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}
Pub 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":"2025-07-22","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}
Pub Date : 2025-07-22DOI: 10.1016/j.apnum.2025.07.011
Xian Zhang, Ya Min, Minfu Feng
This paper presents a weak Galerkin (WG) finite element method based on the variational approach for data assimilation of the unsteady convection-dominated Oseen equation. The WG scheme uses piecewise polynomials of degrees k() and respectively for the approximations of the velocity and the pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and initial value. It is proved that the velocity error in the -norm has a Reynolds-robust error bound with quasi-optimal convergence order in the convection-dominated region. To solve the discrete optimality system efficiently, the conjugate gradient iterative algorithm is developed, which also preserves the globally divergence-free property of WG scheme. Numerical experiments are provided to verify the obtained theoretical results.
{"title":"Robust globally divergence-free weak Galerkin variational data assimilation method for convection-dominated Oseen equations","authors":"Xian Zhang, Ya Min, Minfu Feng","doi":"10.1016/j.apnum.2025.07.011","DOIUrl":"10.1016/j.apnum.2025.07.011","url":null,"abstract":"<div><div>This paper presents a weak Galerkin (WG) finite element method based on the variational approach for data assimilation of the unsteady convection-dominated Oseen equation. The WG scheme uses piecewise polynomials of degrees <em>k</em>(<span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>) and <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> respectively for the approximations of the velocity and the pressure in the interior of elements, and uses piecewise polynomials of degree <em>k</em> for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and initial value. It is proved that the velocity error in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm has a Reynolds-robust error bound with quasi-optimal convergence order <span><math><mi>k</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span> in the convection-dominated region. To solve the discrete optimality system efficiently, the conjugate gradient iterative algorithm is developed, which also preserves the globally divergence-free property of WG scheme. Numerical experiments are provided to verify the obtained theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 22-42"},"PeriodicalIF":2.2,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686025","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 : 2025-07-22DOI: 10.1016/j.apnum.2025.07.009
Victor A. Kovtunenko , Yves Renard
A class of elastodynamic contact problems for fluid-driven cracks stemming from hydro-fracking application is considered in the framework of finite element approximation. The dynamic contact problem aims at finding a non-negative fracture opening and a mean fluid pressure which are controlled by the volume of pumped fracturing fluid. Well-posedness of the fully discrete variational problem is proved rigorously by using the Lagrange multiplier and penalty methods for the minimization problem subjected to both: unilateral and non-local constraints. Numerical solution of the dynamic nonlinear equation is computed in 2D experiments using the semi-smooth Newton and the generalized Hilber–Hughes–Taylor α-method.
{"title":"FEM approximation of dynamic contact problem for fracture under fluid volume control using generalized HHT-α and semi-smooth Newton methods","authors":"Victor A. Kovtunenko , Yves Renard","doi":"10.1016/j.apnum.2025.07.009","DOIUrl":"10.1016/j.apnum.2025.07.009","url":null,"abstract":"<div><div>A class of elastodynamic contact problems for fluid-driven cracks stemming from hydro-fracking application is considered in the framework of finite element approximation. The dynamic contact problem aims at finding a non-negative fracture opening and a mean fluid pressure which are controlled by the volume of pumped fracturing fluid. Well-posedness of the fully discrete variational problem is proved rigorously by using the Lagrange multiplier and penalty methods for the minimization problem subjected to both: unilateral and non-local constraints. Numerical solution of the dynamic nonlinear equation is computed in 2D experiments using the semi-smooth Newton and the generalized Hilber–Hughes–Taylor <strong><em>α</em></strong>-method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 148-158"},"PeriodicalIF":2.4,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144750144","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 : 2025-07-18DOI: 10.1016/j.apnum.2025.07.007
Xinran Huang, Haiyan Su, Xinlong Feng
The mass-conservative finite element method (FEM) is considered for the micropolar Navier-Stokes equations (MNSE), which couple the Navier-Stokes equations (NSE) with the angular momentum equation. A fully divergence-free algorithm is proposed for the MNSE. The Raviart-Thomas element is employed for discretizing the velocity field, ensuring that its divergence-free property is maintained. Furthermore, the interior penalty discontinuous Galerkin (IPDG) method is utilized in order to guarantee the -continuity of velocity. Some implicit-explicit treatments are used to address the convection terms. We also provide energy stability proof and pressure robust error estimation for the proposed scheme. Finally, the accuracy and effectiveness of the proposed algorithm are validated through several 2D/3D numerical experiments.
