Pub Date : 2024-10-22DOI: 10.1016/j.camwa.2024.10.018
Jinjin Zhang , Yanru Su , Xinfeng Gao , Xuemin Tu
Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete norm, an norm, and an semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.
{"title":"Overlapping domain decomposition methods for finite volume discretizations","authors":"Jinjin Zhang , Yanru Su , Xinfeng Gao , Xuemin Tu","doi":"10.1016/j.camwa.2024.10.018","DOIUrl":"10.1016/j.camwa.2024.10.018","url":null,"abstract":"<div><div>Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm, an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> norm, and an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529038","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 : 2024-10-21DOI: 10.1016/j.camwa.2024.10.023
Zhijun Tan
This paper constructs and analyzes an α-robust and new two-grid finite element method (FEM) with nonuniform L2- formula and its fast algorithms for nonlinear time-fractional diffusion equations. The method incorporates a nonuniform L2- formula to achieve temporal second-order accuracy and address the initial solution singularity. By employing a spatial two-grid FEM, computational costs are reduced. Utilizing the cut-off technique and an auxiliary function, the condition on the nonlinear term is lessened to meet the local Lipschitz requirement. We further devise the associated fast algorithms for two-grid nonuniform L2- FEM. To prevent roundoff errors, we introduce an innovative fast algorithm to precisely calculate the kernel coefficients. An α-robust analysis of the stability and optimal error estimates in terms of -norm and -norm for the fully discrete scheme is presented. The derived error bound remains stable as the order of the fractional derivative . Furthermore, a new two-grid algorithm and its corresponding fast algorithm are proposed to decrease the computational expenses by eliminating redundancy in discrete convolutional summation. Numerical experiments support our theoretical results, confirming that two-grid FEMs offer greater efficiency in comparison to FEM.
{"title":"An α-robust and new two-grid nonuniform L2-1σ FEM for nonlinear time-fractional diffusion equation","authors":"Zhijun Tan","doi":"10.1016/j.camwa.2024.10.023","DOIUrl":"10.1016/j.camwa.2024.10.023","url":null,"abstract":"<div><div>This paper constructs and analyzes an <em>α</em>-robust and new two-grid finite element method (FEM) with nonuniform L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> formula and its fast algorithms for nonlinear time-fractional diffusion equations. The method incorporates a nonuniform L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> formula to achieve temporal second-order accuracy and address the initial solution singularity. By employing a spatial two-grid FEM, computational costs are reduced. Utilizing the cut-off technique and an auxiliary function, the condition on the nonlinear term is lessened to meet the local Lipschitz requirement. We further devise the associated fast algorithms for two-grid nonuniform L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> FEM. To prevent roundoff errors, we introduce an innovative fast algorithm to precisely calculate the kernel coefficients. An <em>α</em>-robust analysis of the stability and optimal error estimates in terms of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm for the fully discrete scheme is presented. The derived error bound remains stable as the order of the fractional derivative <span><math><mi>α</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>. Furthermore, a new two-grid algorithm and its corresponding fast algorithm are proposed to decrease the computational expenses by eliminating redundancy in discrete convolutional summation. Numerical experiments support our theoretical results, confirming that two-grid FEMs offer greater efficiency in comparison to FEM.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529039","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 : 2024-10-21DOI: 10.1016/j.camwa.2024.10.014
Cheng Deng , Hui Zheng , Rongping Zhang , Liangyong Gong , Xiangcou Zheng
This study introduces a local radial basis function collocation method (LRBFCM) to analyzing structural deformation in deep excavation within a two dimensional geotechnical model. To mitigate the size effect caused by a large length-to-width ratio, a technique known as the ‘direct method’ is employed. This method effectively reduces the influence of the shape parameter, thereby improving the accuracy of the partial derivative calculations in LRBFCM. The combination of LRBFCM with the direct method is applied to the deep excavation problem, which consists of both the soil and support structures. The soil is modeled using the Drucker-Prager (D-P) elastic-plastic model, while an elastic model is employed for the support structure. Elastic-plastic discretization is performed using incremental theory. The proposed approach is validated through four different examples, comparing the results with numerical solutions obtained from traditional finite element methods (FEM). This study advocates the use of the direct method to optimize the distribution of local influence nodes, particularly in cases involving large length-to-width ratios. The combination of LRBFCM with incremental theory is shown to be effective for addressing elastic-plastic problems.
