial magnetic field on natural convection in nanofluid-filled inclined cavities","authors":"F.S. Oğlakkaya , C. Bozkaya","doi":"10.1016/j.camwa.2026.01.025","DOIUrl":"10.1016/j.camwa.2026.01.025","url":null,"abstract":"<div><div>This study examines unsteady thermal convection of an Al<sub>2</sub>O<sub>3</sub>-water nanofluid in a differentially heated, wavy-walled inclined enclosure under a partially applied magnetic field. Utilizing a two-level time integration scheme combined with the dual reciprocity boundary element method (DRBEM) in space, the research investigates the impact of key parameters, including a wide range of Rayleigh and Hartmann numbers, magnetic field width, cavity inclination angle, number of undulations of wavy walls, and nanofluid solid volume fraction, on the flow dynamics and heat transfer. DRBEM approach, which focuses only on the boundary discretization, enables efficient numerical analysis while reducing computational load. Results presented through streamlines, isotherms, and average Nusselt number, reveal that increasing Hartmann number suppresses the convective motion, leading to a reduction of average Nusselt number, while increasing the Rayleigh number or nanoparticle concentration intensifies the heat transfer rate in enclosures with both flat and wavy-walls. The highest thermal performance is obtained when the enclosure with flat walls is tilted by a right angle under the presence of partially applied magnetic field for various combinations of the governing parameters. This research provides a comprehensive understanding of how multi-physical parameters and a partially applied magnetic field influence thermal convection, particularly within complex geometries, thereby contributing to advancements in the design and analysis of thermal systems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 184-203"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146110516","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 : 2026-04-01Epub Date: 2026-02-01DOI: 10.1016/j.camwa.2026.01.030
Chun-Hua Zhang , Long Kuang , Wen-Ping Yuan , Xiang Wang
This paper presents a high-order operator splitting method incorporating a weighted essentially non-oscillatory (WENO) scheme for solving the Allen-Cahn equation. We employ the Strang operator splitting technique to decompose the original equation into linear and nonlinear subequations. The linear subequation is discretized using a sixth-order WENO scheme for spatial derivatives and a third-order Runge-Kutta method for the time direction, while the nonlinear subequation admits an analytical solution. This approach yields a high-order WENO-operator splitting (WENO-OS) scheme for the Allen-Cahn equation. In theory, the stability and convergence of the proposed scheme have been rigorously analyzed. Numerical experiments have verified that the proposed scheme can achieve sixth-order accuracy in space, second-order accuracy in time, verify stability condition and energy decline characteristic.
{"title":"Stability and convergence of high-order WENO-OS scheme for the Allen-Cahn equation","authors":"Chun-Hua Zhang , Long Kuang , Wen-Ping Yuan , Xiang Wang","doi":"10.1016/j.camwa.2026.01.030","DOIUrl":"10.1016/j.camwa.2026.01.030","url":null,"abstract":"<div><div>This paper presents a high-order operator splitting method incorporating a weighted essentially non-oscillatory (WENO) scheme for solving the Allen-Cahn equation. We employ the Strang operator splitting technique to decompose the original equation into linear and nonlinear subequations. The linear subequation is discretized using a sixth-order WENO scheme for spatial derivatives and a third-order Runge-Kutta method for the time direction, while the nonlinear subequation admits an analytical solution. This approach yields a high-order WENO-operator splitting (WENO-OS) scheme for the Allen-Cahn equation. In theory, the stability and convergence of the proposed scheme have been rigorously analyzed. Numerical experiments have verified that the proposed scheme can achieve sixth-order accuracy in space, second-order accuracy in time, verify stability condition and energy decline characteristic.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 171-183"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146110519","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 : 2026-04-01Epub Date: 2026-01-19DOI: 10.1016/j.camwa.2026.01.017
Jinming Zhang , Zhanjing Tao , Jun Zhu , Jianxian Qiu
In this paper, we propose a finite difference hybrid weighted essentially non-oscillatory (WENO) scheme to solve the hyperbolic conservation laws. A simplified troubled-cell indicator is designed for the hybrid scheme. The multi-resolution WENO (MR-WENO) reconstruction is applied in the troubled-cells, while a simple linear reconstruction is used in the remaining regions. The new hybrid scheme inherits the excellent characteristics of the original MR-WENO scheme [1], and can reduce the expensive computational cost of the WENO reconstruction. Compared to the previous hybrid WENO scheme [2] which used the high-degree polynomial in the troubled-cell indicator, the new scheme can reduce the percentage of the troubled-cells, leading to higher computational efficiency. Moreover, our scheme can effectively identify the troubled-cells, and has better resolution for certain problems than the previous scheme. Extensive numerical examples demonstrate the accuracy, efficiency and high resolution of the proposed method.