{"title":"H(div)-conforming IPDG FEM with pointwise divergence-free velocity field for the micropolar Navier-Stokes equations","authors":"Xinran Huang, Haiyan Su, Xinlong Feng","doi":"10.1016/j.apnum.2025.07.007","DOIUrl":"10.1016/j.apnum.2025.07.007","url":null,"abstract":"<div><div>The mass-conservative finite element method (FEM) is considered for the micropolar Navier-Stokes equations (MNSE), which couple the Navier-Stokes equations (NSE) with the angular momentum equation. A fully divergence-free algorithm is proposed for the MNSE. The Raviart-Thomas element is employed for discretizing the velocity field, ensuring that its divergence-free property is maintained. Furthermore, the interior penalty discontinuous Galerkin (IPDG) method is utilized in order to guarantee the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-continuity of velocity. Some implicit-explicit treatments are used to address the convection terms. We also provide energy stability proof and pressure robust error estimation for the proposed scheme. Finally, the accuracy and effectiveness of the proposed algorithm are validated through several 2D/3D numerical experiments.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 109-127"},"PeriodicalIF":2.4,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144720944","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 work presents a new amplification methodology based on the widely used Gauss-Legendre implicit Runge-Kutta integrations by addressing the phase lag and amplification factor. The novel methodology focuses on these two elements, which are the complex amplifiers associated with the GLIRK integrations.
To enhance the amplifier capabilities of the GLIRK integrations, we introduce two novel equations that clarify the relationships between the amplification factor and phase lag. This paper culminates in the improvement of two well-defined GLIRK integrations, each carefully designed to eliminate both the phase lag and the amplification factor in practical applications. The examination of absolute stability regions in the complex plane, as well as stability regions in the z-v plane, is relevant to the new GLIRK integrations presented.
To satisfy the admissibility of the new methodology, we establish a competitive environment alongside the classical GLIRK integration.
This competitive space includes numerical examples that demonstrate the low cost of the new amplified GLIRK integrations in addressing stiff problems with high frequency. Ultimately, this cost-effectiveness and superiority become increasingly evident as the frequency of the stiff problems increases.
{"title":"A novel amplifying methodology in Gauss-Legendre IRK integrations to cope with high-frequency stiff problems","authors":"Sanaz Hami Hassan Kiyadeh , Hosein Saadat , Ramin Goudarzi Karim , Ali Safaie , Fayyaz Khodadosti","doi":"10.1016/j.apnum.2025.07.006","DOIUrl":"10.1016/j.apnum.2025.07.006","url":null,"abstract":"<div><div>This work presents a new amplification methodology based on the widely used Gauss-Legendre implicit Runge-Kutta integrations by addressing the phase lag and amplification factor. The novel methodology focuses on these two elements, which are the complex amplifiers associated with the GLIRK integrations.</div><div>To enhance the amplifier capabilities of the GLIRK integrations, we introduce two novel equations that clarify the relationships between the amplification factor and phase lag. This paper culminates in the improvement of two well-defined GLIRK integrations, each carefully designed to eliminate both the phase lag and the amplification factor in practical applications. The examination of absolute stability regions in the complex plane, as well as stability regions in the <em>z</em>-<em>v</em> plane, is relevant to the new GLIRK integrations presented.</div><div>To satisfy the admissibility of the new methodology, we establish a competitive environment alongside the classical GLIRK integration.</div><div>This competitive space includes numerical examples that demonstrate the low cost of the new amplified GLIRK integrations in addressing stiff problems with high frequency. Ultimately, this cost-effectiveness and superiority become increasingly evident as the frequency of the stiff problems increases.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 43-57"},"PeriodicalIF":2.2,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694462","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 : 2025-07-16DOI: 10.1016/j.apnum.2025.07.008
Sita Charkrit
This article introduces a novel recursive algorithm for obtaining explicit solutions to initial value problems of Lane-Emden type equations. By combining the traditional power series method with Adomian polynomials, expressed in terms of solution coefficients, the algorithm achieves high accuracy and converges rapidly to the exact solution within only a few iterations. This formulation not only simplifies the solution process but also improves computational efficiency over several existing semi-analytical approaches by requiring fewer iterations to reach a desired level of accuracy. Additionally, the Padé approximation is applied to the power series solution to accelerate convergence and expand the convergence region, allowing the solution to remain accurate over a wider interval. Error analysis using absolute and residual errors confirms that the proposed method, both independently and in combination with Padé approximants, outperforms existing methods in terms of precision and applicability. Several examples illustrate the method’s accuracy, efficiency, and reliability in solving nonlinear singular initial value problems.
