Pub Date : 2026-10-01Epub Date: 2026-02-14DOI: 10.1016/j.cam.2026.117454
Guo Qiu Wang , Wei Liang
Building upon the concept of discretely orthogonal bases, this paper develops a generalized interpolation framework, with the classical Lagrange interpolation method serving as a special case. Specifically, for an arbitrary number of specific non-equidistant interpolation nodes, this paper constructs corresponding discretely orthogonal polynomial bases, whose associated orthogonal matrices coincide with the well-known Discrete Cosine Transforms (DCTs). Using these polynomial bases, we show that when interpolation nodes are chosen as extended Chebyshev nodes, the interpolation of continuous functions converge in the square-integrable sense. Furthermore, we prove that the resulting interpolation functions based on extended Chebyshev nodes exhibit uniform convergence in the Hölder continuity class. These results not only provide a rigorous theoretical foundation for polynomial-based signal representation in digital conditioning of sensors, but also suggest a viable candidate for spectral-type approach for numerical schemes for partial differential equations (PDEs).
{"title":"Non-orthogonal interpolation on closed interval and convergence","authors":"Guo Qiu Wang , Wei Liang","doi":"10.1016/j.cam.2026.117454","DOIUrl":"10.1016/j.cam.2026.117454","url":null,"abstract":"<div><div>Building upon the concept of discretely orthogonal bases, this paper develops a generalized interpolation framework, with the classical Lagrange interpolation method serving as a special case. Specifically, for an arbitrary number of specific non-equidistant interpolation nodes, this paper constructs corresponding discretely orthogonal polynomial bases, whose associated orthogonal matrices coincide with the well-known Discrete Cosine Transforms (DCTs). Using these polynomial bases, we show that when interpolation nodes are chosen as extended Chebyshev nodes, the interpolation of continuous functions converge in the square-integrable sense. Furthermore, we prove that the resulting interpolation functions based on extended Chebyshev nodes exhibit uniform convergence in the Hölder continuity class. These results not only provide a rigorous theoretical foundation for polynomial-based signal representation in digital conditioning of sensors, but also suggest a viable candidate for spectral-type approach for numerical schemes for partial differential equations (PDEs).</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117454"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387183","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 : 2026-10-01Epub Date: 2026-02-10DOI: 10.1016/j.cam.2026.117418
Dean Chou , Ifrah Iqbal , Yasser Alrashedi , Theyab Alrashdi , Hamood Ur Rehman
In this research, we examine the equal-width equation, a basic model for one-dimensional wave propagation in nonlinear fluid dynamics. Using the Kudryashov method, we obtain explicit soliton solutions that reflect the equation’s inherent nonlinear nature, modeling different hydrodynamic phenomena like shallow water waves and internal solitons. The solutions are graphically represented using three-dimensional (3D), contour, density, and two-dimensional (2D) plots to gain further insight into wave evolution. To confirm the analytical solutions, we apply the differential transform method (DTM) for numerical simulations, allowing for comparative analysis between theoretical solitons and their discrete approximations. In addition, stability and modulation instability analyses are conducted to determine the robustness of these wave structures under small perturbations, important for understanding turbulence and energy dissipation in fluids. Furthermore, we perform a bifurcation analysis through the building of phase portraits and vector fields, uncovering complex dynamical behaviors like periodic and chaotic motion in nonlinear fluid systems. In order to expand our investigation, we add a periodic perturbation to investigate chaotic wave interactions, represented through phase space trajectories and time series plots. The perturbed system presents a perturbation with elements of intensity δ and frequency ϕ, enabling us to study how small periodic perturbations influence the dynamical behavior and stability of the nonlinear wave solutions. Finally, we investigate multistability and carry out sensitivity analysis, evaluating how initial conditions affect solution trajectories in a fluid system. Our results are helping toward a deeper understanding of nonlinear wave mechanics and their repercussions in fluid physics. This work addresses the lack of a unified framework by combining exact soliton solutions, numerical validation, and nonlinear dynamical analysis for the equal-width equation.
