Pub Date : 2025-12-04DOI: 10.1016/j.camwa.2025.11.013
Jinxiu Zhang , Xuehua Yang , Song Wang
This paper develops and analyzes a linearized compact three-levelfinite difference (CTLFD) scheme to solve the two-dimensional (2-D) Kuramoto-Tsuzuki dynamics dominated by strong nonlinear characteristics. This method combines the compact difference method (CDM) in space with the Crank-Nicolson (C-N) scheme in time, achieving an overall convergence rate of , where τ, hx, and hy represent the time and space step sizes, respectively. Nonlinear terms are linearized in a semi-implicit manner to enhance stability and computational efficiency. A rigorous stability and error analysis is carried out using an energy technique together with mathematical induction, confirming boundedness, uniqueness, and the optimal convergence for the numerical solution under two discrete norms. Finally, three sets of numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and robustness of the proposed scheme.
{"title":"A compact difference method for the 2-D Kuramoto-Tsuzuki complex equation with Neumann boundary characterized by strong nonlinear effects","authors":"Jinxiu Zhang , Xuehua Yang , Song Wang","doi":"10.1016/j.camwa.2025.11.013","DOIUrl":"10.1016/j.camwa.2025.11.013","url":null,"abstract":"<div><div>This paper develops and analyzes a linearized compact three-levelfinite difference (CTLFD) scheme to solve the two-dimensional (2-D) Kuramoto-Tsuzuki dynamics dominated by strong nonlinear characteristics. This method combines the compact difference method (CDM) in space with the Crank-Nicolson (C-N) scheme in time, achieving an overall convergence rate of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>h</mi><mi>x</mi><mn>4</mn></msubsup><mo>+</mo><msubsup><mi>h</mi><mi>y</mi><mn>4</mn></msubsup><mo>)</mo></mrow></mrow></math></span>, where <em>τ, h<sub>x</sub></em>, and <em>h<sub>y</sub></em> represent the time and space step sizes, respectively. Nonlinear terms are linearized in a semi-implicit manner to enhance stability and computational efficiency. A rigorous stability and error analysis is carried out using an energy technique together with mathematical induction, confirming boundedness, uniqueness, and the optimal convergence for the numerical solution under two discrete norms. Finally, three sets of numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and robustness of the proposed scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 1-19"},"PeriodicalIF":2.5,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658796","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 : 2025-12-04DOI: 10.1016/j.camwa.2025.11.012
Minghua Chen , Jiankang Shi , Zhen Song , Yubin Yan , Zhi Zhou
In this paper, we introduce a time discretization scheme for solving the stochastic subdiffusion equation based on the two-fold integral-differential and two step backward differentiation formula (ID2-BDF2). We prove that this scheme attains a convergence rate of for with α ∈ (0, 1) and γ ∈ [0, 1]. Our approach regularizes the additive noise through a two-fold integral-differential (ID2) calculus and discretizes the equation using BDF2 convolution quadrature, achieving superlinear convergence in solving the stochastic subdiffusion. Furthermore, we extend the scheme to solve the stochastic fractional wave equation, proving that the scheme achieves a convergence rate of for α ∈ (1, 2) and γ ∈ [0, 1]. Numerical examples are presented to validate the theoretical results for the linear problem. The numerical observations further indicate that the same convergence rates also apply to stochastic semilinear time-fractional equations.
