Pub Date : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.apnum.2025.08.006
Lan Wang , Yiyang Luo , Meng Chen , Pengfei Zhu
An efficient fourth-order numerical scheme is developed for the Gross-Pitaevskii equation. The spatial direction is approximated by a fourth-order compact scheme and the temporal direction is discretized by a fourth-order splitting & composition method. This scheme not only preserves the symplectic structure and the discrete mass conservation law exactly but also maintains the discrete energy conservation law in some special case. Some numerical experiments confirm our theoretical expectation.
{"title":"A highly accurate symplectic-preserving scheme for Gross-Pitaevskii equation","authors":"Lan Wang , Yiyang Luo , Meng Chen , Pengfei Zhu","doi":"10.1016/j.apnum.2025.08.006","DOIUrl":"10.1016/j.apnum.2025.08.006","url":null,"abstract":"<div><div>An efficient fourth-order numerical scheme is developed for the Gross-Pitaevskii equation. The spatial direction is approximated by a fourth-order compact scheme and the temporal direction is discretized by a fourth-order splitting & composition method. This scheme not only preserves the symplectic structure and the discrete mass conservation law exactly but also maintains the discrete energy conservation law in some special case. Some numerical experiments confirm our theoretical expectation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 41-52"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145044374","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":"2026-01-01","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 : 2026-01-01Epub Date: 2025-09-12DOI: 10.1016/j.apnum.2025.09.006
Jialing Wang , Anxin Kong , Tingchun Wang , Wenjun Cai
This work presents two implicit and linear finite difference schemes that simultaneously preserve both mass and energy conservation properties for the two-dimensional Schrödinger–Poisson equations. The conservation, existence, uniqueness, as well as the convergence to the exact solution with the order in discrete and norms are established for these two schemes, where and represent temporal and spatial step sizes. In contrast to the existing analysis techniques that rely on an a priori estimate of numerical solutions or impose restrictions on initial data, our approaches guarantee the unconditional convergence for SP equations with both attractive and repulsive forces. Besides the standard energy method, our analytical framework employs the cut-off method for the implicit scheme and the mathematical induction argument for the linear scheme, where the “lifting” technique is utilized in the two schemes to eliminate the constraints on grid ratios. Numerical experiments are provided to illustrate discrete conservation properties and validate the achieved convergence results.
{"title":"Point-wise error estimates of two mass- and energy-preserving schemes for two-dimensional Schrödinger–Poisson equations","authors":"Jialing Wang , Anxin Kong , Tingchun Wang , Wenjun Cai","doi":"10.1016/j.apnum.2025.09.006","DOIUrl":"10.1016/j.apnum.2025.09.006","url":null,"abstract":"<div><div>This work presents two implicit and linear finite difference schemes that simultaneously preserve both mass and energy conservation properties for the two-dimensional Schrödinger–Poisson equations. The conservation, existence, uniqueness, as well as the convergence to the exact solution with the order <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>h</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>h</mi><mi>y</mi><mn>2</mn></msubsup><mo>)</mo></mrow></math></span> in discrete <span><math><msup><mi>L</mi><mn>2</mn></msup></math></span> and <span><math><msup><mi>L</mi><mi>∞</mi></msup></math></span> norms are established for these two schemes, where <span><math><mi>τ</mi></math></span> and <span><math><mrow><msub><mi>h</mi><mi>x</mi></msub><mo>,</mo><msub><mi>h</mi><mi>y</mi></msub></mrow></math></span> represent temporal and spatial step sizes. In contrast to the existing analysis techniques that rely on an <em>a priori</em> <span><math><msup><mi>L</mi><mi>∞</mi></msup></math></span> estimate of numerical solutions or impose restrictions on initial data, our approaches guarantee the unconditional convergence for SP equations with both attractive and repulsive forces. Besides the standard energy method, our analytical framework employs the cut-off method for the implicit scheme and the mathematical induction argument for the linear scheme, where the “lifting” technique is utilized in the two schemes to eliminate the constraints on grid ratios. Numerical experiments are provided to illustrate discrete conservation properties and validate the achieved convergence results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"219 ","pages":"Pages 202-218"},"PeriodicalIF":2.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145106358","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 : 2025-12-01Epub 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":"2025-12-01","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}
Pub Date : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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}
Pub Date : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub Date: 2025-07-14DOI: 10.1016/j.apnum.2025.07.005
Kwanghyuk Park , Xinjuan Chen , Dongjin Lee , Jiaxi Gu , Jae-Hun Jung
In this work, we develop the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. Supervised learning is employed with the training data consisting of three-point stencils and the corresponding WENO3-JS weights as labels. We design two loss functions, one built on the mean squared error and the other from the mean squared logarithmic error. Each loss function consists of two components, where the first enforces the model to maintain the essentially non-oscillatory behavior while the second reduces the dissipation around discontinuities and improves the performance in smooth regions. We choose the shallow neural network (SNN) for computational efficiency with the Delta layer pre-processing the input. The resulting WENO3-SNN schemes outperform the classical WENO3-JS and WENO3-Z in one-dimensional examples, and show comparable sometimes superior simulations to WENO3-JS and WENO3-Z in two-dimensional examples.
