Pub Date : 2025-08-26DOI: 10.1016/j.apnum.2025.08.008
Tao Guo , Yiqun Li , Wenlin Qiu
We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson’s rule and the six-point Simpson’s formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.
{"title":"Compact difference method for Euler-Bernoulli beams and plates with nonlinear nonlocal damping","authors":"Tao Guo , Yiqun Li , Wenlin Qiu","doi":"10.1016/j.apnum.2025.08.008","DOIUrl":"10.1016/j.apnum.2025.08.008","url":null,"abstract":"<div><div>We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson’s rule and the six-point Simpson’s formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 1-18"},"PeriodicalIF":2.4,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926147","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 : 2025-08-16DOI: 10.1016/j.apnum.2025.08.005
Wenbin Li, Tinggang Zhao, Zhenyu Zhao
The biharmonic equation is commonly encountered in various fields such as elasticity theory, fluid dynamics, and image processing. Solving it on irregular domain presents a significant challenge. In this paper, Fourier extension method is used to solve the biharmonic equation on arbitrary domain. The method involves the oversampling collocation technique with the truncated singular value decomposition regularization, which comes out a spectral convergence rate for the smooth solution. This method only uses the function values on equidistant nodes and has the characteristics of less computation, strong universality and better accuracy. The effectiveness of the proposed method is demonstrated by a variety of numerical experiments.
{"title":"A biharmonic solver based on Fourier extension with oversampling technique for arbitrary domain","authors":"Wenbin Li, Tinggang Zhao, Zhenyu Zhao","doi":"10.1016/j.apnum.2025.08.005","DOIUrl":"10.1016/j.apnum.2025.08.005","url":null,"abstract":"<div><div>The biharmonic equation is commonly encountered in various fields such as elasticity theory, fluid dynamics, and image processing. Solving it on irregular domain presents a significant challenge. In this paper, Fourier extension method is used to solve the biharmonic equation on arbitrary domain. The method involves the oversampling collocation technique with the truncated singular value decomposition regularization, which comes out a spectral convergence rate for the smooth solution. This method only uses the function values on equidistant nodes and has the characteristics of less computation, strong universality and better accuracy. The effectiveness of the proposed method is demonstrated by a variety of numerical experiments.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 261-274"},"PeriodicalIF":2.4,"publicationDate":"2025-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144888764","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 : 2025-08-15DOI: 10.1016/j.apnum.2025.08.004
Mojtaba Fardi , Mahmoud A. Zaky , Babak Azarnavid
This study presents an accurate meshless method for the efficient solution of nonlinear time-fractional convection-diffusion systems in complex two- and three-dimensional geometries. The proposed approach combines spatial discretization using a Bernoulli polynomial kernel function with temporal discretization via the backward differentiation formula. By employing positive definite kernels, the method achieves high spatial accuracy, while the use of the backward differentiation formula ensures high-order temporal accuracy. Convergence conditions and error bounds are rigorously analyzed using the Mittag-Leffler function. Error estimates are derived based on the spectral properties of the associated matrices, and inequalities describing error propagation over time are established. The method is tested on a variety of benchmark problems, including the Brusselator model and nonlinear coupled convection-diffusion systems, across both 2D and 3D domains. Extensive numerical experiments are carried out on various geometries-such as rectangular, circular, and spherical shapes-demonstrating the method’s robustness and accuracy in handling both regular and irregular computational domains.
{"title":"Fractional convection-diffusion systems in complex 2D and 3D geometries: A Bernoulli polynomial-based kernel method","authors":"Mojtaba Fardi , Mahmoud A. Zaky , Babak Azarnavid","doi":"10.1016/j.apnum.2025.08.004","DOIUrl":"10.1016/j.apnum.2025.08.004","url":null,"abstract":"<div><div>This study presents an accurate meshless method for the efficient solution of nonlinear time-fractional convection-diffusion systems in complex two- and three-dimensional geometries. The proposed approach combines spatial discretization using a Bernoulli polynomial kernel function with temporal discretization via the backward differentiation formula. By employing positive definite kernels, the method achieves high spatial accuracy, while the use of the backward differentiation formula ensures high-order temporal accuracy. Convergence conditions and error bounds are rigorously analyzed using the Mittag-Leffler function. Error estimates are derived based on the spectral properties of the associated matrices, and inequalities describing error propagation over time are established. The method is tested on a variety of benchmark problems, including the Brusselator model and nonlinear coupled convection-diffusion systems, across both 2D and 3D domains. Extensive numerical experiments are carried out on various geometries-such as rectangular, circular, and spherical shapes-demonstrating the method’s robustness and accuracy in handling both regular and irregular computational domains.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 275-297"},"PeriodicalIF":2.4,"publicationDate":"2025-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896067","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}
This study presents an a posteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator accurately captured control approximation errors, it had limitations in terms of guiding refinement localization in critical circumstances. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach’s effectiveness in achieving accurate solutions while optimizing computational efficiency. Numerical experiments showcase the effectiveness of the derived error estimators.
