Applied Numerical Mathematics最新文献

{"title":"Gradient descent-based parameter-free methods for solving coupled matrix equations and studying an application in dynamical systems","authors":"Akbar Shirilord,&nbsp;Mehdi Dehghan","doi":"10.1016/j.apnum.2025.01.011","DOIUrl":"10.1016/j.apnum.2025.01.011","url":null,"abstract":"<div><div>This paper explores advanced gradient descent-based parameter-free methods for solving coupled matrix equations and examines their applications in dynamical systems. We focus on the coupled matrix equations<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>A</mi><mi>X</mi><mo>+</mo><mi>Y</mi><mi>B</mi><mo>=</mo><mi>C</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>D</mi><mi>X</mi><mo>+</mo><mi>Y</mi><mi>E</mi><mo>=</mo><mi>F</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>A</mi><mo>,</mo><mi>D</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>,</mo><mi>B</mi><mo>,</mo><mi>E</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> are given matrices, and <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> are the unknown matrices to be determined. We propose a novel gradient descent-based approach with parameter-free, enhancing convergence through an accelerated technique related to momentum methods. A comprehensive analysis of the convergence and characteristics of these methods is provided. Our convergence analysis demonstrates that if the spectrum of a block matrix constructed from the matrices <em>A</em>, <em>B</em>, <em>D</em>, and <em>E</em> is confined within a horizontal ellipse in the complex plane, centered at <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span> with a major axis length of 3 and a minor axis length of 1, then the accelerated momentum method will converge to the exact solution of the discussed model. The numerical results indicate that proposed methods significantly improve efficiency, showing faster convergence and reduced computational time compared to traditional approaches. Additionally, we apply these methods to linear dynamic systems, demonstrating their effectiveness in real-world scenarios.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 29-59"},"PeriodicalIF":2.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155008","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 analysis of a mixed finite element method for the stationary convective Brinkman–Forchheimer problem","authors":"Sergio Caucao ,&nbsp;Gabriel N. Gatica ,&nbsp;Luis F. Gatica","doi":"10.1016/j.apnum.2025.01.007","DOIUrl":"10.1016/j.apnum.2025.01.007","url":null,"abstract":"<div><div>We consider a Banach spaces-based mixed variational formulation that has been recently proposed for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations, and develop a reliable and efficient residual-based <em>a posteriori</em> error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, we utilize the global inf-sup condition of the problem, combined with appropriate small data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and the local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-spaces, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. In particular, the case of flow through a 2D porous medium with fracture networks is considered.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 158-178"},"PeriodicalIF":2.2,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141885","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 robust mapping spectral method for elastic equations in curved fan-shaped domains","authors":"Yuling Guo ,&nbsp;Zhongqing Wang ,&nbsp;Chao Zhang","doi":"10.1016/j.apnum.2025.01.008","DOIUrl":"10.1016/j.apnum.2025.01.008","url":null,"abstract":"<div><div>In this paper, we introduce a robust mapping Legendre spectral-Galerkin method for addressing elastic problems in simply-connected, fan-shaped domains with curved boundaries. By employing a polar coordinate transformation, we map the fan-shaped domain onto a rectangle, which transforms the original elastic equation into a variable coefficient equation. We then develop a Legendre spectral-Galerkin scheme for this variable coefficient equation. Additionally, we demonstrate the optimal convergence of the numerical solution in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm as the Lamé coefficient <em>λ</em> remains bounded. Numerical examples illustrate the high accuracy and robustness of our method, even as <em>λ</em> approaches infinity.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 144-157"},"PeriodicalIF":2.2,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141888","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":"Variable-time-step weighted IMEX FEMs for nonlinear evolution equations","authors":"Meng Li ,&nbsp;Dan Wang ,&nbsp;Junjun Wang ,&nbsp;Xiaolong Zhao","doi":"10.1016/j.apnum.2025.01.005","DOIUrl":"10.1016/j.apnum.2025.01.005","url":null,"abstract":"<div><div>In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter <span><math><mi>θ</mi><mo>=</mo><mn>1</mn></math></span>, the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) <span><span>[15]</span></span> and Liao et al. (2021) <span><span>[29]</span></span>.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 123-143"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141889","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 average bound preserving time-implicit discretizations for convection-diffusion-reaction equation","authors":"Fengna Yan ,&nbsp;Yinhua Xia","doi":"10.1016/j.apnum.2025.01.006","DOIUrl":"10.1016/j.apnum.2025.01.006","url":null,"abstract":"<div><div>We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 103-122"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141891","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":"Singular fractional integro-differential equations with non-constant coefficients","authors":"Kaido Lätt,&nbsp;Arvet Pedas,&nbsp;Hanna Britt Soots","doi":"10.1016/j.apnum.2025.01.004","DOIUrl":"10.1016/j.apnum.2025.01.004","url":null,"abstract":"<div><div>We investigate a class of singular fractional integro-differential equations with non-constant coefficients. After reformulating the original problem as a cordial Volterra integral equation we study the unique solvability of the underlying problem. We construct a numerical method based on collocation techniques to find an approximation to the solution of the original problem and analyse the convergence and the convergence order of the proposed method. Additionally, we present the results of some numerical experiments.