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Unconditional energy stability and maximum principle preserving scheme for the Allen-Cahn equation 艾伦-卡恩方程的无条件能量稳定性和最大原则保持方案
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-12 DOI: 10.1007/s11075-024-01880-2
Zhuangzhi Xu, Yayun Fu
{"title":"Unconditional energy stability and maximum principle preserving scheme for the Allen-Cahn equation","authors":"Zhuangzhi Xu, Yayun Fu","doi":"10.1007/s11075-024-01880-2","DOIUrl":"https://doi.org/10.1007/s11075-024-01880-2","url":null,"abstract":"<p>In this paper, we propose a novel fully implicit numerical scheme that satisfies both nonlinear energy stability and maximum principle for the space fractional Allen-Cahn equation. Especially, the fully implicit second-order scheme in time has never been proved to preserve the maximum principle before. For the resulting nonlinear scheme, we also propose a nonlinear iterative algorithm, which is uniquely solvable, convergent, and can preserve discrete maximum principle in each iterative step. Then we provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are presented to verify the theoretical results.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unconditionally positivity-preserving approximations of the Aït-Sahalia type model: Explicit Milstein-type schemes 艾特-萨哈利亚模型的无条件保正近似：米尔斯坦型显式方案
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-10 DOI: 10.1007/s11075-024-01861-5
Yingsong Jiang, Ruishu Liu, Xiaojie Wang, Jinghua Zhuo
{"title":"Unconditionally positivity-preserving approximations of the Aït-Sahalia type model: Explicit Milstein-type schemes","authors":"Yingsong Jiang, Ruishu Liu, Xiaojie Wang, Jinghua Zhuo","doi":"10.1007/s11075-024-01861-5","DOIUrl":"https://doi.org/10.1007/s11075-024-01861-5","url":null,"abstract":"<p>The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain <span>((0, infty ))</span>, which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term <span>(alpha _{-1} x^{-1})</span> and a corrective mapping <span>(Phi _h)</span> in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size <span>(h&gt;0)</span>) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fractional Legendre wavelet approach resolving multi-scale optimal control problems involving Caputo-Fabrizio derivative 用分数 Legendre 小波方法解决涉及 Caputo-Fabrizio 导数的多尺度优化控制问题
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-10 DOI: 10.1007/s11075-024-01871-3
Akanksha Singh, Ankur Kanaujiya, Jugal Mohapatra
{"title":"Fractional Legendre wavelet approach resolving multi-scale optimal control problems involving Caputo-Fabrizio derivative","authors":"Akanksha Singh, Ankur Kanaujiya, Jugal Mohapatra","doi":"10.1007/s11075-024-01871-3","DOIUrl":"https://doi.org/10.1007/s11075-024-01871-3","url":null,"abstract":"<p>This article provides an effective numerical approach using the fractional integral operational matrix method for a fractional Legendre wavelet to deal with multi-dimensional fractional optimal control problems. We proposed operational matrices and implemented them to simplify multi-dimensional fractional optimal control problems into a set of equations, utilizing well-known formulas such as the Caputo-Fabrizio operator with a non-singular kernel defined for calculating fractional derivatives and integrals of fractional Legendre wavelets. Finally, the Lagrange multiplier technique is applied, and we get the state and control functions. The convergence analysis and error bounds of the proposed scheme are established. To check the veracity of the presented method, we tested numerical examples using the fractional Legendre wavelet method and obtained the cost function value based on identifying state and control functions.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"28 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity 表现出弱初始奇异性的二维时间分数对流扩散方程的高阶全离散方法的误差分析
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-09 DOI: 10.1007/s11075-024-01877-x
Anshima Singh, Sunil Kumar
{"title":"Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity","authors":"Anshima Singh, Sunil Kumar","doi":"10.1007/s11075-024-01877-x","DOIUrl":"https://doi.org/10.1007/s11075-024-01877-x","url":null,"abstract":"<p>This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time (<span>(t=0)</span>) is encountered in the considered problem. To overcome this, we consider the high-order L2-1<span>(_sigma )</span> formula on a suitably designed non-uniform fitted mesh, to discretize the time-fractional derivative. Further, a high-order two-dimensional compact operator is developed to approximate the spatial variables. Moreover, an alternating direction implicit (ADI) approach is designed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, is conducted comprehensively. More