Computers & Mathematics with Applications最新文献

{"title":"High order difference schemes for nonlinear Riesz space variable-order fractional diffusion equations","authors":"Qiu-Ya Wang","doi":"10.1016/j.camwa.2025.04.010","DOIUrl":"10.1016/j.camwa.2025.04.010","url":null,"abstract":"<div><div>This article aims at studying new finite difference methods for one-dimensional and two-dimensional nonlinear Riesz space variable-order (VO) fractional diffusion equations. In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Grünwald-difference (4WSGD) operators for Riemann-Liouville constant-order fractional derivatives, which have a free parameter and have at least third order accuracy, we derive variable-order 4WSGD operators for space Riesz variable-order fractional derivatives. In order that the fully discrete schemes exhibit robust stability and can handle the nonlinear term efficiently, we employ the implicit Euler (IE) method to discretize the time derivative, which leads to IE-4WSGD schemes. The stability and convergence properties of the IE-4WSGD schemes are analyzed theoretically. Additionally, a parameter selection strategy is derived for 4WSGD schemes and banded preconditioners are put forward to accelerate the GMRES methods for solving the discretization linear systems. Numerical results demonstrate the effectiveness of the proposed IE-4WSGD schemes and preconditioners.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"188 ","pages":"Pages 221-243"},"PeriodicalIF":2.9,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143854843","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":"Analysis of transient free surface seepage flow using numerical manifold method","authors":"Zhen Jia,&nbsp;Hong Zheng","doi":"10.1016/j.camwa.2025.04.011","DOIUrl":"10.1016/j.camwa.2025.04.011","url":null,"abstract":"<div><div>In the analysis of transient seepage flow with free surfaces, not only the free surfaces but also the boundary conditions vary with time, introducing significant challenges to those traditional mesh-based numerical methods. Although the numerical manifold method (NMM) has shown great advantages in tracking time-independent free surface seepage flow due to its dual cover systems – the mathematical cover and the physical cover, in the analysis of transient free surface seepage flow it will encounter the inheritance issue of degrees of freedom between two consecutive time steps, which is still an open issue for all the partition of unity based methods such as the extended finite element method (XFEM) and the generalized finite element method (GFEM). It is shown in this study that the issue can be easily overcome if a different discretization order from the classical discretization order is adopted, <em>i.e.</em>, time discretization is carried out before to spatial discretization. By analyzing typical transient seepage examples, the positions of the transient free surfaces predicted by the proposed method are excellently consistent with analytical solutions or experimental results. At the same time, it also points out the errors and possible consequences of some literature concerning the handling of sudden drops in upstream water level. The results demonstrate that the proposed procedure not only effectively predicts the evolution of free surfaces but also accurately addresses transient seepage problems, including those with complex drainage systems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"189 ","pages":"Pages 129-143"},"PeriodicalIF":2.9,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839089","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 second-order, unconditionally invariant-set-preserving scheme for the FitzHugh-Nagumo equation","authors":"Yiyi Liu ,&nbsp;Xueqing Teng ,&nbsp;Xiaoqiang Yan ,&nbsp;Hong Zhang","doi":"10.1016/j.camwa.2025.04.013","DOIUrl":"10.1016/j.camwa.2025.04.013","url":null,"abstract":"<div><div>In this paper, we present and analyze a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme for the FitzHugh-Nagumo equation. Utilizing a second-order finite-difference space discretization, we derive the fully discrete numerical scheme by incorporating both the stabilization technique and the ETDRK2 scheme for temporal approximation. The smallest invariant set of the FitzHugh-Nagumo equation is presented. We demonstrate that the proposed scheme unconditionally preserves the invariant set without any time-step constraint. The convergence in both time and space is verified to achieve second-order accuracy. Numerical experiments are carried out to illustrate the efficiency, stability, and structure-preserving property of the proposed scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"189 ","pages":"Pages 161-175"},"PeriodicalIF":2.9,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844179","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":"Higher order multipoint flux mixed finite element methods for parabolic equation","authors":"Guoliang Liu,&nbsp;Wenwen