Applied Numerical Mathematics最新文献

{"title":"Energetic spectral-element time marching methods for phase-field nonlinear gradient systems","authors":"Shiqin Liu ,&nbsp;Haijun Yu","doi":"10.1016/j.apnum.2024.06.021","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.021","url":null,"abstract":"<div><p>We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606786","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":"An efficient hybrid conjugate gradient method with an adaptive strategy and applications in image restoration problems","authors":"Zibo Chen ,&nbsp;Hu Shao ,&nbsp;Pengjie Liu ,&nbsp;Guoxin Li ,&nbsp;Xianglin Rong","doi":"10.1016/j.apnum.2024.06.020","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.020","url":null,"abstract":"<div><p>In this study, we introduce a novel hybrid conjugate gradient method with an adaptive strategy called asHCG method. The asHCG method exhibits the following characteristics. (i) Its search direction guarantees sufficient descent property without dependence on any line search. (ii) It possesses strong convergence for the uniformly convex function using a weak Wolfe line search, and under the same line search, it achieves global convergence for the general function. (iii) Employing the Armijo line search, it provides an approximate guarantee for worst-case complexity for the uniformly convex function. The numerical results demonstrate promising and encouraging performances in both unconstrained optimization problems and image restoration problems.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542140","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":"An optimized algorithm for numerical solution of coupled Burgers equations","authors":"Anurag Kaur ,&nbsp;V. Kanwar ,&nbsp;Higinio Ramos","doi":"10.1016/j.apnum.2024.06.019","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.019","url":null,"abstract":"<div><p>Investigation of the solutions of the coupled viscous Burgers system is crucial for realizing and understanding some physical phenomena in applied sciences. Particularly, Burgers equations are used in the modeling of fluid mechanics and nonlinear acoustics. In the present study, a modified meshless quadrature method based on radial basis functions is used to discretize the partial derivatives in the spatial variable. A technique to find the best value of the shape parameter is introduced. A high-resolution optimized hybrid block method is then used to solve the problem in the temporal variable. To validate the proposed method, several test problems are considered and the simulated results are compared with exact solutions and previous works. Moreover, a sensitivity analysis for parameter <em>c</em> is conducted, and the unconditional stability of the proposed algorithm has been validated.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542142","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":"On the stability of θ-methods for DDEs and PDDEs","authors":"Alejandro Rodríguez-Fernández ,&nbsp;Jesús Martín-Vaquero","doi":"10.1016/j.apnum.2024.06.018","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.018","url":null,"abstract":"<div><p>In this paper, the stability of <em>θ</em>-methods for delay differential equations is studied based on the test equation <span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>A</mi><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>B</mi><mi>y</mi><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></math></span>, where <em>τ</em> is a constant delay and <em>A</em> is a positive definite matrix. It is mainly considered the case where the matrices <em>A</em> and <em>B</em> are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices <em>A</em> and <em>B</em> are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141542141","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":"The sine and cosine diffusive representations for the Caputo fractional derivative","authors":"Hassan Khosravian-Arab ,&nbsp;Mehdi Dehghan","doi":"10.1016/j.apnum.2024.06.017","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.017","url":null,"abstract":"<div><p>In recent years, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order <span><math><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn></math></span>. An error analysis of the newly presented methods together with some numerical examples is provided at the end.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482826","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 threshold stability of a nonlinear age-structured reaction diffusion heroin transmission model","authors":"X. Liu ,&nbsp;M. Zhang ,&nbsp;Z.W. Yang","doi":"10.1016/j.apnum.2024.06.016","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.016","url":null,"abstract":"<div><p>This paper deals with the numerical threshold stability of a nonlinear age-space structured heroin transmission model. A semi-discrete system is established by spatially domain discretization of the original nonlinear age-space structured model. A threshold value is proposed in stability analysis of the semi-discrete system and named as a numerical basic reproduction number. Besides the role it plays in numerical threshold stability analysis, the numerical basic reproduction number can preserve qualitative properties of the exact basic reproduction number and converge to the latter while stepsizes vanish. A fully discrete system is established via a time discretization of the semi-discrete system, in which an implicit-explicit technique is implemented to ensure the preservation of the biological meanings (such as positivity) without CFL restriction. Some numerical experiments are exhibited in the end to confirm the conclusions and explore the final state.