{"title":"Structure deformation analysis of the deep excavation based on the local radial basis function collocation method","authors":"Cheng Deng , Hui Zheng , Rongping Zhang , Liangyong Gong , Xiangcou Zheng","doi":"10.1016/j.camwa.2024.10.014","DOIUrl":"10.1016/j.camwa.2024.10.014","url":null,"abstract":"<div><div>This study introduces a local radial basis function collocation method (LRBFCM) to analyzing structural deformation in deep excavation within a two dimensional geotechnical model. To mitigate the size effect caused by a large length-to-width ratio, a technique known as the ‘direct method’ is employed. This method effectively reduces the influence of the shape parameter, thereby improving the accuracy of the partial derivative calculations in LRBFCM. The combination of LRBFCM with the direct method is applied to the deep excavation problem, which consists of both the soil and support structures. The soil is modeled using the Drucker-Prager (D-P) elastic-plastic model, while an elastic model is employed for the support structure. Elastic-plastic discretization is performed using incremental theory. The proposed approach is validated through four different examples, comparing the results with numerical solutions obtained from traditional finite element methods (FEM). This study advocates the use of the direct method to optimize the distribution of local influence nodes, particularly in cases involving large length-to-width ratios. The combination of LRBFCM with incremental theory is shown to be effective for addressing elastic-plastic problems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529037","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 : 2024-10-21DOI: 10.1016/j.camwa.2024.10.021
Vishnu Prakash K, Ganesh Natarajan
A new and simple implicit Green-Gauss gradient (IGG) scheme for unstructured meshes is proposed exploiting the ideas from the linearity-preserving U-MUSCL scheme to define values at cell faces. We construct an implicit one-parameter family of gradient schemes referred to as IGG(χ) where χ is a free-parameter. The computed gradients are at least first-order accurate on generic polygonal meshes and second-order accurate on uniform Cartesian meshes except when for which fourth-order accuracy can be realised. A theoretical analysis is carried out to understand the effect of the control parameter χ on accuracy and resolution of the gradients and numerical experiments on various mesh topologies confirm the theoretical findings. Finite volume simulations of the Poisson and Euler equations on Cartesian and unstructured meshes further highlight that the IGG(χ) is a versatile gradient scheme that gives second-order accurate solutions with the iterative convergence of the solver dependent on the choice of the χ parameter. The framework described in this study can also be employed to devise an implicit least-squares gradient scheme that applies equally well to unstructured finite volume and meshfree solvers.
{"title":"IGG(χ): A new and simple implicit gradient scheme on unstructured meshes","authors":"Vishnu Prakash K, Ganesh Natarajan","doi":"10.1016/j.camwa.2024.10.021","DOIUrl":"10.1016/j.camwa.2024.10.021","url":null,"abstract":"<div><div>A new and simple implicit Green-Gauss gradient (IGG) scheme for unstructured meshes is proposed exploiting the ideas from the linearity-preserving U-MUSCL scheme to define values at cell faces. We construct an implicit one-parameter family of gradient schemes referred to as IGG(<em>χ</em>) where <em>χ</em> is a free-parameter. The computed gradients are at least first-order accurate on generic polygonal meshes and second-order accurate on uniform Cartesian meshes except when <span><math><mi>χ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span> for which fourth-order accuracy can be realised. A theoretical analysis is carried out to understand the effect of the control parameter <em>χ</em> on accuracy and resolution of the gradients and numerical experiments on various mesh topologies confirm the theoretical findings. Finite volume simulations of the Poisson and Euler equations on Cartesian and unstructured meshes further highlight that the IGG(<em>χ</em>) is a versatile gradient scheme that gives second-order accurate solutions with the iterative convergence of the solver dependent on the choice of the <em>χ</em> parameter. The framework described in this study can also be employed to devise an implicit least-squares gradient scheme that applies equally well to unstructured finite volume and meshfree solvers.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529733","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 : 2024-10-18DOI: 10.1016/j.camwa.2024.10.011
Hua-Yu Liu, Xiao-Wei Gao, Jun Lv
In this paper, a novel implementation of the virtual element method is proposed, which employs fundamental solutions (Green functions) instead of polynomials. Instead of constructing explicit shape functions for polygonal elements, abstract basis functions are employed, which are only computable on the boundaries of elements. With the help of the projection into the space of fundamental solutions, the incomputable domain integration is eliminated. In addition, the source points of the fundamental solutions are moved outside the elements to ensure the boundness of the basis functions. Compared with the conventional implementation which projects into the space of polynomials, the numerical results demonstrate that the proposed method exhibits significant superiority in singular problems or when the number of nodes in elements is large.