{"title":"Time discretization schemes for stochastic subdiffusion and fractional wave equations with integrated additive noise","authors":"Minghua Chen , Jiankang Shi , Zhen Song , Yubin Yan , Zhi Zhou","doi":"10.1016/j.camwa.2025.11.012","DOIUrl":"10.1016/j.camwa.2025.11.012","url":null,"abstract":"<div><div>In this paper, we introduce a time discretization scheme for solving the stochastic subdiffusion equation based on the two-fold integral-differential and two step backward differentiation formula (ID2-BDF2). We prove that this scheme attains a convergence rate of <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mi>α</mi><mo>+</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math></span> for <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>α</mi><mo>+</mo><mi>γ</mi><mo><</mo><mn>2</mn></mrow></math></span> with <em>α</em> ∈ (0, 1) and <em>γ</em> ∈ [0, 1]. Our approach regularizes the additive noise through a two-fold integral-differential (ID2) calculus and discretizes the equation using BDF2 convolution quadrature, achieving superlinear convergence in solving the stochastic subdiffusion. Furthermore, we extend the scheme to solve the stochastic fractional wave equation, proving that the scheme achieves a convergence rate of <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mi>min</mi><mo>{</mo><mn>2</mn><mo>,</mo><mi>α</mi><mo>+</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>}</mo></mrow></msup><mo>)</mo></mrow></math></span> for <em>α</em> ∈ (1, 2) and <em>γ</em> ∈ [0, 1]. Numerical examples are presented to validate the theoretical results for the linear problem. The numerical observations further indicate that the same convergence rates also apply to stochastic semilinear time-fractional equations.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 155-169"},"PeriodicalIF":2.5,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145689359","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}
A local meshless radial basis function-finite difference (RBF-FD) method, leveraging polyharmonic spline (PHSn) kernel, is introduced for solving elliptic interface problems with non-homogeneous jump conditions on curved surfaces. The design of the proposed method is based on the splitting of surface region according to the interface, and the quasi-uniformed meshless fitted nodes on the interface and surface. The meshless fitted setting ensures the efficiency of scattered node generation and the simplicity of discretization. Additionally, we ensure the positivity of the basis functions through polyharmonic spline-coupled polynomials, thus making the interpolation matrix invertible. By incorporating the shape parameter-free PHSn function within the RBF-FD framework, our method yields remarkable spatial accuracy, matrix sparsity, and high-order polynomial convergence of second-, third-, and fourth-order in discretizing the surface interface problem. In the numerical experiments section, we comprehensively demonstrate the superiority of our method in terms of accuracy through a series of rigorous tests. Additionally, we explore the influence of diverse factors, including stencil size, polynomial degree, polyharmonic spline exponent, quasi-uniformed scattering distance, and the diffusion coefficient, on the overall efficiency of our method.
{"title":"An RBF-FD method for solving elliptic interface problems with non-homogeneous jump conditions on curved surfaces","authors":"Shengye Zhang, Xufeng Xiao, Yuanyang Qiao, Xinlong Feng","doi":"10.1016/j.camwa.2025.11.011","DOIUrl":"10.1016/j.camwa.2025.11.011","url":null,"abstract":"<div><div>A local meshless radial basis function-finite difference (RBF-FD) method, leveraging polyharmonic spline (PHSn) kernel, is introduced for solving elliptic interface problems with non-homogeneous jump conditions on curved surfaces. The design of the proposed method is based on the splitting of surface region according to the interface, and the quasi-uniformed meshless fitted nodes on the interface and surface. The meshless fitted setting ensures the efficiency of scattered node generation and the simplicity of discretization. Additionally, we ensure the positivity of the basis functions through polyharmonic spline-coupled polynomials, thus making the interpolation matrix invertible. By incorporating the shape parameter-free PHSn function within the RBF-FD framework, our method yields remarkable spatial accuracy, matrix sparsity, and high-order polynomial convergence of second-, third-, and fourth-order in discretizing the surface interface problem. In the numerical experiments section, we comprehensively demonstrate the superiority of our method in terms of accuracy through a series of rigorous tests. Additionally, we explore the influence of diverse factors, including stencil size, polynomial degree, polyharmonic spline exponent, quasi-uniformed scattering distance, and the diffusion coefficient, on the overall efficiency of our method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 113-129"},"PeriodicalIF":2.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657529","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 : 2025-12-01DOI: 10.1016/j.camwa.2025.11.010
Zhen Yin , Ke Liang
The in-plane stiffness of honeycomb structures is several orders of magnitude lower than the material itself, making them an ideal sandwich structure for flexible deformation mechanisms of high-speed aircrafts in a high-temperature environment. Finite element method is widely used to simulate the in-and-out-of-plane large deformation of honeycomb structures; however the high computational cost is still a major bottleneck for fast analysis and optimization design. The existing reduced-order methods are mainly applicable to buckling problems, rather than the in-plane and out-of-plane large deformation case of honeycomb structures. In this work, a geometrically nonlinear reduced-order method considering both the thermal expansion and temperature-dependent material properties is proposed for in-and-out-of-plane large deformation analysis of honeycomb structures subjected to a high temperature field. The dual reduced-order models with only one degree of freedom are constructed for the temperature rise and mechanical loading phases, respectively, based on the third-order equilibrium equations and perturbation method. The constructional efficiency of the reduced system is largely improved by zeroing the fourth-order strain energy variation using the two-field Hellinger-Reissner variational principle and three-dimensional (3D) solid element. The nonlinear predictor solved by the reduced-order model can be corrected when its numerical accuracy is not satisfactory in path-following analysis. Various numerical examples demonstrate that the proposed method has a superior path-following capability for 3D finite element analysis of honeycomb structures.