{"title":"Adaptive SIPG method for approximations of parabolic boundary control problems with bilateral box constraints on Neumann boundary","authors":"Ram Manohar , B․ V․ Rathish Kumar , Kedarnath Buda , Rajen Kumar Sinha","doi":"10.1016/j.apnum.2025.08.002","DOIUrl":"10.1016/j.apnum.2025.08.002","url":null,"abstract":"<div><div>This study presents an a posteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator accurately captured control approximation errors, it had limitations in terms of guiding refinement localization in critical circumstances. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach’s effectiveness in achieving accurate solutions while optimizing computational efficiency. Numerical experiments showcase the effectiveness of the derived error estimators.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 170-201"},"PeriodicalIF":2.4,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106357","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 : 2025-08-11DOI: 10.1016/j.apnum.2025.08.003
Ramon Codina , Hauke Gravenkamp , Sheraz Ahmed Khan
In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.
{"title":"A posteriori error estimates for the finite element approximation of the convection–diffusion–reaction equation based on the variational multiscale concept","authors":"Ramon Codina , Hauke Gravenkamp , Sheraz Ahmed Khan","doi":"10.1016/j.apnum.2025.08.003","DOIUrl":"10.1016/j.apnum.2025.08.003","url":null,"abstract":"<div><div>In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 238-260"},"PeriodicalIF":2.4,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886791","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 : 2025-08-11DOI: 10.1016/j.apnum.2025.07.015
Kai Zhu, Min Zhong
This paper presents a Kaczmarz type two-point gradient algorithm with the general convex penalty functional (KTPG-), for efficient reconstruction of conductivity in acousto-electric tomography (AET). The algorithm optimizes a convex functional with flexible non-smooth regularization terms, such as -like and total variation-like, to handle sparse and discontinuous conductivity distributions. By cyclically processing the measurement equations and incorporating an acceleration strategy, the proposed method achieves high computational efficiency while ensuring convergence. Numerical experiments on both synthetic and realistic phantoms demonstrate the method’s superior accuracy, strong noise robustness, and ability to resolve fine details. Beyond AET, the KTPG- framework can be applied to a wide range of nonlinear inverse problems involving systems of equations, showcasing its potential for broader applications in science and engineering.
{"title":"2D and 3D reconstructions in acousto-electric tomography via two-point gradient Kaczmarz-type algorithm","authors":"Kai Zhu, Min Zhong","doi":"10.1016/j.apnum.2025.07.015","DOIUrl":"10.1016/j.apnum.2025.07.015","url":null,"abstract":"<div><div>This paper presents a Kaczmarz type two-point gradient algorithm with the general convex penalty functional <span><math><mstyle><mi>Θ</mi></mstyle></math></span> (KTPG-<span><math><mstyle><mi>Θ</mi></mstyle></math></span>), for efficient reconstruction of conductivity in acousto-electric tomography (AET). The algorithm optimizes a convex functional with flexible non-smooth regularization terms, such as <span><math><msup><mi>L</mi><mn>1</mn></msup></math></span>-like and total variation-like, to handle sparse and discontinuous conductivity distributions. By cyclically processing the measurement equations and incorporating an acceleration strategy, the proposed method achieves high computational efficiency while ensuring convergence. Numerical experiments on both synthetic and realistic phantoms demonstrate the method’s superior accuracy, strong noise robustness, and ability to resolve fine details. Beyond AET, the KTPG-<span><math><mstyle><mi>Θ</mi></mstyle></math></span> framework can be applied to a wide range of nonlinear inverse problems involving systems of equations, showcasing its potential for broader applications in science and engineering.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 220-237"},"PeriodicalIF":2.4,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886790","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 : 2025-08-06DOI: 10.1016/j.apnum.2025.08.001
Yongli Hou , Yi Liu , Yanqiu Wang
We propose a boundary-corrected weak Galerkin mixed finite element method for solving elliptic interface problems in 2D domains with curved interfaces. The method is formulated on body-fitted polygonal meshes, where interface edges are straight and may not align exactly with the curved physical interface. To address this discrepancy, a boundary value correction technique is employed to transfer the interface conditions from the physical interface to the approximate interface using a Taylor expansion approach. The Neumann interface condition is then weakly imposed in the variational formulation. This approach eliminates the need for numerical integration on curved elements, thereby reducing implementation complexity. We establish optimal-order convergence in the energy norm for arbitrary-order discretizations. Numerical results are provided to support the theoretical findings.