precisely, it is shown that method is convergent of order <span>(mathcal Oleft( {N_t^{-min {3-alpha ,theta alpha ,1+alpha ,2}}}+h_x^4+h_y^4right) )</span>, where <span>(alpha in (0,1))</span> represents the order of the fractional derivative, <span>(theta )</span> is a parameter which is utilized in the construction of the fitted mesh, <span>(N_t)</span> is the temporal discretization parameter, and <span>(h_x)</span> and <span>(h_y)</span> represent the spatial mesh widths. The numerical outcomes for three test problems, each featuring the nonsmooth solution, verified the theoretical findings. Further, the proposed method on fitted meshes exhibits superior numerical accuracy in comparison to the existing methods.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"51 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of high order numerical methods for solving fourth order nonlinear boundary value problems 构建求解四阶非线性边界值问题的高阶数值方法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-08 DOI: 10.1007/s11075-024-01879-9
Quang A Dang, Thanh Huong Nguyen, Vinh Quang Vu
{"title":"Construction of high order numerical methods for solving fourth order nonlinear boundary value problems","authors":"Quang A Dang, Thanh Huong Nguyen, Vinh Quang Vu","doi":"10.1007/s11075-024-01879-9","DOIUrl":"https://doi.org/10.1007/s11075-024-01879-9","url":null,"abstract":"<p>In this paper, we construct numerical methods of fourth, sixth and eighth orders convergence for solving fully fourth order nonlinear differential equation with the Dirichlet boundary conditions. The methods are based on the use of the trapezoidal quadrature formula with corrections for computing integrals at each iteration of the continuous iterative method for finding the solutions of the BVP. We get the error estimates for the actually obtained numerical solutions of the problem. Many numerical examples confirm the theoretical conclusions and show the efficiency of the proposed methods in comparison with some existing methods.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"36 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Uzawa-DOS method for solving saddle-point problems 解决鞍点问题的乌泽-DOS 方法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-06 DOI: 10.1007/s11075-024-01873-1
Ghodrat Ebadi, Khosro Mehrabi, Predrag S. Stanimirović
{"title":"An Uzawa-DOS method for solving saddle-point problems","authors":"Ghodrat Ebadi, Khosro Mehrabi, Predrag S. Stanimirović","doi":"10.1007/s11075-024-01873-1","DOIUrl":"https://doi.org/10.1007/s11075-024-01873-1","url":null,"abstract":"<p>Based on the diagonal and off-diagonal splitting (DOS) iteration scheme (Dehghan et al. Filomat <b>31</b>(5), 1441–1452 2017), we offer an iteration procedure called Uzawa-DOS to solve a class of saddle-point problems (SPPs). Each iteration of this iterative method involves two subsystems with diagonal and lower triangular matrices. Due to the simple structure of involved coefficient matrices, two linear subsystems are solvable exactly, which is a notable precedence of the Uzawa-DOS method and can make it inexpensive to execute. Theoretical analysis verifies convergence of the proposed method under appropriate conditions. The suggested method is validated by numerical experiments.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"368 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Inertial randomized Kaczmarz algorithms for solving coherent linear systems 求解相干线性系统的惯性随机卡兹马兹算法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-05 DOI: 10.1007/s11075-024-01872-2
Songnian He, Ziting Wang, Qiao-Li Dong
{"title":"Inertial randomized Kaczmarz algorithms for solving coherent linear systems","authors":"Songnian He, Ziting Wang, Qiao-Li Dong","doi":"10.1007/s11075-024-01872-2","DOIUrl":"https://doi.org/10.1007/s11075-024-01872-2","url":null,"abstract":"<p>In this paper, by regarding the two-subspace Kaczmarz method as an alternated inertial randomized Kaczmarz algorithm we present a better convergence rate estimate under a mild condition. Furthermore, we accelerate the alternated inertial randomized Kaczmarz algorithm and introduce a multi-step inertial randomized Kaczmarz algorithm which is proved to have a faster convergence rate. Numerical experiments support the theory results and illustrate that the multi-inertial randomized Kaczmarz algorithm significantly outperform the two-subspace Kaczmarz method in solving coherent linear systems.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"198 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Caputo fractional derivative of $$alpha $$ -fractal spline 分形样条曲线的卡普托分数导数
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-05 DOI: 10.1007/s11075-024-01875-z
T. M. C. Priyanka, A. Gowrisankar, M. Guru Prem Prasad, Yongshun Liang, Jinde Cao