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482827","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":"An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations","authors":"Saurabh Kumar ,&nbsp;Vikas Gupta ,&nbsp;Dia Zeidan","doi":"10.1016/j.apnum.2024.06.014","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.014","url":null,"abstract":"<div><p>In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141438115","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 new class of quadrature rules for estimating the error in Gauss quadrature","authors":"Aleksandar V. Pejčev ,&nbsp;Lothar Reichel ,&nbsp;Miodrag M. Spalević ,&nbsp;Stefan M. Spalević","doi":"10.1016/j.apnum.2024.06.011","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.011","url":null,"abstract":"<div><p>The need to evaluate Gauss quadrature rules arises in many applications in science and engineering. It often is important to be able to estimate the quadrature error when applying an <em>ℓ</em>-point Gauss rule, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, where <em>f</em> is an integrand of interest. Such an estimate often is furnished by applying another quadrature rule, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, with <span><math><mi>k</mi><mo>&gt;</mo><mi>ℓ</mi></math></span> nodes, and using the difference <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> or its magnitude as an estimate for the quadrature error in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> or its magnitude. The classical approach to estimate the error in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is to let <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, with <span><math><mi>k</mi><mo>=</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span>, be the Gauss-Kronrod quadrature rule associated with <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>. However, it is well known that the Gauss-Kronrod rule associated with a Gauss rule <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> might not exist for certain measures that determine the Gauss rule and for certain numbers of nodes. This prompted M. M. Spalević <span>[1]</span> to develop generalized averaged Gauss rules, <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, with <span><math><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></math></span> nodes for estimating the error in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>. Similarly as for <span><math><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-node Gauss-Kronrod rules, <em>ℓ</em> nodes of the rule <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> agree with the nodes of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. However, generalized averaged Gauss rules are not internal for some measures. They therefore may not be applicable when the integrand only is define","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434087","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 phase field method for convective phase change problem preserving maximum bound principle","authors":"Hui Yao","doi":"10.1016/j.apnum.2024.06.012","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.012","url":null,"abstract":"<div><p>Numerical simulations of convective solid-liquid phase change problems have long been a complex problem due to the movement of the solid-liquid interface layer, which leads to a free boundary problem. This work develops a convective phase change heat transfer model based on the phase field method. The governing equations consist of the incompressible Navier-Stokes-Boussinesq equations, the heat transfer equation, and the Allen-Cahn equation. The Navier-Stokes equations are penalised for imposing zero velocity within the solid region. For numerical methods, the mini finite element approach (<span>P1b-P1</span>) is used to solve the momentum equation spatially, the temperature and the phase field are approximated by the <span>P1b</span> elements. In the temporal discretization, the phase field and the temperature are decoupled from the momentum equation by using the finite difference method, forming a solvable linear system. A maximum bound principle for the phase field is derived, coming with an estimation of the tolerance of the time step size, which depends on the temperature range. This estimation guides the time step choice in the simulation. The program is developed within the <span>FreeFem++</span> framework, drawing on our previous work on phase field methods <span>[1]</span> and a mushy-region method toolbox for heat transfer <span>[2]</span>. The accuracy and effectiveness of the proposed method have been validated through real-world cases of melting and solidification with linear or nonlinear buyangcy force, respectively. The simulation results are in agreement with experiments in references.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001582/pdfft?md5=2072fce0ac49fb6e14195fb698625ef4&pid=1-s2.0-S0168927424001582-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-rate convergent multistep collocation techniques to a first-kind Volterra integral equation along with the proportional vanishing delay","authors":"Aws Mushtaq Mudheher,&nbsp;S. Pishbin,&nbsp;P. Darania,&nbsp;Shadi Malek Bagomghaleh","doi":"10.1016/j.apnum.2024.06.015","DOIUrl":"10.1016/j.apnum.2024.06.015","url":null,"abstract":"<div><p>In the present study, we construct a considerably fast convergent multistep collocation technique in order to solve Volterra integral equations, especially first-kind ones with variable vanishing delays. Through a robust theoretical analysis, the optimal global convergence of the numerically achieved solutions to their exact counterparts has been demonstrated with the corresponding high orders. The allusion to the strategy of reformulating a first-kind Volterra integral equation into a second-kind Volterra functional integral equation, assists us for the establishment of regularity, existence and uniqueness features of analytical solution over under consideration equation. The existence and uniqueness of numerical solution have also been shown. Eventually, some test problems have been provided to evaluate effectiveness of the proposed multistep collocation technique.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141405620","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}