{"title":"Partitioned time stepping method for time-fractional Stokes-Darcy model with the Beavers-Joseph-Saffman interface conditions","authors":"Yuting Xiang , Haibiao Zheng","doi":"10.1016/j.camwa.2024.11.033","DOIUrl":"10.1016/j.camwa.2024.11.033","url":null,"abstract":"<div><div>This paper proposes a numerical method for solving the time-fractional Stokes-Darcy problem using a partitioned time stepping algorithm with the Beavers-Joseph-Saffman condition. The stability of the method is established under a moderate time step restriction, <span><math><mi>τ</mi><mo>≤</mo><mi>C</mi></math></span> where <em>C</em> represents physical parameters, by utilizing a discrete fractional Gronwall type inequality. Additionally, error estimates are derived to provide insight into the accuracy of the proposed method. Numerical experiments that support the theoretical results are presented.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 154-178"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142804453","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}
{"title":"A family of second-order time stepping methods for a nonlinear fluid-fluid interaction model","authors":"Yiru Chen , Yun-Bo Yang , Lijie Mei","doi":"10.1016/j.camwa.2025.01.010","DOIUrl":"10.1016/j.camwa.2025.01.010","url":null,"abstract":"<div><div>In this paper, we present a fully discrete finite element scheme for the nonlinear fluid-fluid interaction model, which consists of two Navier-Stokes equations coupled by some nonlinear interface. The presented fully discrete scheme is based on a type of implicit-explicit (IMEX) second-order time-stepping schemes in temporal discretization and mixed finite element in spatial discretization. The scheme is a combination of a linearization treatment for the advection term, explicit treatment for nonlinear interface conditions by a scalar auxiliary variable method, together with stabilization terms which are proportional to discrete curvature of the solutions in both velocity and pressure. Because of the scalar auxiliary variable method, we only require solving a sequence of linear differential equation with constant coefficients at each time step. Unconditional stability is proved and convergence analysis is derived. Finally, the derived theoretical results are supported by three numerical examples.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"182 ","pages":"Pages 1-23"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986207","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}
{"title":"An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms","authors":"Carmelo Clavero , Ram Shiromani","doi":"10.1016/j.camwa.2025.01.011","DOIUrl":"10.1016/j.camwa.2025.01.011","url":null,"abstract":"<div><div>In this work we are interested in constructing a uniformly convergent method to solve a 2D elliptic singularly perturbed weakly system of convection-diffusion type. We assume that small positive parameters appear at both the diffusion and the convection terms of the partial differential equation. Moreover, we suppose that both the diffusion and the convection parameters can be distinct and also they can have a different order of magnitude. Then, the nature of the overlapping regular or parabolic boundary layers, which, in general, appear in the exact solution, is much more complicated. To solve the continuous problem, we use the classical upwind finite difference scheme, which is defined on piecewise uniform Shishkin meshes, which are given in a different way depending on the value and the ratio between the four singular perturbation parameters which appear in the continuous problem. So, the numerical algorithm is an almost first order uniformly convergent method. The numerical results obtained with our algorithm for a test problem are presented; these results corroborate in practice the good behavior and the uniform convergence of the algorithm, aligning with the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 287-322"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020193","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}
{"title":"A time-fractional optimal transport: Formulation and algorithm","authors":"Yiqun Li, Hong Wang, Wuchen Li","doi":"10.1016/j.camwa.2025.01.009","DOIUrl":"10.1016/j.camwa.2025.01.009","url":null,"abstract":"<div><div>The time-fractional optimal transport (OT) models are developed to describe the anomalous transport of the agents such that their densities are transported from the initial density distribution to the terminal one with the minimal cost. The general-proximal primal-dual hybrid gradient (G-prox PDHG) algorithm is applied to solve the OT formulations, in which a preconditioner induced by the numerical approximation to the time-fractional PDE is derived to accelerate the convergence of the algorithm. Numerical experiments for OT problems between Gaussian distributions are carried out to investigate the performance of the OT formulations. Those numerical experiments also demonstrate the effectiveness and flexibility of our proposed algorithm.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 261-278"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986208","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}
{"title":"Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian","authors":"Zhaopeng Hao , Zhiqiang Cai , Zhongqiang Zhang","doi":"10.1016/j.camwa.2024.11.027","DOIUrl":"10.1016/j.camwa.2024.11.027","url":null,"abstract":"<div><div>We study the numerical evaluation of the integral fractional Laplacian and its application in solving fractional diffusion equations. We derive a pseudo-spectral formula for the integral fractional Laplacian operator based on fractional order-dependent, generalized multi-quadratic radial basis functions (RBFs) to address efficient computation of the hyper-singular integral. We apply the proposed formula to solving fractional diffusion equations and design a simple, easy-to-implement and nearly integration-free meshless method. We discuss the convergence of the novel meshless method through equivalent Galerkin formulations. We carry out numerical experiments to demonstrate the accuracy and efficiency of the proposed approach compared to the existing method using Gaussian RBFs.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 197-213"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142804449","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}
