Applied Numerical Mathematics最新文献

{"title":"On greedy randomized coordinate updating iteration methods for solving symmetric eigenvalue problems","authors":"Zhong-Zhi Bai","doi":"10.1016/j.apnum.2025.04.010","DOIUrl":"10.1016/j.apnum.2025.04.010","url":null,"abstract":"<div><div>In order to compute the smallest eigenvalue and its corresponding eigenvector of a large-scale, real, and symmetric matrix, we propose a class of greedy randomized coordinate updating iteration methods based on the principle that the indices of larger entries in absolute value of the current residual are selected with a higher probability and, with respect to this index set, the next iterate is updated from the current iterate along with the selected coordinate such that the corresponding entry of the residual is annihilated, resulting in fast convergence rates of the proposed iteration methods. Under appropriate conditions, we prove the convergence of both sequences of the Rayleigh-quotients and the acute angles between the iterates and the eigenvector in terms of the expectation. By numerical experiments, we show that this class of greedy randomized coordinate updating iteration methods are advantageous over the parameterized power method and the coordinate descent method in both iteration counts and computing times. Moreover, with theoretical analysis and computational performance, we confirm that the convergence property of this class of iteration methods can be improved significantly by suitably choosing the arbitrary nonnegative parameter involved in the greedy probability criterion.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 76-97"},"PeriodicalIF":2.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069908","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":"Data-dependent moving least squares","authors":"David Levin ,&nbsp;José M. Ramón ,&nbsp;Juan Ruiz-Álvarez ,&nbsp;Dionisio F. Yáñez","doi":"10.1016/j.apnum.2025.05.002","DOIUrl":"10.1016/j.apnum.2025.05.002","url":null,"abstract":"<div><div>In this paper, we address a data-dependent modification of the moving least squares (MLS) problem. We propose a novel approach by replacing the traditional weight functions with new functions that assign smaller weights to nodes that are close to discontinuities, while still assigning smaller weights to nodes that are far from the point of approximation. Through this adjustment, we are able to mitigate the undesirable Gibbs phenomenon that appears close to the discontinuities in the classical MLS approach, and reduce the smearing of discontinuities in the final approximation of the original data. The core of our method involves accurately identifying those nodes affected by the presence of discontinuities using smoothness indicators, a concept derived from the data-dependent WENO method. Our formulation results in a data-dependent weighted least squares problem where the weights depend on two factors: the distances between nodes and the point of approximation, and the smoothness of the data in a region of predetermined radius around the nodes. We explore the design of the new data-dependent approximant, analyze its properties including polynomial reproduction, accuracy, and smoothness, and study its impact on diffusion and the Gibbs phenomenon. Numerical experiments are conducted to validate the theoretical findings, and we conclude with some insights and potential directions for future research.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 56-75"},"PeriodicalIF":2.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143946842","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 posteriori error estimates and time adaptivity for fully discrete finite element method for the incompressible Navier-Stokes equations","authors":"Shuo Yang,&nbsp;Hongjiong Tian","doi":"10.1016/j.apnum.2025.05.001","DOIUrl":"10.1016/j.apnum.2025.05.001","url":null,"abstract":"<div><div>In this paper, we study a posteriori error estimates for the incompressible Navier-Stokes equations in a convex polygonal domain. The semi-implicit variable step-size two-step backward differentiation formula (BDF2) is employed for the time discretization and the Taylor–Hood finite element method (FEM) is used for the space discretization. We prove energy stability of semi-implicit variable step-size BDF2 FEM under different Courant Friedreich Lewy (CFL)-type conditions by utilizing different embeddings for the nonlinear term. Two appropriate reconstructions of the approximate solution are proposed to obtain the time discretization error. Resorting to the energy stability and the quadratic reconstructions, we obtain a posteriori upper and lower error bounds for the fully discrete approximation. We further develop a time adaptive algorithm for efficient time step control based on the time error estimators. Several numerical experiments are performed to verify our theoretical results and demonstrate the efficiency of the time adaptive algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 17-38"},"PeriodicalIF":2.