{"title":"Caputo fractional derivative of $$alpha $$ -fractal spline","authors":"T. M. C. Priyanka, A. Gowrisankar, M. Guru Prem Prasad, Yongshun Liang, Jinde Cao","doi":"10.1007/s11075-024-01875-z","DOIUrl":"https://doi.org/10.1007/s11075-024-01875-z","url":null,"abstract":"<p>The Caputo fractional derivative of a real continuous function <i>g</i> distinguishes from the other fractional derivative methods with the demand for the existence of its first order derivative <span>(g')</span>. This attribute leads to the investigation of Caputo fractional derivative of <span>(alpha )</span>-fractal splines rather than just a continuous non-differentiable <span>(alpha )</span>-fractal function. A bounded linear operator corresponding to the Caputo fractional derivative of fractal version is reported. In addition, a new family of fractal perturbations is proposed in association with the fractional derivative. Thereafter, a numerical approach is used to determine the exact Caputo fractional derivative of fractal functions in terms of Legendre polynomials.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"134 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations 论自动卷积 Volterra 积分微分方程 Galerkin 方法的收敛性
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-04 DOI: 10.1007/s11075-024-01874-0
Yuping Li, Hui Liang, Huifang Yuan
{"title":"On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations","authors":"Yuping Li, Hui Liang, Huifang Yuan","doi":"10.1007/s11075-024-01874-0","DOIUrl":"https://doi.org/10.1007/s11075-024-01874-0","url":null,"abstract":"<p>The Galerkin method is proposed for initial value problem of auto-convolution Volterra integro-differential equation (AVIDE). The solvability of the Galerkin method is discussed, and the uniform boundedness of the numerical solution is provided by defining a discrete weighted exponential norm. In particular, it is proved that the quadrature Galerkin method obtained from the Galerkin method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the continuous piecewise polynomial collocation method. For the Galerkin approximated solution in continuous piecewise polynomial space of degree <span>(varvec{m})</span>, at first, the <span>(varvec{m})</span> global convergence order is obtained. By defining a projection operator, the convergence is improved, and the optimal <span>(varvec{m+1})</span> global convergence order is gained, as well as <span>(varvec{2m})</span> local convergence order at mesh points. Furthermore, all the above analysis for uniform mesh can be extended to typical quasi-uniform meshes. Some numerical experiments are given to illustrate the theoretical results.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"2014 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A new splitting mixed finite element analysis of the viscoelastic wave equation 粘弹性波方程的新型分裂混合有限元分析
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-07-04 DOI: 10.1007/s11075-024-01876-y
Jiansong Zhang, Liping Gao, Yuanshuo Kong, Mei Wang, Guanqi Yang
{"title":"A new splitting mixed finite element analysis of the viscoelastic wave equation","authors":"Jiansong Zhang, Liping Gao, Yuanshuo Kong, Mei Wang, Guanqi Yang","doi":"10.1007/s11075-024-01876-y","DOIUrl":"https://doi.org/10.1007/s11075-024-01876-y","url":null,"abstract":"<p>This paper aims to propose a new splitting mixed finite element method (MFE) for solving viscoelastic wave equations and give convergence analysis. First, by introducing two new variables <span>(q=u_t)</span> and <span>(varvec{sigma }=A(x)nabla u+B(x)nabla u_t)</span>, a new system of first-order differential-integral equations is derived from the original second-order viscoelastic wave equation. Then, the semi-discrete and fully-discrete splitting MFE schemes are proposed by using the MFE spaces and the second-order time discetization. By the two schemes the approximate solutions for the unknowns <i>u</i>, <span>(u_t)</span> and <span>(sigma )</span> are obtained simultaneously. It is proved that the semi-discrete and fully-discrete schemes have the optimal error estimates in <span>(L^2)</span>-norm. Meanwhile, it is proved that the fully-discrete SMFE scheme based on the Raviart-Thomas mixed finite element spaces and the uniform rectangular mesh partitions is super convergent. Finally, numerical experiments to compute the <span>(L^2)</span> errors for approximating <i>u</i>, <i>q</i> and <span>(varvec{sigma })</span> and their convergence rates are presented, and the theoretical analysis on error estimates and convergence is then confirmed.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"125 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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