{"title":"The fundamental solution element method based on irregular polygonal meshes","authors":"Hua-Yu Liu, Xiao-Wei Gao, Jun Lv","doi":"10.1016/j.camwa.2024.10.011","DOIUrl":"10.1016/j.camwa.2024.10.011","url":null,"abstract":"<div><div>In this paper, a novel implementation of the virtual element method is proposed, which employs fundamental solutions (Green functions) instead of polynomials. Instead of constructing explicit shape functions for polygonal elements, abstract basis functions are employed, which are only computable on the boundaries of elements. With the help of the projection into the space of fundamental solutions, the incomputable domain integration is eliminated. In addition, the source points of the fundamental solutions are moved outside the elements to ensure the boundness of the basis functions. Compared with the conventional implementation which projects into the space of polynomials, the numerical results demonstrate that the proposed method exhibits significant superiority in singular problems or when the number of nodes in elements is large.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529036","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 : 2024-10-17DOI: 10.1016/j.camwa.2024.10.019
Won-Kwang Park
In this paper, we consider a limited-aperture inverse scattering problem for a fast identification of small dielectric objects from two-dimensional Fresnel experimental dataset. To this end, we apply the Kirchhoff migration (KM) imaging technique and design an imaging function from the generated multi-static response matrix. Using the integral equation-based representation formula for the scattered field, we theoretically investigate the applicability of the KM by formulating the imaging function as a uniformly convergent infinite series of integer-order Bessel functions of the first kind. Numerical simulation results using the experimental Fresnel dataset are presented to support the theoretical result.
在本文中，我们考虑了从二维菲涅尔实验数据集中快速识别小型介质物体的有限孔径反向散射问题。为此，我们应用了基尔霍夫迁移（KM）成像技术，并根据生成的多静态响应矩阵设计了一个成像函数。利用基于积分方程的散射场表示公式，我们将成像函数表述为一阶整数贝塞尔函数的均匀收敛无穷级数，从理论上研究了 KM 的适用性。我们还利用菲涅尔实验数据集给出了数值模拟结果，以支持理论结果。
{"title":"Unveiling novel insights into Kirchhoff migration for a fast and effective object detection from experimental Fresnel dataset","authors":"Won-Kwang Park","doi":"10.1016/j.camwa.2024.10.019","DOIUrl":"10.1016/j.camwa.2024.10.019","url":null,"abstract":"<div><div>In this paper, we consider a limited-aperture inverse scattering problem for a fast identification of small dielectric objects from two-dimensional Fresnel experimental dataset. To this end, we apply the Kirchhoff migration (KM) imaging technique and design an imaging function from the generated multi-static response matrix. Using the integral equation-based representation formula for the scattered field, we theoretically investigate the applicability of the KM by formulating the imaging function as a uniformly convergent infinite series of integer-order Bessel functions of the first kind. Numerical simulation results using the experimental Fresnel dataset are presented to support the theoretical result.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445939","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 : 2024-10-17DOI: 10.1016/j.camwa.2024.10.007
Jianghong Zhang , Fuzheng Gao , Jintao Cui
In this paper, we propose a two-grid algorithm for solving parabolic equation with nonlinear compressibility coefficient, spatially discretized by the weak Galerkin finite element method. The optimal error estimates are established. We further show that both grid solutions can achieve the same accuracy as long as the grid size satisfies . Compared with Newton iteration, the two-grid algorithm could greatly reduce the computational cost. We verify the effectiveness of the algorithm by performing numerical experiments.
{"title":"Two-grid weak Galerkin finite element method for nonlinear parabolic equations","authors":"Jianghong Zhang , Fuzheng Gao , Jintao Cui","doi":"10.1016/j.camwa.2024.10.007","DOIUrl":"10.1016/j.camwa.2024.10.007","url":null,"abstract":"<div><div>In this paper, we propose a two-grid algorithm for solving parabolic equation with nonlinear compressibility coefficient, spatially discretized by the weak Galerkin finite element method. The optimal error estimates are established. We further show that both grid solutions can achieve the same accuracy as long as the grid size satisfies <span><math><mi>H</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>. Compared with Newton iteration, the two-grid algorithm could greatly reduce the computational cost. We verify the effectiveness of the algorithm by performing numerical experiments.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445940","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 : 2024-10-17DOI: 10.1016/j.camwa.2024.10.009
Riedson Baptista , Isaac P. dos Santos , Lucia Catabriga
In this work, we present a nonlinear variational multiscale finite element method for solving both stationary and transient incompressible Navier-Stokes equations. The method is founded on a two-level decomposition of the approximation space, where a nonlinear artificial viscosity operator is exclusively added to the unresolved scales. It can be regarded as a self-adaptive method, since the amount of subgrid viscosity is automatically introduced according to the residual of the equation, in its strong form, associated with the resolved scales. Two variants for the subgrid viscosity are presented: one considering only the residual of the momentum equation and the other also incorporating the residual of the conservation of mass. To alleviate the computational cost typical of two-scale methods, the microscale space is defined through polynomial functions that vanish on the boundary of the elements, known as bubble functions. We compared the numerical and computational performance of the method with the results obtained by the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation combined with the Pressure Stabilizing/Petrov-Galerkin (PSPG) method through a set of 2D reference problems.