{"title":"Bifurcation, chaotic behaviour, multistability and sensitivity analysis: Exact and numerical analysis of nonlinear dispersive wave equation","authors":"Dean Chou , Ifrah Iqbal , Yasser Alrashedi , Theyab Alrashdi , Hamood Ur Rehman","doi":"10.1016/j.cam.2026.117418","DOIUrl":"10.1016/j.cam.2026.117418","url":null,"abstract":"<div><div>In this research, we examine the equal-width equation, a basic model for one-dimensional wave propagation in nonlinear fluid dynamics. Using the Kudryashov method, we obtain explicit soliton solutions that reflect the equation’s inherent nonlinear nature, modeling different hydrodynamic phenomena like shallow water waves and internal solitons. The solutions are graphically represented using three-dimensional (3D), contour, density, and two-dimensional (2D) plots to gain further insight into wave evolution. To confirm the analytical solutions, we apply the differential transform method (DTM) for numerical simulations, allowing for comparative analysis between theoretical solitons and their discrete approximations. In addition, stability and modulation instability analyses are conducted to determine the robustness of these wave structures under small perturbations, important for understanding turbulence and energy dissipation in fluids. Furthermore, we perform a bifurcation analysis through the building of phase portraits and vector fields, uncovering complex dynamical behaviors like periodic and chaotic motion in nonlinear fluid systems. In order to expand our investigation, we add a periodic perturbation to investigate chaotic wave interactions, represented through phase space trajectories and time series plots. The perturbed system presents a perturbation with elements of intensity <em>δ</em> and frequency <em>ϕ</em>, enabling us to study how small periodic perturbations influence the dynamical behavior and stability of the nonlinear wave solutions. Finally, we investigate multistability and carry out sensitivity analysis, evaluating how initial conditions affect solution trajectories in a fluid system. Our results are helping toward a deeper understanding of nonlinear wave mechanics and their repercussions in fluid physics. This work addresses the lack of a unified framework by combining exact soliton solutions, numerical validation, and nonlinear dynamical analysis for the equal-width equation.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117418"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387184","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}
This paper presents a C-FISTA-type proximal point algorithm for solving strongly quasiconvex pseudomonotone equilibrium problems. Our proposed method consists of two momentum terms, a correction term, and the proximal point algorithm. We establish the convergence of our proposed method under standard assumptions. Furthermore, we obtain the sublinear and linear convergence rates of our proposed method. Finally, we present a numerical test for solving equilibrium problems to illustrate the effectiveness and versatility of our proposed method.
{"title":"A C-FISTA-type proximal point algorithm for strongly quasiconvex pseudomonotone equilibrium problems","authors":"Grace Nnennaya Ogwo , Chinedu Izuchukwu , Yekini Shehu","doi":"10.1016/j.cam.2026.117504","DOIUrl":"10.1016/j.cam.2026.117504","url":null,"abstract":"<div><div>This paper presents a C-FISTA-type proximal point algorithm for solving strongly quasiconvex pseudomonotone equilibrium problems. Our proposed method consists of two momentum terms, a correction term, and the proximal point algorithm. We establish the convergence of our proposed method under standard assumptions. Furthermore, we obtain the sublinear and linear convergence rates of our proposed method. Finally, we present a numerical test for solving equilibrium problems to illustrate the effectiveness and versatility of our proposed method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117504"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386988","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 : 2026-10-01Epub Date: 2026-02-22DOI: 10.1016/j.cam.2026.117476
Na-Na Wang , Ji-Cheng Li
In this paper, a class of product-type (PT) preconditioners for generalized saddle point problems recently proposed in [N. Wang, J. Li, A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems, IMA J. Numer. Anal., (2023) 43, 2998–3025] are extended to solve the double saddle point problems arising from the modeling of liquid crystal directors. By combining augmented Lagrangian (AL) technique, two specific block PT preconditioners are developed, which are applied appropriately with the efficient conjugate gradient (CG) and conjugate residual (CR) methods although neither the preconditioners nor the double saddle point systems are symmetric positive definite (SPD). This is the biggest advantage and novelty of the proposed preconditioners. The proposed preconditioned CG (PCG) and preconditioned CR (PCR) methods actually belong to the categories of nonstandard inner product CG and nonstandard inner product CR methods, respectively. Moreover, the PCG and PCR algorithms and their convergence theorems are given. Theoretical and experimental analysis shows that the spectra of the preconditioned matrices are contained within real and positive intervals which are very sharp if the involved parameters are chosen appropriately. In addition, the practically useful values for parameters are easy to obtain. Numerical experiments are presented to illustrate the rapidity, effectiveness and numerical stability of the proposed preconditioners and show the advantages of the proposed preconditioners over the existing state-of-the-art preconditioners for double saddle point problems.