{"title":"A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network","authors":"Kwanghyuk Park , Xinjuan Chen , Dongjin Lee , Jiaxi Gu , Jae-Hun Jung","doi":"10.1016/j.apnum.2025.07.005","DOIUrl":"10.1016/j.apnum.2025.07.005","url":null,"abstract":"<div><div>In this work, we develop the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. Supervised learning is employed with the training data consisting of three-point stencils and the corresponding WENO3-JS weights as labels. We design two loss functions, one built on the mean squared error and the other from the mean squared logarithmic error. Each loss function consists of two components, where the first enforces the model to maintain the essentially non-oscillatory behavior while the second reduces the dissipation around discontinuities and improves the performance in smooth regions. We choose the shallow neural network (SNN) for computational efficiency with the Delta layer pre-processing the input. The resulting WENO3-SNN schemes outperform the classical WENO3-JS and WENO3-Z in one-dimensional examples, and show comparable sometimes superior simulations to WENO3-JS and WENO3-Z in two-dimensional examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 1-21"},"PeriodicalIF":2.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144656379","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 : 2025-12-01Epub Date: 2025-07-22DOI: 10.1016/j.apnum.2025.07.010
Jiajia Pan , Huiyuan Li
Lower bound approximation and super-convergence of the weak Galerkin spectral element method for second-order elliptic eigenvalue problems are comprehensively investigated in this paper. At first, we establish the approximation spaces with diverse polynomial degrees of weak functions and weak gradients by using the one-to-one mapping from the reference element to each physical element. General weak Galerkin triangular/quadrilateral spectral element approximation schemes are then proposed for the eigenvalue problem of the second-order elliptic operators. A study on the well-posedness of our schemes is carried out, resulting in the constraint conditions on the polynomial degrees of the discrete weak function space and the discrete weak gradient space. Further, qualitative numerical analysis and numerical investigation are performed on a series of polynomial degree configurations for the weak function space and the weak gradient space. We obtain in the sequel the super-convergence of the numerical eigenvalues with the weak Galerkin spectral element methods for the first time, and discover some lower bound approximation scenario that has never been reported before in literature.
{"title":"Weak Galerkin spectral element methods for elliptic eigenvalue problems: Lower bound approximation and superconvergence","authors":"Jiajia Pan , Huiyuan Li","doi":"10.1016/j.apnum.2025.07.010","DOIUrl":"10.1016/j.apnum.2025.07.010","url":null,"abstract":"<div><div>Lower bound approximation and super-convergence of the weak Galerkin spectral element method for second-order elliptic eigenvalue problems are comprehensively investigated in this paper. At first, we establish the approximation spaces with diverse polynomial degrees of weak functions and weak gradients by using the one-to-one mapping from the reference element to each physical element. General weak Galerkin triangular/quadrilateral spectral element approximation schemes are then proposed for the eigenvalue problem of the second-order elliptic operators. A study on the well-posedness of our schemes is carried out, resulting in the constraint conditions on the polynomial degrees of the discrete weak function space and the discrete weak gradient space. Further, qualitative numerical analysis and numerical investigation are performed on a series of polynomial degree configurations for the weak function space and the weak gradient space. We obtain in the sequel the super-convergence of the numerical eigenvalues with the weak Galerkin spectral element methods for the first time, and discover some lower bound approximation scenario that has never been reported before in literature.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 182-200"},"PeriodicalIF":2.4,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831172","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号