Applied Numerical Mathematics最新文献

{"title":"A finite difference scheme for (2+1)D cubic-quintic nonlinear Schrödinger equations with nonlinear damping","authors":"Anh Ha Le ,&nbsp;Toan T. Huynh ,&nbsp;Quan M. Nguyen","doi":"10.1016/j.apnum.2024.07.008","DOIUrl":"10.1016/j.apnum.2024.07.008","url":null,"abstract":"<div><p>Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete <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. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001867/pdfft?md5=29380bcb28827a3fc015cb9476b8a842&pid=1-s2.0-S0168927424001867-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141841491","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 order numerical method for a subdiffusion problem","authors":"Carla Jesus,&nbsp;Ercília Sousa","doi":"10.1016/j.apnum.2024.07.006","DOIUrl":"10.1016/j.apnum.2024.07.006","url":null,"abstract":"<div><p>We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order <span><math><mn>1</mn><mo>−</mo><mi>α</mi></math></span>, <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001788/pdfft?md5=1f5d4bf4c00b7e9a89694a19dc45c8c8&pid=1-s2.0-S0168927424001788-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141844659","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":"The enriched multinode Shepard collocation method for solving elliptic problems with singularities","authors":"Francesco Dell'Accio ,&nbsp;Filomena Di Tommaso ,&nbsp;Elisa Francomano","doi":"10.1016/j.apnum.2024.07.005","DOIUrl":"10.1016/j.apnum.2024.07.005","url":null,"abstract":"<div><p>In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001776/pdfft?md5=bad5b96a018721e6a24777c89fe88152&pid=1-s2.0-S0168927424001776-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950761","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":"Approximation of the Hilbert transform on the half–line","authors":"Donatella Occorsio ,&nbsp;Woula Themistoclakis","doi":"10.1016/j.apnum.2024.07.004","DOIUrl":"10.1016/j.apnum.2024.07.004","url":null,"abstract":"<div><p>The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001764/pdfft?md5=412b695fdae44005e406f5ee943aff4b&pid=1-s2.0-S0168927424001764-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950760","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":"Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs","authors":"Aditi Tomar ,&nbsp;Lok Pati Tripathi ,&nbsp;Amiya K. Pani","doi":"10.1016/j.apnum.2024.07.003","DOIUrl":"10.1016/j.apnum.2024.07.003","url":null,"abstract":"<div><p>Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>- and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norms are derived for the problem with initial data <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>∩</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> error estimate is obtained for 2D problems that too with the initial condition is in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>∩</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. All results proved in this paper are valid uniformly as <span><math><mi>α</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>, where <em>α</em> is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141846520","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 treatment of singular ODEs using finite difference and collocation methods","authors":"Matthias Hohenegger ,&nbsp;Giuseppina Settanni ,&nbsp;Ewa B. Weinmüller ,&nbsp;Mered Wolde","doi":"10.1016/j.apnum.2024.07.002","DOIUrl":"10.1016/j.apnum.2024.07.002","url":null,"abstract":"<div><p>Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class.</p><p>In this paper, we shall compare the performance of the code <span>HOFiD_bvp</span> based on the high order finite difference schemes and <span>bvpsuite2.0</span> based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001740/pdfft?md5=54ac20f2fb292c9a9ee04286910e6e1d&pid=1-s2.0-S0168927424001740-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141711577","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":"Conformal structure-preserving SVM methods for the nonlinear Schrödinger equation with weakly linear damping term","authors":"Xin Li ,&nbsp;Luming Zhang","doi":"10.1016/j.apnum.2024.06.024","DOIUrl":"10.1016/j.apnum.2024.06.024","url":null,"abstract":"<div><p>In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141716221","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":"Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients","authors":"Yan Li,&nbsp;Nan Deng,&nbsp;Wanrong Cao","doi":"10.1016/j.apnum.2024.07.001","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.07.001","url":null,"abstract":"<div><p>In our previous works <span>[1]</span> and <span>[2]</span>, we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606785","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":"Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of convection-diffusion type","authors":"Yanhua Liu,&nbsp;Xuesong Wang,&nbsp;Yao Cheng","doi":"10.1016/j.apnum.2024.06.023","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.023","url":null,"abstract":"<div><p>We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593236","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":"Low rank approximation in the computation of first kind integral equations with TauToolbox","authors":"Paulo B. Vasconcelos ,&nbsp;Laurence Grammont ,&nbsp;Nilson J. Lima","doi":"10.1016/j.apnum.2024.06.022","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.022","url":null,"abstract":"<div><p><span>Tau Toolbox</span> is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called <span>polynomial</span>, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.</p><p>In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.</p><p>Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.</p><p>The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001703/pdfft?md5=49b2fb7a69e48f47a313e9bf4b9ddaa9&pid=1-s2.0-S0168927424001703-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593237","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}