{"title":"A boundary-corrected weak Galerkin mixed finite method for elliptic interface problems with curved interfaces","authors":"Yongli Hou , Yi Liu , Yanqiu Wang","doi":"10.1016/j.apnum.2025.08.001","DOIUrl":"10.1016/j.apnum.2025.08.001","url":null,"abstract":"<div><div>We propose a boundary-corrected weak Galerkin mixed finite element method for solving elliptic interface problems in 2D domains with curved interfaces. The method is formulated on body-fitted polygonal meshes, where interface edges are straight and may not align exactly with the curved physical interface. To address this discrepancy, a boundary value correction technique is employed to transfer the interface conditions from the physical interface to the approximate interface using a Taylor expansion approach. The Neumann interface condition is then weakly imposed in the variational formulation. This approach eliminates the need for numerical integration on curved elements, thereby reducing implementation complexity. We establish optimal-order convergence in the energy norm for arbitrary-order discretizations. Numerical results are provided to support the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 201-219"},"PeriodicalIF":2.4,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144842654","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 : 2025-07-29DOI: 10.1016/j.apnum.2025.07.014
C. Caballero-Cárdenas , M.J. Castro , C. Chalons , T. Morales de Luna , M.L. Muñoz-Ruiz
This work addresses the design of semi-implicit numerical schemes that are fully exactly well-balanced for the two-layer shallow water system, meaning that they are capable of preserving every possible steady state, and not only the lake-at-rest ones. The proposed approach exhibits better performance compared to standard explicit methods in low-Froude number regimes, where wave propagation speeds significantly exceed flow velocities, thereby reducing the computational cost associated with long-time simulations. The methodology relies on a combination of splitting strategies and relaxation techniques to construct first- and second-order semi-implicit schemes that satisfy the fully exactly well-balanced property.
{"title":"Semi-implicit fully exactly well-balanced schemes for the two-layer shallow water system","authors":"C. Caballero-Cárdenas , M.J. Castro , C. Chalons , T. Morales de Luna , M.L. Muñoz-Ruiz","doi":"10.1016/j.apnum.2025.07.014","DOIUrl":"10.1016/j.apnum.2025.07.014","url":null,"abstract":"<div><div>This work addresses the design of semi-implicit numerical schemes that are fully exactly well-balanced for the two-layer shallow water system, meaning that they are capable of preserving every possible steady state, and not only the lake-at-rest ones. The proposed approach exhibits better performance compared to standard explicit methods in low-Froude number regimes, where wave propagation speeds significantly exceed flow velocities, thereby reducing the computational cost associated with long-time simulations. The methodology relies on a combination of splitting strategies and relaxation techniques to construct first- and second-order semi-implicit schemes that satisfy the fully exactly well-balanced property.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 128-147"},"PeriodicalIF":2.4,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144720945","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 : 2025-07-24DOI: 10.1016/j.apnum.2025.07.013
Yiying Fang , Ying Jiang , Jiafeng Su
We propose a fast Fourier–Galerkin method for solving boundary integral equations (BIEs) on smooth, non-axisymmetric toroidal surfaces. Our approach begins by analyzing the structure of the integral kernel, revealing an exponential decay pattern in the Fourier coefficients after a shear transformation. Leveraging this decay, we design a truncation strategy that compresses the dense representation matrix into a sparse form with only nonzero entries, where N denotes the degrees of freedom. We rigorously prove that the truncated system retains the stability of the original Fourier–Galerkin formulation and achieves a quasi-optimal convergence rate of , with p denoting the regularity of the exact solution. Numerical experiments corroborate our theoretical results, demonstrating both high accuracy and computational efficiency. Furthermore, we extend the proposed strategy to BIEs defined on surfaces diffeomorphic to the sphere, confirming the sparsity structure remains exploitable under broader geometric settings.