{"title":"Solving incompressible Navier-Stokes equations: A nonlinear multiscale approach","authors":"Riedson Baptista , Isaac P. dos Santos , Lucia Catabriga","doi":"10.1016/j.camwa.2024.10.009","DOIUrl":"10.1016/j.camwa.2024.10.009","url":null,"abstract":"<div><div>In this work, we present a nonlinear variational multiscale finite element method for solving both stationary and transient incompressible Navier-Stokes equations. The method is founded on a two-level decomposition of the approximation space, where a nonlinear artificial viscosity operator is exclusively added to the unresolved scales. It can be regarded as a self-adaptive method, since the amount of subgrid viscosity is automatically introduced according to the residual of the equation, in its strong form, associated with the resolved scales. Two variants for the subgrid viscosity are presented: one considering only the residual of the momentum equation and the other also incorporating the residual of the conservation of mass. To alleviate the computational cost typical of two-scale methods, the microscale space is defined through polynomial functions that vanish on the boundary of the elements, known as bubble functions. We compared the numerical and computational performance of the method with the results obtained by the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation combined with the Pressure Stabilizing/Petrov-Galerkin (PSPG) method through a set of 2D reference problems.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445941","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 : 2024-10-17DOI: 10.1016/j.camwa.2024.10.016
Anatoly A. Alikhanov , Mohammad Shahbazi Asl , Dongfang Li
The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.
{"title":"A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application","authors":"Anatoly A. Alikhanov , Mohammad Shahbazi Asl , Dongfang Li","doi":"10.1016/j.camwa.2024.10.016","DOIUrl":"10.1016/j.camwa.2024.10.016","url":null,"abstract":"<div><div>The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445307","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 : 2024-10-15DOI: 10.1016/j.camwa.2024.10.010
Jiantao Jiang , Zhimin Zhang
In this paper, we propose a spectral-Galerkin approximation for the quad-curl eigenvalue problem within spherical geometries. Utilizing vector spherical harmonics in conjunction with the Laplace-Beltrami operator, we decompose the quad-curl eigenvalue problem into two distinct categories of fourth-order equations: corresponding to the transverse electric (TE) and transverse magnetic (TM) modes. A thorough analysis is provided for the TE mode. The TM mode, however, is characterized by a system of coupled fourth-order equations that are subject to a divergence-free condition. We develop two separate sets of vector basis functions tailored for the coupled system in both solid spheres and spherical shells. Moreover, we design a parameterized technique aimed at eliminating spurious eigenpairs. Numerical examples are presented to demonstrate the high precision achieved by the proposed method. We also include graphs to illustrate the localization of the eigenfunctions. Furthermore, we employ Bessel functions to analyze the quad-curl problem, revealing the intrinsic connection between the eigenvalues and the zeros of combinations of Bessel functions.
在本文中，我们提出了球面几何中四卷线特征值问题的谱-加勒金近似方法。利用矢量球面谐波和拉普拉斯-贝尔特拉米算子，我们将四曲面特征值问题分解为两类不同的四阶方程：对应于横向电（TE）和横向磁（TM）模式。我们对 TE 模式进行了深入分析。然而，TM 模式的特征是一个耦合的四阶方程系统，该系统受制于无发散条件。我们开发了两套独立的矢量基函数，分别适用于实心球和球壳的耦合系统。此外，我们还设计了一种参数化技术，旨在消除虚假特征对。我们列举了一些数值示例，以证明所提出的方法能达到很高的精度。我们还通过图表说明了特征函数的定位。此外，我们还利用贝塞尔函数分析了四曲面问题，揭示了特征值与贝塞尔函数组合零点之间的内在联系。
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