Pub Date : 2025-04-15DOI: 10.1016/j.camwa.2025.04.007
Xianyang Zhao, Jin Zhang
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.
{"title":"Uniform convergence of finite element method on Vulanović-Bakhvalov mesh for singularly perturbed convection–diffusion equation in 2D","authors":"Xianyang Zhao, 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}
Chew, Goldberger & 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.
{"title":"Chew, Goldberger & Low equations: Eigensystem analysis and applications to one-dimensional test problems","authors":"Chetan Singh , Deepak Bhoriya , Anshu Yadav , Harish Kumar , 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 & 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}
Pub Date : 2025-04-14DOI: 10.1016/j.camwa.2025.04.003
Sachin Kumar, Srinivasan Natesan
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 -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.
{"title":"A novel numerical scheme for Black-Scholes PDEs modeling pricing securities","authors":"Sachin Kumar, 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}
Pub Date : 2025-04-14DOI: 10.1016/j.camwa.2025.04.002
Raed Ali Mara'Beh , J.M. Mantas , P. González , Raymond J. Spiteri
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 physics-based splitting and a splitting based on a dynamic linearization of the resulting system of ODEs, called jacobian splitting 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.
{"title":"Performance comparison of variable-stepsize IMEX SBDF methods on advection-diffusion-reaction models","authors":"Raed Ali Mara'Beh , J.M. Mantas , P. González , 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}
Pub Date : 2025-04-14DOI: 10.1016/j.camwa.2025.04.009
Xinyu Diao, Bo Yu
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 and , where γ and β are orders of two Caputo fractional derivatives, τ, and 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.
本文旨在提出解决二维分数奥尔德罗伊德-B 模型的高效数值算法。首先，构建了两种收敛阶数分别为 O(τmin{3-γ,2-β,1+γ-2β}+hx4+hy4)和 O(τmin{3-γ,2-β}+hx4+hy4)的紧凑交替方向隐式（ADI）方法，其中γ和β是两个卡普托分数导数的阶数，τ、hx 和 hy 分别是时间步长和空间步长。其次，严格利用能量估计技术研究了所提出的紧凑 ADI 方法的收敛性分析。最后，实现了两种紧凑型 ADI 方法，以确认收敛分析的有效性。两种紧凑型 ADI 方法的收敛阶数分别在时间和空间方向上进行了测试，CPU 计算时间与直接紧凑型方案进行了比较，以证明衍生紧凑型 ADI 方法的效率，数值结果也与现有文献进行了比较。所有数值模拟结果都以表格形式列出，体现了衍生紧凑型 ADI 方法的有效性。
{"title":"Two efficient compact ADI methods for the two-dimensional fractional Oldroyd-B model","authors":"Xinyu Diao, 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}
Pub Date : 2025-04-11DOI: 10.1016/j.camwa.2025.04.004
J.H. Adler , A.S. Andrei , T.J. Atherton
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.
{"title":"Nonlinear methods for shape optimization problems in liquid crystal tactoids","authors":"J.H. Adler , A.S. Andrei , 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}
Pub Date : 2025-04-08DOI: 10.1016/j.camwa.2025.03.038
Min-Li Zeng , Martin Stoll
In this paper, we explore efficient methods for discretized linear systems that arise from eddy current optimal control problems utilizing an all-at-once approach. We propose a novel low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. First, we reformulate the resulting discretized linear system into a matrix equation format. Then, by employing the KPIK algorithm, we derive a low-rank approximation solution. This new approach is referred to as the splitting-based Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method exhibits the potential for efficiently tackle large and sparse discretized systems, while also significantly reducing both storage requirements and computational time. Next, theoretical results regarding the existence of low-rank solutions are provided. Furthermore, numerical experiments are conducted to demonstrate the effectiveness of the proposed low-rank matrix equation method in comparison to several established classical efficient techniques.
{"title":"A splitting-based KPIK method for eddy current optimal control problems in an all-at-once approach","authors":"Min-Li Zeng , Martin Stoll","doi":"10.1016/j.camwa.2025.03.038","DOIUrl":"10.1016/j.camwa.2025.03.038","url":null,"abstract":"<div><div>In this paper, we explore efficient methods for discretized linear systems that arise from eddy current optimal control problems utilizing an all-at-once approach. We propose a novel low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. First, we reformulate the resulting discretized linear system into a matrix equation format. Then, by employing the KPIK algorithm, we derive a low-rank approximation solution. This new approach is referred to as the splitting-based Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method exhibits the potential for efficiently tackle large and sparse discretized systems, while also significantly reducing both storage requirements and computational time. Next, theoretical results regarding the existence of low-rank solutions are provided. Furthermore, numerical experiments are conducted to demonstrate the effectiveness of the proposed low-rank matrix equation method in comparison to several established classical efficient techniques.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 1-15"},"PeriodicalIF":2.9,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791834","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 : 2025-04-08DOI: 10.1016/j.camwa.2025.04.001
Chang-Ock Lee , Youngkyu Lee , Byungeun Ryoo
Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known to be much faster than Physics informed neural networks. However, classical ELM is still computationally expensive when a high level of representation is desired in the solution as this requires solving a large least squares system. In this paper, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. We introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. A method for initializing the hidden layer parameters suitable for high approximation quality in large systems is also proposed. Numerical results that verify the acceleration performance of the proposed method with respect to the number of subdomains are presented.
