Pub Date : 2025-07-08DOI: 10.1016/j.apnum.2025.07.003
Alicia Cordero , Renso V. Rojas-Hiciano , Juan R. Torregrosa , Maria P. Vassileva
We present an efficient strategy to increase, under certain conditions, the order of convergence of iterative methods to solve nonlinear systems of equations. We analytically compare the new accelerator with others and establish the conditions under which this technique is more efficient. We perform an analysis of the efficiency of some one-step accelerators that increase the convergence order by two units. New concepts about efficiency are introduced which allow us to compare different iterative schemes from other points of view. We demonstrate that our proposal is a good alternative to the existing ones. As a consequence, we propose two new maximally efficient, damped Newton-Traub type schemes of order 5 and 6. These are an improvement of two other maximally efficient methods. Their numerical performance is better than that of known methods of the same order, and we find that it is a very economical way to achieve high order. Some numerical examples confirm the theoretical results.
{"title":"High-level convergence order accelerators of iterative methods for nonlinear problems","authors":"Alicia Cordero , Renso V. Rojas-Hiciano , Juan R. Torregrosa , Maria P. Vassileva","doi":"10.1016/j.apnum.2025.07.003","DOIUrl":"10.1016/j.apnum.2025.07.003","url":null,"abstract":"<div><div>We present an efficient strategy to increase, under certain conditions, the order of convergence of iterative methods to solve nonlinear systems of equations. We analytically compare the new accelerator with others and establish the conditions under which this technique is more efficient. We perform an analysis of the efficiency of some one-step accelerators that increase the convergence order by two units. New concepts about efficiency are introduced which allow us to compare different iterative schemes from other points of view. We demonstrate that our proposal is a good alternative to the existing ones. As a consequence, we propose two new maximally efficient, damped Newton-Traub type schemes of order 5 and 6. These are an improvement of two other maximally efficient methods. Their numerical performance is better than that of known methods of the same order, and we find that it is a very economical way to achieve high order. Some numerical examples confirm the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 390-411"},"PeriodicalIF":2.2,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580373","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}
Pub Date : 2025-07-05DOI: 10.1016/j.apnum.2025.06.017
I. Higueras , J.I. Montijano , L. Rández
In this paper, we study explicit Runge-Kutta (RK) methods for solving high-dimensional systems of ordinary differential equations (ODEs), with oscillatory or periodic solutions, that can be implemented with a few memory registers. We will refer to these schemes as Low-Storage Exponentially Fitted explicit Runge-Kutta methods (LSEFRK).
In order to obtain them, we first study second-order and third-order low-storage (LS) schemes that can be implemented with two memory registers per step of the van der Houwen- and Williamson-type. Next, we construct optimal LSEFRK methods by imposing exponential fitting conditions along with accuracy and stability properties. In this way, new optimal three-stage third-order and five-stage fourth-order LSEFRK schemes are constructed for each type of LS method.
The performance of these new schemes is tested by solving some high-dimensional differential systems with periodic solutions. Comparison with other non-LS exponentially fitted and low-storage non-EF RK methods from the literature shows that the new LSEFRK schemes outperform the efficiency of RK methods that only satisfy either the LS or the EF condition.
本文研究了求解具有振荡或周期解的高维常微分方程系统的显式龙格-库塔（RK）方法，这种方法可以用少量内存寄存器实现。我们将这些方案称为低存储指数拟合显式龙格-库塔方法（LSEFRK）。为了获得它们，我们首先研究了二阶和三阶低存储（LS）方案，这些方案可以用van der Houwen和williamson型的每步两个存储寄存器来实现。接下来，我们通过施加指数拟合条件以及精度和稳定性来构造最优LSEFRK方法。在此基础上，针对每种LS方法构造了新的最优三阶和五阶LSEFRK方案。通过求解一些具有周期解的高维微分系统，验证了这些新格式的性能。与文献中其他非LS指数拟合和低存储非EFRK方法的比较表明，新的LSEFRK方案优于仅满足LS或EF条件的RK方法的效率。
{"title":"Low-storage exponentially fitted explicit Runge-Kutta methods","authors":"I. Higueras , J.I. Montijano , L. Rández","doi":"10.1016/j.apnum.2025.06.017","DOIUrl":"10.1016/j.apnum.2025.06.017","url":null,"abstract":"<div><div>In this paper, we study explicit Runge-Kutta (RK) methods for solving high-dimensional systems of ordinary differential equations (ODEs), with oscillatory or periodic solutions, that can be implemented with a few memory registers. We will refer to these schemes as Low-Storage Exponentially Fitted explicit Runge-Kutta methods (LSEFRK).</div><div>In order to obtain them, we first study second-order and third-order low-storage (LS) schemes that can be implemented with two memory registers per step of the van der Houwen- and Williamson-type. Next, we construct optimal LSEFRK methods by imposing exponential fitting conditions along with accuracy and stability properties. In this way, new optimal three-stage third-order and five-stage fourth-order LSEFRK schemes are constructed for each type of LS method.</div><div>The performance of these new schemes is tested by solving some high-dimensional differential systems with periodic solutions. Comparison with other non-LS exponentially fitted and low-storage non-EF RK methods from the literature shows that the new LSEFRK schemes outperform the efficiency of RK methods that only satisfy either the LS or the EF condition.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 372-389"},"PeriodicalIF":2.2,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580372","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}
Pub Date : 2025-07-05DOI: 10.1016/j.apnum.2025.06.018
Jiaxiang Cai , Yushun Wang
We propose a class of high-order schemes preserving original conservation/dissipation energy law and constraints for the constrained conservative/dissipative system. These schemes are efficient, i.e., only require solving linear system with constant coefficients at each time step, plus an algebraic optimization problem which consumes negligible cost. The proposed schemes are applied to conservative semiclassical nonlinear Schrödinger equation, as well as dissipative three-component ternary Cahn-Hilliard phase-field model. Some numerical experiments are conducted to validate applicability, effectiveness and accuracy of the proposed schemes.