Xu,&nbsp;Xindong Li","doi":"10.1016/j.camwa.2025.04.012","DOIUrl":"10.1016/j.camwa.2025.04.012","url":null,"abstract":"<div><div>In this paper, we consider higher order multipoint flux mixed finite element methods for parabolic problems. The methods are based on enhanced Raviart-Thomas spaces with bubbles. The tensor-product Gauss-Lobatto quadrature rule is employed, which enables local velocity elimination and results in a symmetric, positive definite cell-based system for pressures. We construct two fully discrete schemes for the problems, including the backward Euler scheme and Crank-Nicolson scheme. Theoretical analysis shows optimal order convergence for pressure and velocity on <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-perturbed meshes. Numerical experiments are presented to verify the theoretical results and demonstrate the superiority of the proposed method compared to classical mixed finite element methods.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"189 ","pages":"Pages 144-160"},"PeriodicalIF":2.9,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839090","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":"Uniform convergence of finite element method on Vulanović-Bakhvalov mesh for singularly perturbed convection–diffusion equation in 2D","authors":"Xianyang Zhao,&nbsp;Jin Zhang","doi":"10.1016/j.camwa.2025.04.007","DOIUrl":"10.1016/j.camwa.2025.04.007","url":null,"abstract":"<div><div>This paper investigates the uniform convergence of arbitrary order finite element methods on Vulanović-Bakhvalov mesh. We carefully design a new interpolation based on exponential layer structure, which not only overcomes the difficulties caused by the mesh step width, but also ensures the Dirichlet boundary condition. We successfully demonstrate the uniform convergence of the optimal order in the energy norm. The results of numerical experiments strongly validate our analysis.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"188 ","pages":"Pages 183-194"},"PeriodicalIF":2.9,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143828578","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":"Chew, Goldberger & Low equations: Eigensystem analysis and applications to one-dimensional test problems","authors":"Chetan Singh ,&nbsp;Deepak Bhoriya ,&nbsp;Anshu Yadav ,&nbsp;Harish Kumar ,&nbsp;Dinshaw S. Balsara","doi":"10.1016/j.camwa.2025.04.008","DOIUrl":"10.1016/j.camwa.2025.04.008","url":null,"abstract":"<div><div>Chew, Goldberger &amp; Low (CGL) equations describe one of the simplest plasma flow models that allow anisotropic pressure, i.e., pressure is modeled using a symmetric tensor described by two scalar pressure components, one parallel to the magnetic field, another perpendicular to the magnetic field. The system of equations is a non-conservative hyperbolic system. In this work, we analyze the eigensystem of the CGL equations. We present the eigenvalues and the complete set of right eigenvectors. We also prove the linear degeneracy of some of the characteristic fields. Using the eigensystem for CGL equations, we propose HLL and HLLI Riemann solvers for the CGL system. Furthermore, we present the AFD-WENO schemes up to the seventh order in one dimension and demonstrate the performance of the schemes on several one-dimensional test cases.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"188 ","pages":"Pages 195-220"},"PeriodicalIF":2.9,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143828415","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 novel numerical scheme for Black-Scholes PDEs modeling pricing securities","authors":"Sachin Kumar,&nbsp;Srinivasan Natesan","doi":"10.1016/j.camwa.2025.04.003","DOIUrl":"10.1016/j.camwa.2025.04.003","url":null,"abstract":"<div><div>This article introduces an efficient numerical method for solving the Black-Scholes partial differential equation (PDE) that governs European options. The methodology employs the backward Euler scheme to discretize the time derivative and incorporates the non-symmetric interior penalty Galerkin method for handling the spatial derivatives. The study aims to determine optimal order error estimates in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and discrete energy norm. In addition, the proposed method is used to determine Greeks in option pricing. We validate the theoretical results presented in this work with numerical experiments.