{"title":"Two block product-type preconditioners for double saddle point problems","authors":"Na-Na Wang , Ji-Cheng Li","doi":"10.1016/j.cam.2026.117476","DOIUrl":"10.1016/j.cam.2026.117476","url":null,"abstract":"<div><div>In this paper, a class of product-type (PT) preconditioners for generalized saddle point problems recently proposed in [N. Wang, J. Li, A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems, IMA J. Numer. Anal., (2023) 43, 2998–3025] are extended to solve the double saddle point problems arising from the modeling of liquid crystal directors. By combining augmented Lagrangian (AL) technique, two specific block PT preconditioners are developed, which are applied appropriately with the efficient conjugate gradient (CG) and conjugate residual (CR) methods although neither the preconditioners nor the double saddle point systems are symmetric positive definite (SPD). This is the biggest advantage and novelty of the proposed preconditioners. The proposed preconditioned CG (PCG) and preconditioned CR (PCR) methods actually belong to the categories of nonstandard inner product CG and nonstandard inner product CR methods, respectively. Moreover, the PCG and PCR algorithms and their convergence theorems are given. Theoretical and experimental analysis shows that the spectra of the preconditioned matrices are contained within real and positive intervals which are very sharp if the involved parameters are chosen appropriately. In addition, the practically useful values for parameters are easy to obtain. Numerical experiments are presented to illustrate the rapidity, effectiveness and numerical stability of the proposed preconditioners and show the advantages of the proposed preconditioners over the existing state-of-the-art preconditioners for double saddle point problems.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117476"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386993","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 : 2026-10-01Epub Date: 2026-02-26DOI: 10.1016/j.cam.2026.117473
F. Arenas , R. Pérez , M. Gonzalez-Lima , C.A. Arias
In this paper we present a centered Newton type algorithm for solving the nonlinear complementarity problem by a reformulation of the problem as a nonlinear system of equations with nonnegativity constraints. The proposed algorithm considers centered Newton directions projected over the feasible set in order to maintain iterate feasibility. We present theoretical and numerical results for the proposal.
{"title":"A centered Newton method for nonlinear complementarity problem","authors":"F. Arenas , R. Pérez , M. Gonzalez-Lima , C.A. Arias","doi":"10.1016/j.cam.2026.117473","DOIUrl":"10.1016/j.cam.2026.117473","url":null,"abstract":"<div><div>In this paper we present a centered Newton type algorithm for solving the nonlinear complementarity problem by a reformulation of the problem as a nonlinear system of equations with nonnegativity constraints. The proposed algorithm considers centered Newton directions projected over the feasible set in order to maintain iterate feasibility. We present theoretical and numerical results for the proposal.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117473"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386997","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 : 2026-10-01Epub Date: 2026-02-10DOI: 10.1016/j.cam.2026.117422
Shashi Kant Mishra , Dheerendra Singh
In this paper, we consider a nonsmooth mathematical programming problem and establish the characterization of solution sets. We also give a normal cone condition for nonsmooth mathematical programming problems to obtain optimality conditions using Lagrange multipliers and tangential subdifferentials. We also provide some examples in support of our results.
{"title":"Characterizations of solution sets of nonsmooth mathematical programming problems","authors":"Shashi Kant Mishra , Dheerendra Singh","doi":"10.1016/j.cam.2026.117422","DOIUrl":"10.1016/j.cam.2026.117422","url":null,"abstract":"<div><div>In this paper, we consider a nonsmooth mathematical programming problem and establish the characterization of solution sets. We also give a normal cone condition for nonsmooth mathematical programming problems to obtain optimality conditions using Lagrange multipliers and tangential subdifferentials. We also provide some examples in support of our results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117422"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387119","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 : 2026-10-01Epub Date: 2026-02-19DOI: 10.1016/j.cam.2026.117459
Saba Asgarzadeh , M.R. Eslahchi
In this research, we introduce new classes of regularization matrices constructed using generalized operational matrices of the Caputo fractional derivative. Experimental results confirm the performance and effectiveness of the proposed method. In another part of this study, the obtained matrices are applied to classification tasks in machine learning. The results demonstrate improved classification accuracy and more effective data representation.
{"title":"Regularization via generalized operational matrices: Theory and applications in machine learning classification","authors":"Saba Asgarzadeh , M.R. Eslahchi","doi":"10.1016/j.cam.2026.117459","DOIUrl":"10.1016/j.cam.2026.117459","url":null,"abstract":"<div><div>In this research, we introduce new classes of regularization matrices constructed using generalized operational matrices of the Caputo fractional derivative. Experimental results confirm the performance and effectiveness of the proposed method. In another part of this study, the obtained matrices are applied to classification tasks in machine learning. The results demonstrate improved classification accuracy and more effective data representation.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117459"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387130","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 : 2026-10-01Epub Date: 2026-02-12DOI: 10.1016/j.cam.2026.117406
Pengyuan Liu , Zhaodong Xu , Zhiqiang Sheng
In this paper, we propose a positivity-preserving subspace method, termed PSNNW, which is based on neural networks formulated in the weak form for solving diffusion equations. The method employs a monotonic positivity-preserving nonlinear functions to transform the original equations into mathematically equivalent forms. The numerical solution of the transformed equation is subsequently computed using a subspace neural network method in the weak form designed for nonlinear problems. In this method, neural networks are employed to train and generate basis functions, which are then incorporated into iterative schemes, such as Picard iteration, to solve the problem within the Galerkin framework. Owing to the positivity-preserving transformation, the numerical solution of the original equation is guaranteed to remain positive. Numerical experiments demonstrate that the proposed method yields nonnegative solutions with high accuracy, confirming its simplicity and effectiveness in preserving positivity.