{"title":"High-order structure-preserving approaches for constrained conservative or dissipative systems","authors":"Jiaxiang Cai , Yushun Wang","doi":"10.1016/j.apnum.2025.06.018","DOIUrl":"10.1016/j.apnum.2025.06.018","url":null,"abstract":"<div><div>We propose a class of high-order schemes preserving original conservation/dissipation energy law and constraints for the constrained conservative/dissipative system. These schemes are efficient, i.e., only require solving linear system with constant coefficients at each time step, plus an algebraic optimization problem which consumes negligible cost. The proposed schemes are applied to conservative semiclassical nonlinear Schrödinger equation, as well as dissipative three-component ternary Cahn-Hilliard phase-field model. Some numerical experiments are conducted to validate applicability, effectiveness and accuracy of the proposed schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 340-354"},"PeriodicalIF":2.2,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570253","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}
Pub Date : 2025-07-04DOI: 10.1016/j.apnum.2025.06.016
M.H. Heydari , D. Baleanu , M. Bayram
This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.
{"title":"Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel","authors":"M.H. Heydari , D. Baleanu , M. Bayram","doi":"10.1016/j.apnum.2025.06.016","DOIUrl":"10.1016/j.apnum.2025.06.016","url":null,"abstract":"<div><div>This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 355-371"},"PeriodicalIF":2.2,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570254","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}
Pub Date : 2025-07-03DOI: 10.1016/j.apnum.2025.06.015
Vincenzo Schiano Di Cola , Salvatore Cuomo , Gerardo Severino , Marco Berardi
We consider steady flow generated by a source through a porous medium where, due to its erratic variations in the space, the conductivity K is regarded as a random field. As a consequence, flow variables become stochastic, and we aim at quantifying their uncertainty. To this purpose, we use Monte Carlo simulations, where for each realization the governing flow equation is solved by a finite volume method. This yields a deterministic linear system solved by algebraic multigrid (AMG) techniques. By leveraging analytical solutions valid for homogeneous (constant K) formations, we first compare different AMG solvers, that are subsequently used as trial in order to extend our approach to heterogeneous porous media. Results demonstrate that AMG methods enable achieving, especially at higher iteration counts, an -error lower than other, Gaussian-type, approximations.
{"title":"Algebraic multigrid methods for uncertainty quantification of source-type flows through randomly heterogeneous porous media","authors":"Vincenzo Schiano Di Cola , Salvatore Cuomo , Gerardo Severino , Marco Berardi","doi":"10.1016/j.apnum.2025.06.015","DOIUrl":"10.1016/j.apnum.2025.06.015","url":null,"abstract":"<div><div>We consider steady flow generated by a source through a porous medium where, due to its erratic variations in the space, the conductivity <em>K</em> is regarded as a random field. As a consequence, flow variables become stochastic, and we aim at quantifying their uncertainty. To this purpose, we use Monte Carlo simulations, where for each realization the governing flow equation is solved by a finite volume method. This yields a deterministic linear system solved by algebraic multigrid (AMG) techniques. By leveraging analytical solutions valid for homogeneous (constant <em>K</em>) formations, we first compare different AMG solvers, that are subsequently used as trial in order to extend our approach to heterogeneous porous media. Results demonstrate that AMG methods enable achieving, especially at higher iteration counts, an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-error lower than other, Gaussian-type, approximations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 58-72"},"PeriodicalIF":2.2,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694379","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}
Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct -continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree l, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.