{"title":"A dual reduced-order modeling method for nonlinear thermoelastic analysis of 3D honeycomb structure","authors":"Zhen Yin , Ke Liang","doi":"10.1016/j.camwa.2025.11.010","DOIUrl":"10.1016/j.camwa.2025.11.010","url":null,"abstract":"<div><div>The in-plane stiffness of honeycomb structures is several orders of magnitude lower than the material itself, making them an ideal sandwich structure for flexible deformation mechanisms of high-speed aircrafts in a high-temperature environment. Finite element method is widely used to simulate the in-and-out-of-plane large deformation of honeycomb structures; however the high computational cost is still a major bottleneck for fast analysis and optimization design. The existing reduced-order methods are mainly applicable to buckling problems, rather than the in-plane and out-of-plane large deformation case of honeycomb structures. In this work, a geometrically nonlinear reduced-order method considering both the thermal expansion and temperature-dependent material properties is proposed for in-and-out-of-plane large deformation analysis of honeycomb structures subjected to a high temperature field. The dual reduced-order models with only one degree of freedom are constructed for the temperature rise and mechanical loading phases, respectively, based on the third-order equilibrium equations and perturbation method. The constructional efficiency of the reduced system is largely improved by zeroing the fourth-order strain energy variation using the two-field Hellinger-Reissner variational principle and three-dimensional (3D) solid element. The nonlinear predictor solved by the reduced-order model can be corrected when its numerical accuracy is not satisfactory in path-following analysis. Various numerical examples demonstrate that the proposed method has a superior path-following capability for 3D finite element analysis of honeycomb structures.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 130-154"},"PeriodicalIF":2.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657527","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 : 2025-11-25DOI: 10.1016/j.camwa.2025.11.009
Renjun Gao, Xiangjie Kong, Dongting Cai, Boyi Fu, Junxiang Yang
Reconstruction of an object from points cloud is essential in prosthetics, medical imaging, computer vision, etc. We present an effective algorithm for an Allen–Cahn-type model of reconstruction, employing the Lagrange multiplier approach. Utilizing scattered data points from an object, we reconstruct a narrow shell by solving the governing equation enhanced with an edge detection function derived from the unsigned distance function. The specifically designed edge detection function ensures the energy stability. By reformulating the governing equation through the Lagrange multiplier technique and implementing a Crank–Nicolson time discretization, we can update the solutions in a stable and decoupled manner. The spatial operations are approximated using the finite difference method, and we analytically demonstrate the unconditional stability of the fully discrete scheme. Comprehensive numerical experiments, including reconstructions of complex 3D volumes such as characters from Star Wars, validate the algorithm’s accuracy, stability, and effectiveness. Additionally, we analyze how specific parameter selections influence the level of detail and refinement in the reconstructed volumes. To facilitate the interested readers to understand our algorithm, we share the computational codes and data in https://github.com/cfdyang521/C-3PO/tree/main.