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936933","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":"Numerical analysis of American option pricing in a two-asset jump-diffusion model","authors":"Hao Zhou,&nbsp;Duy-Minh Dang","doi":"10.1016/j.apnum.2025.03.005","DOIUrl":"10.1016/j.apnum.2025.03.005","url":null,"abstract":"<div><div>This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives–a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals.</div><div>We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is then expressed as a 2-D convolution integral involving the Green's function of the PIDE. We derive an infinite series representation of this Green's function, where each term is strictly positive and computable. This series facilitates the numerical approximation of the PIDE solution through a monotone integration method, such as the composite quadrature rule. To further enhance efficiency, we develop an efficient implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.</div><div>The proposed method is proved to be both <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach, highlighting its practical applicability and theoretical soundness.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 98-126"},"PeriodicalIF":2.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069909","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":"Efficient numerical methods based on general linear methods for Volterra integral equations","authors":"Z. Ghahremani ,&nbsp;A. Abdi ,&nbsp;G. Hojjati","doi":"10.1016/j.apnum.2025.04.011","DOIUrl":"10.1016/j.apnum.2025.04.011","url":null,"abstract":"<div><div>This paper presents a family of methods for solving Volterra integral equations of the second kind. The approach combines a special class of general linear methods for ordinary differential equations with the backward differentiation formulas and Gregory quadrature rules. The order of the proposed methods is derived in terms of the number of internal stages in the underlying general linear methods and the order of the proposed starting and finishing methods. Numerical experiments confirm the theoretical results regarding the convergence order and linear stability, demonstrating the efficiency of the proposed family of methods in solving stiff equations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 1-16"},"PeriodicalIF":2.2,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143924447","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":"Stabilization and adaptive FEM for optimal control problems of stationary convection-dominated diffusion equations on surfaces","authors":"Qiuhui Yan,&nbsp;Xufeng Xiao,&nbsp;Xinlong Feng","doi":"10.1016/j.apnum.2025.04.009","DOIUrl":"10.1016/j.apnum.2025.04.009","url":null,"abstract":"<div><div>This paper explores the optimal control problem for the stationary convection-dominated diffusion equation on surfaces, employing stabilization and adaptive finite element methods as numerical approaches. We propose an optimal system on surfaces, demonstrate the existence and uniqueness of the solution, and analyze the relevant finite element theory. To mitigate oscillation phenomena, we adopt the streamline diffusion stabilization method. To enhance computational efficiency and achieve high-resolution numerical solutions with fewer degrees of freedom, we integrate the streamline diffusion method with adaptive mesh refinement. Finally, numerical experiments confirm the advantages of our approach.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 157-176"},"PeriodicalIF":2.2,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903467","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":"High-order well-balanced schemes for shallow models for dry avalanches","authors":"M.J. Castro Díaz ,&nbsp;C. Escalante ,&nbsp;J. Garres-Díaz ,&nbsp;T. Morales de Luna","doi":"10.1016/j.apnum.2025.04.008","DOIUrl":"10.1016/j.apnum.2025.04.008","url":null,"abstract":"<div><div>In this work we consider a depth-averaged model for granular flows with a Coulomb-type friction force described by the <span><math><mi>μ</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> rheology. In this model, the so-called lake-at-rest steady states are of special interest, where velocity is zero and the slope is under a critical threshold defined by the angle of repose of the granular material. It leads to a family with an infinite number of lake-at-rest steady states. We describe a well-balanced reconstruction procedure that allows to define well-balanced finite volume methods for such problem. The technique is generalized to high-order space/time schemes. In particular, the second and third-order schemes are considered in the numerical tests section. An accuracy test is included showing that second and third-order are achieved. A well-balanced test is also considered. The proposed scheme is well-balanced for steady states with non-constant free surface, and it is exactly well-balanced for those steady states given by a simple characterization.