{"title":"A fast Fourier-Galerkin method for solving boundary integral equations on non-axisymmetric toroidal surfaces","authors":"Yiying Fang , Ying Jiang , Jiafeng Su","doi":"10.1016/j.apnum.2025.07.013","DOIUrl":"10.1016/j.apnum.2025.07.013","url":null,"abstract":"<div><div>We propose a fast Fourier–Galerkin method for solving boundary integral equations (BIEs) on smooth, non-axisymmetric toroidal surfaces. Our approach begins by analyzing the structure of the integral kernel, revealing an exponential decay pattern in the Fourier coefficients after a shear transformation. Leveraging this decay, we design a truncation strategy that compresses the dense representation matrix into a sparse form with only <span><math><mi>O</mi><mo>(</mo><mi>N</mi><msup><mrow><mi>ln</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>N</mi><mo>)</mo></math></span> nonzero entries, where <em>N</em> denotes the degrees of freedom. We rigorously prove that the truncated system retains the stability of the original Fourier–Galerkin formulation and achieves a quasi-optimal convergence rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>ln</mi><mo></mo><mi>N</mi><mo>)</mo></math></span>, with <em>p</em> denoting the regularity of the exact solution. Numerical experiments corroborate our theoretical results, demonstrating both high accuracy and computational efficiency. Furthermore, we extend the proposed strategy to BIEs defined on surfaces diffeomorphic to the sphere, confirming the sparsity structure remains exploitable under broader geometric settings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 73-90"},"PeriodicalIF":2.4,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144720943","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 : 2025-07-23DOI: 10.1016/j.apnum.2025.07.012
Andrés Arrarás , Francisco J. Gaspar , Iñigo Jimenez-Ciga , Laura Portero
In recent years, parallelization has become a strong tool to avoid the limits of classical sequential computing. In the present paper, we introduce four space-time parallel methods that combine the parareal algorithm with suitable splitting techniques for the numerical solution of reaction-diffusion problems. In particular, we consider a suitable partition of the elliptic operator that enables the parallelization in space by using splitting time integrators. Those schemes are then chosen as the propagators of the parareal algorithm, a well-known parallel-in-time method. Both first- and second-order time integrators are considered for this task. The resulting space-time parallel methods are applied to integrate reaction-diffusion problems that model Turing pattern formation. This phenomenon appears in chemical reactions due to diffusion-driven instabilities, and rules the pattern formation for animal coat markings. Such reaction-diffusion problems require fine space and time meshes for their numerical integration, so we illustrate the usefulness of the proposed methods by solving several models of practical interest.
{"title":"Space-time parallel solvers for reaction-diffusion problems forming Turing patterns","authors":"Andrés Arrarás , Francisco J. Gaspar , Iñigo Jimenez-Ciga , Laura Portero","doi":"10.1016/j.apnum.2025.07.012","DOIUrl":"10.1016/j.apnum.2025.07.012","url":null,"abstract":"<div><div>In recent years, parallelization has become a strong tool to avoid the limits of classical sequential computing. In the present paper, we introduce four space-time parallel methods that combine the parareal algorithm with suitable splitting techniques for the numerical solution of reaction-diffusion problems. In particular, we consider a suitable partition of the elliptic operator that enables the parallelization in space by using splitting time integrators. Those schemes are then chosen as the propagators of the parareal algorithm, a well-known parallel-in-time method. Both first- and second-order time integrators are considered for this task. The resulting space-time parallel methods are applied to integrate reaction-diffusion problems that model Turing pattern formation. This phenomenon appears in chemical reactions due to diffusion-driven instabilities, and rules the pattern formation for animal coat markings. Such reaction-diffusion problems require fine space and time meshes for their numerical integration, so we illustrate the usefulness of the proposed methods by solving several models of practical interest.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 91-108"},"PeriodicalIF":2.4,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144720946","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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