{"title":"A nonoverlapping domain decomposition method for extreme learning machines: Elliptic problems","authors":"Chang-Ock Lee , Youngkyu Lee , Byungeun Ryoo","doi":"10.1016/j.camwa.2025.04.001","DOIUrl":"10.1016/j.camwa.2025.04.001","url":null,"abstract":"<div><div>Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known to be much faster than Physics informed neural networks. However, classical ELM is still computationally expensive when a high level of representation is desired in the solution as this requires solving a large least squares system. In this paper, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. We introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. A method for initializing the hidden layer parameters suitable for high approximation quality in large systems is also proposed. Numerical results that verify the acceleration performance of the proposed method with respect to the number of subdomains are presented.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"189 ","pages":"Pages 109-128"},"PeriodicalIF":2.9,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791396","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 : 2025-04-08DOI: 10.1016/j.camwa.2025.03.031
Masoud Pendar, Kamal Shanazari
In this work, we apply the local Petrov-Galerkin method based on radial basis functions to solving the two dimensional linear hyperbolic equations with variable coefficients subject to given appropriate initial and boundary conditions. Due to the presence of variable coefficients of the differential operator, special treatment is carried out in order to apply Green's theorem and derive the variational formulation. We use the radial point interpolation method to construct shape functions and a Crank-Nicolson finite difference scheme is employed to approximate the time derivatives. The stability, convergence and error analysis of the method are also discussed and theoretically proven. Some numerical examples are presented to examine the efficiency and accuracy of the proposed method.
{"title":"Local Petrov-Galerkin meshfree method based on radial point interpolation for the numerical solution of 2D linear hyperbolic equations with variable coefficients","authors":"Masoud Pendar, Kamal Shanazari","doi":"10.1016/j.camwa.2025.03.031","DOIUrl":"10.1016/j.camwa.2025.03.031","url":null,"abstract":"<div><div>In this work, we apply the local Petrov-Galerkin method based on radial basis functions to solving the two dimensional linear hyperbolic equations with variable coefficients subject to given appropriate initial and boundary conditions. Due to the presence of variable coefficients of the differential operator, special treatment is carried out in order to apply Green's theorem and derive the variational formulation. We use the radial point interpolation method to construct shape functions and a Crank-Nicolson finite difference scheme is employed to approximate the time derivatives. The stability, convergence and error analysis of the method are also discussed and theoretically proven. Some numerical examples are presented to examine the efficiency and accuracy of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"190 ","pages":"Pages 16-40"},"PeriodicalIF":2.9,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791835","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 : 2025-04-05DOI: 10.1016/j.camwa.2025.03.032
Xiaochen Chu , Xiangyu Shi , Dongyang Shi
The purpose of this article is to explore the superconvergence behavior of the first-order backward-Euler (BE) implicit/explicit fully discrete schemes for the unsteady incompressible MHD equations with low-order mixed finite element method (MFEM) by utilizing the scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) methods. Through dealing with linear terms in implicit format and nonlinear terms in explicit format, the original problem is decomposed into several subproblems, which effectively reduces the amount of calculation. Particularly, a new high-precision estimation is given, which acts as a requisite role in getting the expected results. Following this, combined with a simple, effective and economic interpolation post-processing approach, the superclose and superconvergence error estimates of the decoupled and linearized fully discrete finite element SAV-BE scheme are rigorously derived. And the derivation process is also applicable to the ZEC-BE scheme. Finally, the corresponding numerical simulations are carried out to confirm the accuracy and reliability of our theoretical findings.
{"title":"Superconvergence analysis of the decoupled and linearized mixed finite element methods for unsteady incompressible MHD equations","authors":"Xiaochen Chu , Xiangyu Shi , Dongyang Shi","doi":"10.1016/j.camwa.2025.03.032","DOIUrl":"10.1016/j.camwa.2025.03.032","url":null,"abstract":"<div><div>The purpose of this article is to explore the superconvergence behavior of the first-order backward-Euler (BE) implicit/explicit fully discrete schemes for the unsteady incompressible MHD equations with low-order mixed finite element method (MFEM) by utilizing the scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) methods. Through dealing with linear terms in implicit format and nonlinear terms in explicit format, the original problem is decomposed into several subproblems, which effectively reduces the amount of calculation. Particularly, a new high-precision estimation is given, which acts as a requisite role in getting the expected results. Following this, combined with a simple, effective and economic interpolation post-processing approach, the superclose and superconvergence error estimates of the decoupled and linearized fully discrete finite element SAV-BE scheme are rigorously derived. And the derivation process is also applicable to the ZEC-BE scheme. Finally, the corresponding numerical simulations are carried out to confirm the accuracy and reliability of our theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"188 ","pages":"Pages 160-182"},"PeriodicalIF":2.9,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777067","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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