Applied Numerical Mathematics最新文献

{"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}

{"title":"Traveling wave solutions for the accelerated Frenkel-Kontorova model: The monostable cases","authors":"G. Abi Younes,&nbsp;N. El Khatib,&nbsp;M. Zaydan","doi":"10.1016/j.apnum.2025.04.003","DOIUrl":"10.1016/j.apnum.2025.04.003","url":null,"abstract":"<div><div>In this paper, we consider a system of accelerated and general fully non-linear discrete equations depending on a parameter <em>σ</em> lying inside an interval <span><math><mo>[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>,</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo></math></span>. For <span><math><mi>σ</mi><mo>∈</mo><mo>(</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>,</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></math></span>, our non-linearity is bistable and for <span><math><mi>σ</mi><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>±</mo></mrow></msup></math></span>, it is monostable. Two results are obtained: the first one is to derive properties of the velocity function associated to the existence of traveling waves in the bistable regimes. The second one is to construct traveling waves in the monostable regimes. Our approach is to consider the monostable regimes as the limit of bistable ones. As far as we know, this is the first result concerning traveling waves for accelerated, general and monostable fully-nonlinear discrete system.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 25-48"},"PeriodicalIF":2.2,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838832","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":"Stroboscopic averaging methods to study autoresonance and other problems with slowly varying forcing frequencies","authors":"M.P. Calvo ,&nbsp;J.M. Sanz-Serna ,&nbsp;Beibei Zhu","doi":"10.1016/j.apnum.2025.04.004","DOIUrl":"10.1016/j.apnum.2025.04.004","url":null,"abstract":"<div><div>Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 15-24"},"PeriodicalIF":2.2,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838831","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 rates of Galerkin finite element methods for SWEs with cubic polynomial nonlinearity","authors":"Ruisheng Qi ,&nbsp;Xiaojie Wang","doi":"10.1016/j.apnum.2025.04.001","DOIUrl":"10.1016/j.apnum.2025.04.001","url":null,"abstract":"<div><div>In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin finite element method in space and a novel exponential time integrator combined with the averaged vector field approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difficulties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 112-137"},"PeriodicalIF":2.2,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860247","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 Wachspress-Habetler extension to the HSS iteration method in Rn×n","authors":"Thomas Smotzer,&nbsp;John Buoni","doi":"10.1016/j.apnum.2025.04.002","DOIUrl":"10.1016/j.apnum.2025.04.002","url":null,"abstract":"<div><div>In the study of implicit iterations for the two dimensional heat and Helmhotz equations, one constructs a splitting of the form <span><math><mi>A</mi><mo>=</mo><mi>U</mi><mo>+</mo><mi>V</mi></math></span> where <em>U</em> and <em>V</em> are the difference approximations parallel to the <em>x</em> and <em>y</em> axis, respectively. In the commutative case for <em>U</em> and <em>V</em>, several investigations have taken place. For the noncommutative case, a symmetric positive definite matrix <em>F</em> is found, such that <span><math><mi>U</mi><mi>F</mi><mi>V</mi><mo>=</mo><mi>V</mi><mi>F</mi><mi>U</mi></math></span> and then, the investigations use this to address the non-commutative nature of <em>U</em> with <em>V</em>. The purpose of this paper is to study the same problem type for the commutativity of <em>H</em> and <em>S</em> in the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> splitting of <span><math><mi>A</mi><mo>=</mo><mi>H</mi><mo>+</mo><mi>S</mi></math></span>, where <em>H</em> and <em>S</em> are the symmetric and skew-symmetric parts of <em>A</em>, respectively. Although throughout the literature in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, the <em>H</em> of <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> stands for hermitian, we use it in the symmetric matrix case. One then applies this result to the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> iteration method. Since it is well known that the commutativity of <em>U</em> and <em>V</em> plays an important role in the analysis of <em>ADI</em> methods; especially for the solution of the Helmhotz Equation, one hopes that this commutativity will improve performance of the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> method. This extension is similar to that of Wachspress and Habetler's variation of the Peaceman-Rachford method. Along the way, suitable conditions are found for <em>A</em>, which yield a symmetric non-zero matrix <span><math><mi>P</mi><mo>=</mo><msqrt><mrow><mi>F</mi></mrow></msqrt></math></span> for which <span><math><mi>N</mi><mspace></mspace><mo>=</mo><mspace></mspace><mi>P</mi><mi>A</mi><mi>P</mi></math></span> is a normal matrix.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 49-58"},"PeriodicalIF":2.2,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843197","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":"General enrichments of stable GFEM for interface problems: Theory and extreme learning machine construction","authors":"Dongmei Wang ,&nbsp;Hengguang Li ,&nbsp;Qinghui Zhang","doi":"10.1016/j.apnum.2025.03.009","DOIUrl":"10.1016/j.apnum.2025.03.009","url":null,"abstract":"<div><div>Generalized finite element methods (GFEMs), when applied to interface problems (IPs), need to be enriched with special functions to enhance approximation accuracy. These functions include distance functions, one-side distance functions, level set functions, and exponential forms of level set function. For the IP with geometrically complex interface curves, computation of the distance function or level set function could be challenging, and algorithms of computational geometry are usually involved. Moreover, theoretical analysis on optimal convergence of the GFEM enriched by these functions has not been fully investigated. In this study we propose a general enrichment scheme, based on which all the aforementioned enrichments can be viewed as special examples. We prove that a stable GFEM (SGFEM) with such a new enrichment scheme reaches the optimal convergence rate. Most importantly, the new scheme provides an instruction to construct machine learning (ML) based enrichments, which advances the ability of GFEM to handle geometrically complex interfaces. Two ML methods, deep neural network (DNN) and extreme learning machine (ELM), are studied. Among them, the ELM is highly suggested because it exhibits high accuracy for the interface curve with complex geometries. The learning dimension for the ML is one dimension less than that of the domain so that the proposed ML algorithm can be implemented efficiently. The numerical experiments demonstrate that the SGFEM with the ELM enrichment achieves the optimal convergence rates for the IP, as predicted theoretically.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 143-159"},"PeriodicalIF":2.2,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824352","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 unified space-time finite element scheme for a quasilinear parabolic problem","authors":"I. Toulopoulos","doi":"10.1016/j.apnum.2025.03.006","DOIUrl":"10.1016/j.apnum.2025.03.006","url":null,"abstract":"<div><div>A new approach is presented to obtain stabilized space - time finite element schemes for solving in a unified space-time way a quasilinear parabolic model problem. The procedure consists in introducing first upwind diffusion terms with an appropriate scaling factor in the initial space-time finite element scheme. Then additional interface jump terms are introduced for ensuring the consistency of the final finite element discetzation. A discretization error analysis is presented and a priori error estimates in an appropriate discrete norm are shown. The corresponding convergence rates are optimal with respect to the regularity of the solution and are confirmed through a series of numerical tests.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 127-142"},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768201","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}