{"title":"A hybrid MR-WENO scheme with a simplified troubled-cell indicator for hyperbolic conservation laws","authors":"Jinming Zhang , Zhanjing Tao , Jun Zhu , Jianxian Qiu","doi":"10.1016/j.camwa.2026.01.017","DOIUrl":"10.1016/j.camwa.2026.01.017","url":null,"abstract":"<div><div>In this paper, we propose a finite difference hybrid weighted essentially non-oscillatory (WENO) scheme to solve the hyperbolic conservation laws. A simplified troubled-cell indicator is designed for the hybrid scheme. The multi-resolution WENO (MR-WENO) reconstruction is applied in the troubled-cells, while a simple linear reconstruction is used in the remaining regions. The new hybrid scheme inherits the excellent characteristics of the original MR-WENO scheme [1], and can reduce the expensive computational cost of the WENO reconstruction. Compared to the previous hybrid WENO scheme [2] which used the high-degree polynomial in the troubled-cell indicator, the new scheme can reduce the percentage of the troubled-cells, leading to higher computational efficiency. Moreover, our scheme can effectively identify the troubled-cells, and has better resolution for certain problems than the previous scheme. Extensive numerical examples demonstrate the accuracy, efficiency and high resolution of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 1-14"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145996270","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 : 2026-04-01Epub Date: 2026-02-20DOI: 10.1016/j.camwa.2026.02.009
Alonso J. Bustos , Sergio Caucao , Gabriel N. Gatica
We introduce and analyze new mixed formulations, within Banach space-based frameworks, for numerically solving the model given by the coupling of the Brinkman–Forchheimer equations with a convection-diffusion-reaction phenomenon. Specifically, for the former, we consider a pseudostress-velocity mixed formulation, whereas for the latter we analyze both primal and mixed approaches. In particular, for the mixed one the convection-diffusion-reaction part is reformulated by introducing the diffusion vector as an additional unknown, thus leading to a fully-mixed formulation of the coupling. On the other hand, in the mixed-primal setting, the Dirichlet boundary condition for the concentration is enforced through a suitable Lagrange multiplier, whereas this requirement is avoided in the fully mixed approach. We establish the well-posedness of both formulations using a fixed-point strategy and prove the well-posedness of the uncoupled problems by relying on recently established solvability results for perturbed saddle-point problems in Banach spaces, together with the Banach–Nečas–Babuška theorem and the Babuška–Brezzi theory. Additionally, we provide a discrete analysis for both approaches under specific hypotheses on arbitrary finite element spaces. For instance, for each integer k ≥ 0, we consider tensor and vector Raviart–Thomas subspaces of order k for the pseudostress and the diffusion, respectively, along with piecewise polynomial subspaces of degree ≤ k for the velocity and concentration. This choice yields stable Galerkin schemes for the fully-mixed approach, for which optimal theoretical convergence rates are achieved. Finally, we illustrate the theoretical results through several numerical examples, comparing both approaches and discussing the advantages of each.
{"title":"Mixed-primal and fully-mixed formulations for the convection-diffusion-reaction system based upon Brinkman–Forchheimer equations","authors":"Alonso J. Bustos , Sergio Caucao , Gabriel N. Gatica","doi":"10.1016/j.camwa.2026.02.009","DOIUrl":"10.1016/j.camwa.2026.02.009","url":null,"abstract":"<div><div>We introduce and analyze new mixed formulations, within Banach space-based frameworks, for numerically solving the model given by the coupling of the Brinkman–Forchheimer equations with a convection-diffusion-reaction phenomenon. Specifically, for the former, we consider a pseudostress-velocity mixed formulation, whereas for the latter we analyze both primal and mixed approaches. In particular, for the mixed one the convection-diffusion-reaction part is reformulated by introducing the diffusion vector as an additional unknown, thus leading to a fully-mixed formulation of the coupling. On the other hand, in the mixed-primal setting, the Dirichlet boundary condition for the concentration is enforced through a suitable Lagrange multiplier, whereas this requirement is avoided in the fully mixed approach. We establish the well-posedness of both formulations using a fixed-point strategy and prove the well-posedness of the uncoupled problems by relying on recently established solvability results for perturbed saddle-point problems in Banach spaces, together with the Banach–Nečas–Babuška theorem and the Babuška–Brezzi theory. Additionally, we provide a discrete analysis for both approaches under specific hypotheses on arbitrary finite element spaces. For instance, for each integer <em>k</em> ≥ 0, we consider tensor and vector Raviart–Thomas subspaces of order <em>k</em> for the pseudostress and the diffusion, respectively, along with piecewise polynomial subspaces of degree ≤ <em>k</em> for the velocity and concentration. This choice yields stable Galerkin schemes for the fully-mixed approach, for which optimal theoretical convergence rates are achieved. Finally, we illustrate the theoretical results through several numerical examples, comparing both approaches and discussing the advantages of each.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 222-249"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146777698","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 : 2026-04-01Epub Date: 2026-02-01DOI: 10.1016/j.camwa.2026.01.029
Jian Sun , Wenshuai Wang
A scalable MQRBF-FD framework is developed for full-vector elastic wave simulation in heterogeneous media with a spatially varying stiffness tensor C(x). The method resolves P- and S-wave separation and mode conversion at material interfaces using MQRBF spatial discretization on scattered nodes. Parallel subdomain decomposition with ghost-node continuity enables independent execution of all stages—adaptive node refinement, shape parameter optimization using parallel Adam-BP, localized interpolation, and CG solving. Subdomain-specific hierarchical time-stepping and error-driven ANR reduce the number of computational nodes by 35% in the Marmousi model while preserving sharp interface resolution. Compared with structured FD methods, the proposed approach achieves 39% higher accuracy and 15% lower memory usage at equivalent runtime. Validated across 2D and true 3D benchmarks, it establishes a scalable, high-fidelity parallel platform for seismic imaging and advanced material wave modeling.