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 57-71"},"PeriodicalIF":2.9,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829754","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":"Performance comparison of variable-stepsize IMEX SBDF methods on advection-diffusion-reaction models","authors":"Raed Ali Mara'Beh ,&nbsp;J.M. Mantas ,&nbsp;P. González ,&nbsp;Raymond J. Spiteri","doi":"10.1016/j.camwa.2025.04.002","DOIUrl":"10.1016/j.camwa.2025.04.002","url":null,"abstract":"<div><div>Advection-diffusion-reaction (ADR) models describe transport mechanisms in fluid or solid media. They are often formulated as partial differential equations that are spatially discretized into systems of ordinary differential equations (ODEs) in time for numerical resolution. This paper investigates the performance of variable stepsize, semi-implicit, backward differentiation formula (VSSBDF) methods of up to fourth order for solving ADR models employing two different implicit-explicit splitting approaches: a <em>physics-based</em> splitting and a splitting based on a dynamic linearization of the resulting system of ODEs, called <em>jacobian splitting</em> in this paper. We develop an adaptive time-stepping and error control algorithm for VSSBDF methods up to fourth order based on a step-doubling refinement technique using estimates of the local truncation errors. Through a systematic comparison between physics-based and Jacobian splitting across six ADR test models, we evaluate the performance based on CPU times and corresponding accuracy. Our findings demonstrate the general superiority of Jacobian splitting in several experiments.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 41-56"},"PeriodicalIF":2.9,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829753","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":"Two efficient compact ADI methods for the two-dimensional fractional Oldroyd-B model","authors":"Xinyu Diao,&nbsp;Bo Yu","doi":"10.1016/j.camwa.2025.04.009","DOIUrl":"10.1016/j.camwa.2025.04.009","url":null,"abstract":"<div><div>The objective of this paper is to present efficient numerical algorithms to resolve the two-dimensional fractional Oldroyd-B model. Firstly, two compact alternating direction implicit (ADI) methods are constructed with convergence orders <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mi>min</mi><mo>⁡</mo><mo>{</mo><mn>3</mn><mo>−</mo><mi>γ</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>β</mi><mo>,</mo><mn>1</mn><mo>+</mo><mi>γ</mi><mo>−</mo><mn>2</mn><mi>β</mi><mo>}</mo></mrow></msup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>)</mo></mrow></math></span> and <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mi>min</mi><mo>⁡</mo><mo>{</mo><mn>3</mn><mo>−</mo><mi>γ</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>β</mi><mo>}</mo></mrow></msup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>)</mo></mrow></math></span>, where <em>γ</em> and <em>β</em> are orders of two Caputo fractional derivatives, <em>τ</em>, <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span> are the time and space step sizes, respectively. Secondly, the convergence analyses of the proposed compact ADI methods are investigated strictly utilizing the energy estimation technique. Lastly, the two compact ADI methods are implemented to confirm the effectiveness of the convergence analysis. The convergence orders of the two compact ADI methods are separately tested in the direction of time and space, the CPU times are computed compared with the direct compact scheme to demonstrate the efficiency of the derived compact ADI methods, numerical results are also compared with the existing literature. All the numerical simulation results are listed in tabular forms which manifest the validity of the derived compact ADI methods.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 72-89"},"PeriodicalIF":2.9,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829755","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":"Nonlinear methods for shape optimization problems in liquid crystal tactoids","authors":"J.H. Adler ,&nbsp;A.S. Andrei ,&nbsp;T.J. Atherton","doi":"10.1016/j.camwa.2025.04.004","DOIUrl":"10.1016/j.camwa.2025.04.004","url":null,"abstract":"<div><div>Anisotropic fluids, such as nematic liquid crystals, can form non-spherical equilibrium shapes known as tactoids. Predicting the shape of these structures as a function of material parameters is challenging and paradigmatic of a broader class of problems that combine shape and order. Here, we consider a discrete shape optimization approach with finite elements to find the configuration of two-dimensional and three-dimensional tactoids using the Landau–de Genne framework and a Q-tensor representation. Efficient solution of the resulting constrained energy minimization problem is achieved using a quasi-Newton and nested iteration algorithm. Numerical validation is performed with benchmark solutions and compared against experimental data and earlier work. We explore physically motivated subproblems, whereby the shape and order are separately held fixed, respectively, to explore the role of both and examine material parameter dependence of the convergence. Nested iteration significantly improves both the computational cost and convergence of numerical solutions of these highly deformable materials.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"187 ","pages":"Pages 231-248"},"PeriodicalIF":2.9,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820228","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}