{"title":"A positivity-preserving subspace method based on neural networks for solving diffusion equations in the weak form","authors":"Pengyuan Liu , Zhaodong Xu , Zhiqiang Sheng","doi":"10.1016/j.cam.2026.117406","DOIUrl":"10.1016/j.cam.2026.117406","url":null,"abstract":"<div><div>In this paper, we propose a positivity-preserving subspace method, termed PSNNW, which is based on neural networks formulated in the weak form for solving diffusion equations. The method employs a monotonic positivity-preserving nonlinear functions to transform the original equations into mathematically equivalent forms. The numerical solution of the transformed equation is subsequently computed using a subspace neural network method in the weak form designed for nonlinear problems. In this method, neural networks are employed to train and generate basis functions, which are then incorporated into iterative schemes, such as Picard iteration, to solve the problem within the Galerkin framework. Owing to the positivity-preserving transformation, the numerical solution of the original equation is guaranteed to remain positive. Numerical experiments demonstrate that the proposed method yields nonnegative solutions with high accuracy, confirming its simplicity and effectiveness in preserving positivity.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117406"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147387180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.
{"title":"Bound-preserving adaptive time-stepping methods with energy stability for simulating compressible gas flow in poroelastic media","authors":"Huangxin Chen , Yuxiang Chen , Jisheng Kou , Shuyu Sun","doi":"10.1016/j.cam.2026.117552","DOIUrl":"10.1016/j.cam.2026.117552","url":null,"abstract":"<div><div>In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117552"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386633","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}
The C1-continuity of calculated state curves is crucial for engineering problems subject to acceleration-dependent forces, such as hydrodynamic loads and control forces. Conventional variational integration schemes with prominent energy and momentum preserving properties are favored to calculate the dynamics of mechanical systems, however, can sometimes lose their efficacy due to a lack of continuity. In this work, a novel C1-continuous variational integration scheme is developed within a simple and general construction framework, ensuring the continuity of the generalized coordinates and their first derivative at the discrete-time points. This scheme is constructed by approximating generalized coordinates and velocities using Hermite polynomials within a certain time span with the action integral computed numerically. This framework greatly simplifies the derivation and implementation by avoiding the summation of discrete node variations, and it is also suitable for constructing other variational schemes based on Lagrangian polynomials of various orders. The algorithmic characteristics, including stability, dissipation, period elongation, and convergence order, are theoretically analyzed. The momentum-preserving and nearly energy-preserving properties are numerically demonstrated. Moreover, practical engineering problems subject to acceleration-dependent forces are investigated, which have well confirmed the feasibility of the proposed C1-continuous variational scheme in practical dynamic analyses.
{"title":"A new C1-continuous variational integration scheme for mechanical systems subjected to acceleration-dependent forces","authors":"Ping Zhou , Songhan Zhang , Hui Ren , Zheng Chen , Wei Fan","doi":"10.1016/j.cam.2026.117509","DOIUrl":"10.1016/j.cam.2026.117509","url":null,"abstract":"<div><div>The C<sup>1</sup>-continuity of calculated state curves is crucial for engineering problems subject to acceleration-dependent forces, such as hydrodynamic loads and control forces. Conventional variational integration schemes with prominent energy and momentum preserving properties are favored to calculate the dynamics of mechanical systems, however, can sometimes lose their efficacy due to a lack of continuity. In this work, a novel C<sup>1</sup>-continuous variational integration scheme is developed within a simple and general construction framework, ensuring the continuity of the generalized coordinates and their first derivative at the discrete-time points. This scheme is constructed by approximating generalized coordinates and velocities using Hermite polynomials within a certain time span with the action integral computed numerically. This framework greatly simplifies the derivation and implementation by avoiding the summation of discrete node variations, and it is also suitable for constructing other variational schemes based on Lagrangian polynomials of various orders. The algorithmic characteristics, including stability, dissipation, period elongation, and convergence order, are theoretically analyzed. The momentum-preserving and nearly energy-preserving properties are numerically demonstrated. Moreover, practical engineering problems subject to acceleration-dependent forces are investigated, which have well confirmed the feasibility of the proposed C<sup>1</sup>-continuous variational scheme in practical dynamic analyses.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"484 ","pages":"Article 117509"},"PeriodicalIF":2.6,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147386987","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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