{"title":"An enhanced MQRBF-FD method with parallel computing and multiscale modeling for efficient elastic wave propagation","authors":"Jian Sun , Wenshuai Wang","doi":"10.1016/j.camwa.2026.01.029","DOIUrl":"10.1016/j.camwa.2026.01.029","url":null,"abstract":"<div><div>A scalable MQRBF-FD framework is developed for full-vector elastic wave simulation in heterogeneous media with a spatially varying stiffness tensor <strong>C</strong>(<strong>x</strong>). The method resolves P- and S-wave separation and mode conversion at material interfaces using MQRBF spatial discretization on scattered nodes. Parallel subdomain decomposition with ghost-node continuity enables independent execution of all stages—adaptive node refinement, shape parameter optimization using parallel Adam-BP, localized interpolation, and CG solving. Subdomain-specific hierarchical time-stepping and error-driven ANR reduce the number of computational nodes by 35% in the Marmousi model while preserving sharp interface resolution. Compared with structured FD methods, the proposed approach achieves 39% higher accuracy and 15% lower memory usage at equivalent runtime. Validated across 2D and true 3D benchmarks, it establishes a scalable, high-fidelity parallel platform for seismic imaging and advanced material wave modeling.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"207 ","pages":"Pages 152-170"},"PeriodicalIF":2.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146110518","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 : 2026-04-01Epub Date: 2026-01-20DOI: 10.1016/j.camwa.2026.01.015
Chen Bai , Yunhu Zhang , Hongxing Zheng , Quan Qian
Simulating the metal solidification process is crucial for improving product quality, optimizing manufacturing processes, and developing new materials. Traditional numerical methods like the Finite Element Method (FEM) and Physics-Informed Neural Networks (PINNs) face significant challenges when applied to metal solidification simulations due to inefficiencies and inaccuracies in dealing with multiphysics coupling, nonlinearity, and spatio-temporal complexity. Despite their potential, PINNs require further optimization to accurately capture complex physical phenomena in practical simulations. In this study, we propose a novel method based on PINNs, termed MS-PINN, which integrates Fourier Feature Embedding (FFE), Residual-based Adaptive Resampling (RAD), and Self-adaptive Loss Balanced methods (SAL) to significantly enhance simulation accuracy and efficiency. FFE improves the model’s ability to capture high-frequency features, RAD increases learning efficiency in high-gradient regions, and SAL dynamically adjusts loss function weights to optimize the training process. Experimental results show that MS-PINN outperforms traditional PINNs and other advanced approaches, achieving average error reductions of approximately 81.00% compared to Conv-LSTM, 77.11% compared to TCN, and 61.56% compared to PINN in reconstruction experiments. In predictive experiments, MS-PINN reduces errors by 53.23%, 68.81%, and 72.54% compared to PINN, TCN, and CONV-LSTM methods, respectively. Additionally, we developed a general PDE-solving software, NeuroPDE, based on this method. NeuroPDE has demonstrated success not only in the solidification process of Cu-1wt.%Ag alloy but also in solving Burgers, diffusion, and Navier-Stokes (NS) equations, including turbulent datasets characterized by high Reynolds numbers, and their inverse problems. This highlights NeuroPDE’s versatility and broad applicability in solving complex forward and inverse problems in fluid dynamics and other fields.
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