{"title":"A surface mesh DG-VEM for elliptic membrane shell model","authors":"Qian Yang , Xiaoqin Shen , Jikun Zhao , Zhiming Gao","doi":"10.1016/j.apnum.2025.06.014","DOIUrl":"10.1016/j.apnum.2025.06.014","url":null,"abstract":"<div><div>Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree <em>l</em>, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 298-318"},"PeriodicalIF":2.2,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517713","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}
Pub Date : 2025-06-26DOI: 10.1016/j.apnum.2025.06.010
Jiaying Feng, Changtao Sheng, Chenglong Xu
This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by α-stable Lévy processes with , which was initially proposed and developed by Gobet and Maire in their pioneering works ((2004) [24], and (2005) [25]) for the case . The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-spheres method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both and . Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.
{"title":"Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates","authors":"Jiaying Feng, Changtao Sheng, Chenglong Xu","doi":"10.1016/j.apnum.2025.06.010","DOIUrl":"10.1016/j.apnum.2025.06.010","url":null,"abstract":"<div><div>This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by <em>α</em>-stable Lévy processes with <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, which was initially proposed and developed by Gobet and Maire in their pioneering works ((2004) <span><span>[24]</span></span>, and (2005) <span><span>[25]</span></span>) for the case <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-spheres method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 278-297"},"PeriodicalIF":2.2,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489532","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}
Pub Date : 2025-06-26DOI: 10.1016/j.apnum.2025.06.013
Du Ouyang, Jichang Xiao, Xiaoqun Wang
We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) [1] for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size m, the generalization error under QMC methods exhibits a convergence rate of , where can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is . Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.
{"title":"Generalization error analysis of deep backward dynamic programming for solving nonlinear PDEs","authors":"Du Ouyang, Jichang Xiao, Xiaoqun Wang","doi":"10.1016/j.apnum.2025.06.013","DOIUrl":"10.1016/j.apnum.2025.06.013","url":null,"abstract":"<div><div>We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) <span><span>[1]</span></span> for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size <em>m</em>, the generalization error under QMC methods exhibits a convergence rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 319-339"},"PeriodicalIF":2.2,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522669","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}
Pub Date : 2025-06-25DOI: 10.1016/j.apnum.2025.06.009
Hongwei Li, Xinyue Chen
Numerically solving the fluid-structure model on unbounded domains poses a challenge, due to the unbounded nature of the physical domain. To overcome this challenge, the artificial boundary method is specifically applied to numerically solve the fluid-structure model on unbounded domains, which can be used to analyze fluid-structure interactions in various scientific and engineering fields. Drawing inspiration from the artificial boundary method, we employ artificial boundaries to truncate the unbounded domain, subsequently designing the high order local artificial boundary conditions thereon based on the Padé approximation. Then, the initial value problem on the unbounded domain is reduced into an initial boundary value problem on the computational domain, which can be efficiently solved by adopting the finite difference method. Furthermore, a series of auxiliary variables is introduced specifically to address the issue of mixed derivatives arising in the artificial boundary conditions, and the stability, convergence and solvability of the reduced problem are rigorously analyzed. Numerical experiments are reported to demonstrate the effectiveness of artificial boundary conditions and theoretical analysis.
{"title":"Numerical method and analysis for fluid-structure model on unbounded domains","authors":"Hongwei Li, Xinyue Chen","doi":"10.1016/j.apnum.2025.06.009","DOIUrl":"10.1016/j.apnum.2025.06.009","url":null,"abstract":"<div><div>Numerically solving the fluid-structure model on unbounded domains poses a challenge, due to the unbounded nature of the physical domain. To overcome this challenge, the artificial boundary method is specifically applied to numerically solve the fluid-structure model on unbounded domains, which can be used to analyze fluid-structure interactions in various scientific and engineering fields. Drawing inspiration from the artificial boundary method, we employ artificial boundaries to truncate the unbounded domain, subsequently designing the high order local artificial boundary conditions thereon based on the Padé approximation. Then, the initial value problem on the unbounded domain is reduced into an initial boundary value problem on the computational domain, which can be efficiently solved by adopting the finite difference method. Furthermore, a series of auxiliary variables is introduced specifically to address the issue of mixed derivatives arising in the artificial boundary conditions, and the stability, convergence and solvability of the reduced problem are rigorously analyzed. Numerical experiments are reported to demonstrate the effectiveness of artificial boundary conditions and theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 255-277"},"PeriodicalIF":2.2,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489531","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}
Pub Date : 2025-06-23DOI: 10.1016/j.apnum.2025.06.011
Christophe Charlier , Daniel Eriksson , Jonatan Lenells
We present a numerical scheme for the solution of the initial-value problem for the “bad” Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.
{"title":"Numerical scheme for the solution of the “bad” Boussinesq equation","authors":"Christophe Charlier , Daniel Eriksson , Jonatan Lenells","doi":"10.1016/j.apnum.2025.06.011","DOIUrl":"10.1016/j.apnum.2025.06.011","url":null,"abstract":"<div><div>We present a numerical scheme for the solution of the initial-value problem for the “bad” Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 216-233"},"PeriodicalIF":2.2,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489529","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}
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