{"title":"Three-dimensional narrow volume reconstruction method with unconditional stability based on a phase-field lagrange multiplier approach","authors":"Renjun Gao, Xiangjie Kong, Dongting Cai, Boyi Fu, Junxiang Yang","doi":"10.1016/j.camwa.2025.11.009","DOIUrl":"10.1016/j.camwa.2025.11.009","url":null,"abstract":"<div><div>Reconstruction of an object from points cloud is essential in prosthetics, medical imaging, computer vision, etc. We present an effective algorithm for an Allen–Cahn-type model of reconstruction, employing the Lagrange multiplier approach. Utilizing scattered data points from an object, we reconstruct a narrow shell by solving the governing equation enhanced with an edge detection function derived from the unsigned distance function. The specifically designed edge detection function ensures the energy stability. By reformulating the governing equation through the Lagrange multiplier technique and implementing a Crank–Nicolson time discretization, we can update the solutions in a stable and decoupled manner. The spatial operations are approximated using the finite difference method, and we analytically demonstrate the unconditional stability of the fully discrete scheme. Comprehensive numerical experiments, including reconstructions of complex 3D volumes such as characters from <em>Star Wars</em>, validate the algorithm’s accuracy, stability, and effectiveness. Additionally, we analyze how specific parameter selections influence the level of detail and refinement in the reconstructed volumes. To facilitate the interested readers to understand our algorithm, we share the computational codes and data in <span><span>https://github.com/cfdyang521/C-3PO/tree/main</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 88-112"},"PeriodicalIF":2.5,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145598938","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 : 2025-11-24DOI: 10.1016/j.camwa.2025.11.005
Rui Wang , Cheng Wang , Yuzhe Qin , Zhengru Zhang
In this work, we propose a Keller–Segel–Navier–Stokes (KSNS) model for describing chemotactic phenomena, formulated within the framework of the Energetic Variational Approach (EnVarA). A second-order accurate numerical scheme is developed that rigorously preserves three fundamental properties in discrete sense: the positivity of cell density, mass conservation of cell density, and total energy dissipation. The Keller–Segel subsystem is reformulated as a coupling between an gradient flow with non-constant mobility and an gradient flow, enabling the effective treatment of the nonlinear and singular logarithmic energy potential via a modified Crank-Nicolson scheme. Artificial regularization terms are introduced to enforce positivity preservation. For the fluid dynamics component, we adopt a second-order semi-implicit time discretization. The marker-and-cell (MAC) finite difference approximation is used as the spatial discretization, which ensures a discretely divergence-free velocity field. The proposed numerical method guarantees unique solvability, mass conservation, and total energy stability. Furthermore, through detailed asymptotic expansions and rigorous error analysis, we establish optimal convergence rates. A series of numerical experiments are presented to validate the effectiveness and robustness of both the physical model and the numerical scheme.
{"title":"A second order accurate, and structure-preserving numerical scheme for the thermodynamical consistent Keller–Segel–Navier–Stokes model","authors":"Rui Wang , Cheng Wang , Yuzhe Qin , Zhengru Zhang","doi":"10.1016/j.camwa.2025.11.005","DOIUrl":"10.1016/j.camwa.2025.11.005","url":null,"abstract":"<div><div>In this work, we propose a Keller–Segel–Navier–Stokes (KSNS) model for describing chemotactic phenomena, formulated within the framework of the Energetic Variational Approach (EnVarA). A second-order accurate numerical scheme is developed that rigorously preserves three fundamental properties in discrete sense: the positivity of cell density, mass conservation of cell density, and total energy dissipation. The Keller–Segel subsystem is reformulated as a coupling between an <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> gradient flow with non-constant mobility and an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> gradient flow, enabling the effective treatment of the nonlinear and singular logarithmic energy potential via a modified Crank-Nicolson scheme. Artificial regularization terms are introduced to enforce positivity preservation. For the fluid dynamics component, we adopt a second-order semi-implicit time discretization. The