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 179-192"},"PeriodicalIF":2.2,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142752","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":"Error estimates with polynomial growth O(ε−1) for the HHO method on polygonal meshes of the Allen-Cahn model","authors":"Naresh Kumar ,&nbsp;Ajeet Singh ,&nbsp;Ram Jiwari ,&nbsp;J.Y. Yuan","doi":"10.1016/j.apnum.2025.01.001","DOIUrl":"10.1016/j.apnum.2025.01.001","url":null,"abstract":"<div><div>A novel approach is presented to tackle the Allen-Cahn equation arising from phase separation in alloys, by utilizing the hybrid high-order (HHO) method on polygonal meshes. The primary challenge in this equation lies in employing a straightforward Gronwall inequality-type argument for error estimation with exponential growth factor <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>(</mo><mi>C</mi><mi>T</mi><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup></math></span> as <em>ε</em> approaches zero. The application of the discrete Lyapunov functional and the discrete HHO spectrum estimate of the linearized Allen-Cahn operator <span><math><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mi>H</mi><mi>H</mi><mi>O</mi></mrow></msubsup></math></span> are used to overcome this exponential growth factor and achieve polynomial growth of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for error bounds in error estimations. Rigorous convergence analyses are established for the fully implicit schemes, which are energy stable. However, due to the implicit processing of the nonlinear term, the computational cost significantly increases. To enhance computational efficiency, a static condensation process is hired by using the HHO method, resulting in optimal convergence rates in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm. Finally, various numerical experiments on diverse meshes are conducted to validate our theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 78-102"},"PeriodicalIF":2.2,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141886","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 distributed stochastic forward-backward-forward self-adaptive algorithm for Cartesian stochastic variational inequalities","authors":"Liya Liu ,&nbsp;Xiaolong Qin ,&nbsp;Jen-Chih Yao","doi":"10.1016/j.apnum.2025.01.003","DOIUrl":"10.1016/j.apnum.2025.01.003","url":null,"abstract":"<div><div>In this paper, we consider a Cartesian stochastic variational inequality with a high dimensional solution space. This mathematical formulation captures a wide range of optimization problems including stochastic Nash games and stochastic minimization problems. By combining the advantages of the forward-backward-forward method and the stochastic approximated method, a novel distributed algorithm is developed for addressing this large-scale problem without any kind of monotonicity. A salient feature of the proposed algorithm is to compute two independent queries of a stochastic oracle at each iteration. The main contributions include: (i) The necessary condition imposed on the involved operator is related merely to the Lipschitz continuity, which are quite general. (ii) At each iteration, the suggested algorithm only requires one computation of the projection onto each feasible set, which can be easily evaluated. (iii) The distributed implementation of the stochastic approximation based Armijo-type line search strategy is adopted to weaken the line search condition and define variable adaptive non-monotonic stepsizes, when the Lipschitz constant is unknown. Some theoretical results of the almost sure convergence, the optimal rate statement, and the oracle complexity bound are established with conditions weaker than the conditions of other methods studied in the literature. Finally, preliminary numerical results are presented to show the efficiency and the competitiveness of our algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 17-41"},"PeriodicalIF":2.2,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141893","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 fitted space-time finite element method for an advection-diffusion problem with moving interfaces","authors":"Quang Huy Nguyen ,&nbsp;Van Chien Le ,&nbsp;Phuong Cuc Hoang ,&nbsp;Thi Thanh Mai Ta","doi":"10.1016/j.apnum.2025.01.002","DOIUrl":"10.1016/j.apnum.2025.01.002","url":null,"abstract":"<div><div>This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Nečas-Babuška theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 61-77"},"PeriodicalIF":2.2,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141890","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 numerical study of WENO approximations to sharp propagating fronts for reaction-diffusion systems","authors":"Jiaxi Gu ,&nbsp;Daniel Olmos-Liceaga ,&nbsp;Jae-Hun Jung","doi":"10.1016/j.apnum.2024.12.014","DOIUrl":"10.1016/j.apnum.2024.12.014","url":null,"abstract":"<div><div>Many reaction-diffusion systems exhibit traveling wave solutions that evolve on multiple spatiotemporal scales, where obtaining fast and accurate numerical solutions is challenging. In this work, we employ sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve the reaction-diffusion system for the traveling wave solution with the sharp fronts. It is shown that those WENO methods achieve the expected sixth-order accuracy in the Fisher's, Zeldovich, bistable equations, and the Lotka-Volterra competition-diffusion system. However, we find that the WENO methods converge very slowly in the Newell-Whitehead-Segel equation because of the speed issue, in which one possible way to match the exact speed is to coarsen the spatial grid and decrease the time step simultaneously. It is also seen that the central WENO method could carry the larger time step while preserving the essentially non-oscillatory behavior for the approximations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 1-16"},"PeriodicalIF":2.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141894","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}