{"title":"Explicit solution of Lane-Emden type equations via a novel recurrence and Padé approximation approach","authors":"Sita Charkrit","doi":"10.1016/j.apnum.2025.07.008","DOIUrl":"10.1016/j.apnum.2025.07.008","url":null,"abstract":"<div><div>This article introduces a novel recursive algorithm for obtaining explicit solutions to initial value problems of Lane-Emden type equations. By combining the traditional power series method with Adomian polynomials, expressed in terms of solution coefficients, the algorithm achieves high accuracy and converges rapidly to the exact solution within only a few iterations. This formulation not only simplifies the solution process but also improves computational efficiency over several existing semi-analytical approaches by requiring fewer iterations to reach a desired level of accuracy. Additionally, the Padé approximation is applied to the power series solution to accelerate convergence and expand the convergence region, allowing the solution to remain accurate over a wider interval. Error analysis using absolute and residual errors confirms that the proposed method, both independently and in combination with Padé approximants, outperforms existing methods in terms of precision and applicability. Several examples illustrate the method’s accuracy, efficiency, and reliability in solving nonlinear singular initial value problems.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 159-181"},"PeriodicalIF":2.4,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144767213","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 : 2025-07-14DOI: 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":"2025-07-14","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 : 2025-07-10DOI: 10.1016/j.apnum.2025.07.002
Peixuan Wu, Xiaohui Wu, Yang Wang, Ruqing Wang
In this article, we present a new two-grid discretization for the approximation of semilinear parabolic equation found on virtual element method (VEM). The two-step backward differentiation formula (BDF2) is comtemplated in the time dimension, while the VEM is utilized in spatial dimension. The two-grid VEM primarily computes the numerical solution from solving a nonlinear system on a coarse mesh with size H and then gets the numerical solution to a linear system built by the earlier result on a fine mesh with size h (). Consequently, our proposed scheme not only reduces total computational expense, but also achieves same accuracy as the single-grid VEM. The convergence analysis in and semi- norm for both the VEM and the two-grid VEM methods are provided concretely.
{"title":"Error estimates of a two-grid BDF2 virtual element scheme for semilinear parabolic equation","authors":"Peixuan Wu, Xiaohui Wu, Yang Wang, Ruqing Wang","doi":"10.1016/j.apnum.2025.07.002","DOIUrl":"10.1016/j.apnum.2025.07.002","url":null,"abstract":"<div><div>In this article, we present a new two-grid discretization for the approximation of semilinear parabolic equation found on virtual element method (VEM). The two-step backward differentiation formula (BDF2) is comtemplated in the time dimension, while the VEM is utilized in spatial dimension. The two-grid VEM primarily computes the numerical solution <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> from solving a nonlinear system on a coarse mesh with size <em>H</em> and then gets the numerical solution <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to a linear system built by the earlier result <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> on a fine mesh with size <em>h</em> (<span><math><mi>h</mi><mo>≪</mo><mi>H</mi></math></span>). Consequently, our proposed scheme not only reduces total computational expense, but also achieves same accuracy as the single-grid VEM. The convergence analysis in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and semi-<span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm for both the VEM and the two-grid VEM methods are provided concretely.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 451-475"},"PeriodicalIF":2.2,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631126","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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