marker-and-cell (MAC) finite difference approximation is used as the spatial discretization, which ensures a discretely divergence-free velocity field. The proposed numerical method guarantees unique solvability, mass conservation, and total energy stability. Furthermore, through detailed asymptotic expansions and rigorous error analysis, we establish optimal convergence rates. A series of numerical experiments are presented to validate the effectiveness and robustness of both the physical model and the numerical scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 59-87"},"PeriodicalIF":2.5,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145592984","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 : 2025-11-22DOI: 10.1016/j.camwa.2025.11.006
Meixin Xiong , Liuhong Chen , Yulan Ning , Ju Ming , Zhiwen Zhang
We propose a novel stochastic reduced-order model (SROM) for complex systems by combining statistical analysis tools. Based on the generalizability of distance in the centroidal Voronoi tessellation (CVT) method and the minimization of projection error in proper orthogonal decomposition (POD), we define a time-dependent generalized CVT clustering. Each generalized centroid corresponds to a set of cluster-based POD (CPOD) basis functions. Then, using the clustering results as the training dataset, the classification mechanism of the system input can be obtained by applying the naive Bayesian method. For a given input sample, the predicted label obtained by the classifier is used to determine a set of CPOD basis functions for model reduction. Rigorous error analysis is shown, and a discussion of the Navier-Stokes equation with random parameters is given to provide a context for the application of this SROM. Numerical experiments verify that the accuracy of our SROM is improved compared with the standard POD method.
{"title":"Pre-classification based stochastic reduced-order model for time-dependent complex system","authors":"Meixin Xiong , Liuhong Chen , Yulan Ning , Ju Ming , Zhiwen Zhang","doi":"10.1016/j.camwa.2025.11.006","DOIUrl":"10.1016/j.camwa.2025.11.006","url":null,"abstract":"<div><div>We propose a novel stochastic reduced-order model (SROM) for complex systems by combining statistical analysis tools. Based on the generalizability of distance in the centroidal Voronoi tessellation (CVT) method and the minimization of projection error in proper orthogonal decomposition (POD), we define a time-dependent generalized CVT clustering. Each generalized centroid corresponds to a set of cluster-based POD (CPOD) basis functions. Then, using the clustering results as the training dataset, the classification mechanism of the system input can be obtained by applying the naive Bayesian method. For a given input sample, the predicted label obtained by the classifier is used to determine a set of CPOD basis functions for model reduction. Rigorous error analysis is shown, and a discussion of the Navier-Stokes equation with random parameters is given to provide a context for the application of this SROM. Numerical experiments verify that the accuracy of our SROM is improved compared with the standard POD method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 38-58"},"PeriodicalIF":2.5,"publicationDate":"2025-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145575488","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 : 2025-11-20DOI: 10.1016/j.camwa.2025.11.007
Chenglong Xu , Yining Qiu , Keyan Wang , Bihao Su
This article proposes a prediction-correction method for solving nonlinear partial differential equations (PDEs) in complex multidimensional domains. Our approach employs the Feynman-Kac formula, which establishes a connection between stochastic differential equations and PDEs, along with multidimensional function interpolation and corresponding integration methods based on sparse grid nodes. First, we derive the relationship between solutions of discrete-time PDEs using the Feynman-Kac formula. Next, we establish a discretization scheme through integration over sparse grid points and then construct the expression of each time-layer solution via interpolation on these points. Prediction-correction techniques are applied to address the integration of nonlinear unknown functions. While deriving a prediction-correction numerical method, this paper also presents a rigorous theoretical error analysis of the method. The effectiveness and convergence of the proposed method are validated through two types of examples. Compared with traditional numerical methods, our method enables efficient handling of more complex and multidimensional regions.