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 138-156"},"PeriodicalIF":2.2,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860248","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":"Fast algorithms of compact scheme for solving parabolic equations and their application","authors":"Wenzhuo Xiong ,&nbsp;Xiao Wang ,&nbsp;Xiujun Cheng","doi":"10.1016/j.apnum.2025.04.006","DOIUrl":"10.1016/j.apnum.2025.04.006","url":null,"abstract":"<div><div>Based on existing work on compact scheme, particularly utilizing the Crank-Nicolson scheme for time derivatives and compact difference schemes for spatial derivatives in solving linear parabolic equations, we propose a fast algorithm of the scheme to solve the systems for the first time. Given that the resulting coefficient matrices of the scheme are diagonalizable, we transform the matrix-vector equations into a diagonal component-wise system, utilizing modified discrete cosine transform (MDCT), discrete sine transform (DST), and discrete Fourier transform (DFT) to optimize CPU time and reduce storage requirements. Moreover, the algorithmic technique facilitates a novelty and simple convergence demonstration strategy in the discrete maximum norm that is easily extendable to high-dimensional linear cases. The computational framework is also extendable to three-dimensional (3D) linear case and semi-linear case. Numerical experiments are given to support our findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 90-111"},"PeriodicalIF":2.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855572","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 decoupled linear, mass- and energy-conserving relaxation-type high-order compact finite difference scheme for the nonlinear Schrödinger equation","authors":"Wenrong Zhou ,&nbsp;Hongfei Fu ,&nbsp;Shusen Xie","doi":"10.1016/j.apnum.2025.04.005","DOIUrl":"10.1016/j.apnum.2025.04.005","url":null,"abstract":"<div><div>In this paper, a relaxation-type high-order compact finite difference (RCFD) scheme is proposed for the one-dimensional nonlinear Schrödinger equation. More specifically, the relaxation approach combined with the Crank-Nicolson formula is utilized for time discretization, and fourth-order compact difference method is applied for space discretization. The scheme is linear, decoupled, and can be solved sequentially with respect to the primal and relaxation variables, which avoids solving large-scale nonlinear algebraic systems resulting in fully implicit numerical schemes. Furthermore, the developed scheme is shown to preserve both mass and energy at the discrete level. Most importantly, with the help of a discrete elliptic projection and a cut-off numerical technique, the existence and uniqueness of the high-order RCFD scheme are ensured, and unconditional optimal-order error estimate in discrete maximum-norm is rigorously established. Finally, several numerical experiments are given to support the theoretical findings, and comparisons with other methods are also presented to show the efficiency and effectiveness of our method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 59-89"},"PeriodicalIF":2.2,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855571","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 starting approximations for implicit Runge-Kutta methods applied to ODEs based on the reverse method","authors":"Laurent O. Jay ,&nbsp;Juan I. Montijano","doi":"10.1016/j.apnum.2025.04.007","DOIUrl":"10.1016/j.apnum.2025.04.007","url":null,"abstract":"<div><div>We consider the application of <em>s</em>-stage implicit Runge-Kutta methods to ordinary differential equations (ODEs). We consider starting approximations based on values from the previous step to obtain an accurate initial guess for the internal stages of the current step. To simplify the analysis of those starting approximations we compare the expansions of the starting approximation and of the exact value of the internal stages at the initial value <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the current step and not at the initial value <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> of the previous step. In particular, for the starting approximation we make use of the expansion of the reverse IRK method from the initial value <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the current step with a negative step size. This simplifies considerably the expression of the order conditions. As a consequence it allows us to give more general and precise statements about the existence and uniqueness of a starting approximation of a given order for IRK methods satisfying the simplifying assumptions <span><math><mi>B</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. In particular we show under certain assumptions the nonexistence of starting approximations of order <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> for the type of starting approximations considered.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838931","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}