{"title":"An efficient high-order weak Galerkin finite element approach for Sobolev equation with variable matrix coefficients","authors":"Eric Ngondiep","doi":"10.1016/j.camwa.2025.01.013","DOIUrl":"10.1016/j.camwa.2025.01.013","url":null,"abstract":"<div><div>This paper constructs a high-order weak Galerkin finite element method for solving a two-dimensional Sobolev equation with variable matrix coefficients subjects to initial and boundary conditions. The proposed approach approximates the exact solution in time using interpolation techniques whereas the space derivatives are approximated by weak forms through integration by parts. The new algorithm is unconditionally stable, temporal second order convergence and spatial accurate with convergence order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span>, in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>;</mo><mspace></mspace><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span>-norm, where <em>p</em> is a nonnegative integer and <em>h</em> represents the grid space. The developed computational scheme is faster and more efficient than a broad range of numerical methods deeply studied in the literature for solving Sobolev problems. Some numerical examples are carried out to confirm the theory and to investigate the performance and validity of the constructed high-order numerical scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 279-298"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020194","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}
Takuma Kimura , Teruya Minamoto , Mitsuhiro T. Nakao
{"title":"Constructive error estimates for a full-discretized periodic solution of heat equation by spatial finite-element and time spectral method","authors":"Takuma Kimura , Teruya Minamoto , Mitsuhiro T. Nakao","doi":"10.1016/j.camwa.2025.01.008","DOIUrl":"10.1016/j.camwa.2025.01.008","url":null,"abstract":"<div><div>We consider the constructive a priori error estimates for a full discrete approximation of a periodic solution for the heat equation. Our numerical scheme is based on the finite element semidiscretization in space direction combined with the Fourier expansion in time. We derive the optimal order explicit <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimates which play an important role in the numerical verification method of exact solutions for nonlinear parabolic equations. Several numerical examples which confirm the theoretical results will be presented.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 248-258"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986209","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}
Zhaohua Li , Guang-an Zou , Lina Ma , Xiaofeng Yang
{"title":"An efficient discontinuous Galerkin method for the hydro-dynamically coupled phase-field vesicle membrane model","authors":"Zhaohua Li , Guang-an Zou , Lina Ma , Xiaofeng Yang","doi":"10.1016/j.camwa.2025.01.002","DOIUrl":"10.1016/j.camwa.2025.01.002","url":null,"abstract":"<div><div>This paper presents a linear, fully-decoupled discontinuous Galerkin (DG) method for the flow-coupled phase-field vesicle membrane model. The fully discrete scheme is implemented by combining several reliable numerical techniques. Firstly, we employ the DG method for spatial discretization, which does not require that the numerical solutions are continuous across different grid cells, enabling adaptive treatment of complex vesicle membrane shapes and fluid flows. Secondly, to cope with nonlinear and coupling terms, scalar auxiliary variables (SAV) and implicit-explicit approaches are used, which improve numerical stability and computational efficiency. Finally, for the momentum equation, the pressure-correction method is used to assure the decoupling of velocity and pressure. Through rigorous mathematical proofs, this paper demonstrates the unconditional energy stability of the proposed scheme and derives its optimal error estimation. Numerical examples also show the method's effectiveness in terms of precision, energy stability, and simulated vesicle deformation in thin channels. The results highlight the method's potential application in fluid-coupled phase-field vesicle membrane models, while also providing new methodologies and technological support for numerical simulation and computational research in related domains.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 259-286"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986210","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}
{"title":"Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints","authors":"Peter Gangl , Richard Löscher , Olaf Steinbach","doi":"10.1016/j.camwa.2024.12.021","DOIUrl":"10.1016/j.camwa.2024.12.021","url":null,"abstract":"<div><div>In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal balance between the error and the cost, and, on the discrete level, an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from which we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 242-260"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986212","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}
{"title":"Fourier spectral methods based on restricted Padé approximations for space fractional reaction-diffusion systems","authors":"M. Yousuf , M. Alshayqi , S.S. Alzahrani","doi":"10.1016/j.camwa.2024.12.025","DOIUrl":"10.1016/j.camwa.2024.12.025","url":null,"abstract":"<div><div>By utilizing the power of the Fourier spectral approach and the restricted Padé rational approximations, we have devised two third-order numerical methods to investigate the complex phenomena that arise in multi-dimensional space fractional reaction-diffusion models. The Fourier spectral approach yields a fully diagonal representation of the fractional Laplacian with the ability to extend the methods to multi-dimensional cases with the same computational complexity as one-dimensional and makes it possible to attain spectral convergence. Third-order single-pole restricted Padé approximations of the matrix exponential are utilized in developing the time stepping methods. We also use sophisticated mathematical techniques, namely, discrete sine and cosine transforms, to improve the computational efficiency of the methods. Algorithms are derived from these methods for straight-forward implementation in one- and multidimensional models, accommodating both homogeneous Dirichlet and homogeneous Neumann boundary conditions. The third-order accuracy of these methods is proved analytically and demonstrated numerically. Linear error analysis of these methods is presented, stability regions of both methods are computed, and their graphs are plotted. The computational efficiency, reliability, and effectiveness of the presented methods are demonstrated through numerical experiments. The convergence results are computed to support the theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 228-247"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986211","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}