{"title":"Prediction-correction method for nonlinear partial differential equations based on sparse grid interpolation techniques","authors":"Chenglong Xu , Yining Qiu , Keyan Wang , Bihao Su","doi":"10.1016/j.camwa.2025.11.007","DOIUrl":"10.1016/j.camwa.2025.11.007","url":null,"abstract":"<div><div>This article proposes a prediction-correction method for solving nonlinear partial differential equations (PDEs) in complex multidimensional domains. Our approach employs the Feynman-Kac formula, which establishes a connection between stochastic differential equations and PDEs, along with multidimensional function interpolation and corresponding integration methods based on sparse grid nodes. First, we derive the relationship between solutions of discrete-time PDEs using the Feynman-Kac formula. Next, we establish a discretization scheme through integration over sparse grid points and then construct the expression of each time-layer solution via interpolation on these points. Prediction-correction techniques are applied to address the integration of nonlinear unknown functions. While deriving a prediction-correction numerical method, this paper also presents a rigorous theoretical error analysis of the method. The effectiveness and convergence of the proposed method are validated through two types of examples. Compared with traditional numerical methods, our method enables efficient handling of more complex and multidimensional regions.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 1-19"},"PeriodicalIF":2.5,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555378","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 : 2025-11-20DOI: 10.1016/j.camwa.2025.11.008
Seungmin Lee, Sanghyun Lee
Real-world applications often suffer from incomplete or uncertain data, which can undermine the accuracy of physical models. In our work, we tackle this challenge by applying continuous data assimilation (CDA) to Biot's poroelasticity system, a model that captures how fluid-saturated porous materials, such as soils or biological tissues, deform under stress. We utilize a nudging term in the conservation of mass equation to integrate sparse observational data directly into the pressure field. We then use an iterative fixed-stress method to decouple the pressure and displacement problem, making the process both stable and efficient. Our approach employs finite element methods along with an interpolation strategy that adapts to different data resolutions. Several numerical tests illustrate that the proposed method reliably drives the model's predictions towards the true state, even in complex scenarios involving heterogeneous permeability or nonlinear permeability, and including a three dimensional setup. This study is the first to rigorously apply CDA to Biot's poroelasticity system, offering a promising avenue for more accurate predictive modeling in fields ranging from geomechanics to biomechanics.
{"title":"Convergence and numerical simulations of the continuous data assimilation for Biot's poroelasticity system","authors":"Seungmin Lee, Sanghyun Lee","doi":"10.1016/j.camwa.2025.11.008","DOIUrl":"10.1016/j.camwa.2025.11.008","url":null,"abstract":"<div><div>Real-world applications often suffer from incomplete or uncertain data, which can undermine the accuracy of physical models. In our work, we tackle this challenge by applying continuous data assimilation (CDA) to Biot's poroelasticity system, a model that captures how fluid-saturated porous materials, such as soils or biological tissues, deform under stress. We utilize a nudging term in the conservation of mass equation to integrate sparse observational data directly into the pressure field. We then use an iterative fixed-stress method to decouple the pressure and displacement problem, making the process both stable and efficient. Our approach employs finite element methods along with an interpolation strategy that adapts to different data resolutions. Several numerical tests illustrate that the proposed method reliably drives the model's predictions towards the true state, even in complex scenarios involving heterogeneous permeability or nonlinear permeability, and including a three dimensional setup. This study is the first to rigorously apply CDA to Biot's poroelasticity system, offering a promising avenue for more accurate predictive modeling in fields ranging from geomechanics to biomechanics.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"202 ","pages":"Pages 20-37"},"PeriodicalIF":2.5,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555379","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 : 2025-11-19DOI: 10.1016/j.camwa.2025.11.003
Witold Cecot , Marta Oleksy , Mateusz Dryzek
This paper presents a comparison of various approaches to approximating the unknown functions within the mixed ultra-weak formulation, using Voronoi polygon discretization and stabilization via the Discontinuous Petrov–Galerkin (DPG) methodology. The primary objective is to identify the basis functions that offer the best approximability. We consider meshless functions, which have not previously been applied to fully mixed formulations (stress–displacement in mechanics), and compare several of their variants with polynomial approximations. Numerical tests are conducted for the Laplace equation on both square and L-shaped domains, for regular as well as singular solutions. Our study reveals that, with one exception, the polynomial approximation—particularly the orthogonal Legendre type—exhibits faster convergence, most likely due to its superior interpolation properties in the two-field formulation. The findings of this study can be extended to other elliptic problems, including those in solid mechanics, where mixed formulations and polygonal finite elements offer significant advantages.
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