Pub Date : 2025-12-01Epub 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":"2025-12-01","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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-12-01Epub 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":"2025-12-01","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 : 2025-11-22DOI: 10.1016/j.apnum.2025.11.008
Yongyong Cai , Fenghua Tong
We present a novel structure preserving approximation for solving the Patlak-Keller-Segel equation, combining conventional numerical discretization with a constrained optimization (or projection) based post-processing. To illustrate the idea, we use finite difference with Crank-Nicolson time stepping, followed by a projection step that solves an optimization problem to enforce positivity and mass conservation in the numerical solution. Rigorous error estimates are established with second-order accuracy in both space and time. Numerical experiments support the theoretical results and demonstrate the efficiency of our proposed approach. Extensive numerical tests demonstrate that the positivity preserving and mass conserving properties are crucial in simulating the Patlak-Keller-Segel equation.
{"title":"Positivity preserving and mass conservative projection methods for the Patlak-Keller-Segel equation","authors":"Yongyong Cai , Fenghua Tong","doi":"10.1016/j.apnum.2025.11.008","DOIUrl":"10.1016/j.apnum.2025.11.008","url":null,"abstract":"<div><div>We present a novel structure preserving approximation for solving the Patlak-Keller-Segel equation, combining conventional numerical discretization with a constrained optimization (or projection) based post-processing. To illustrate the idea, we use finite difference with Crank-Nicolson time stepping, followed by a projection step that solves an optimization problem to enforce positivity and mass conservation in the numerical solution. Rigorous error estimates are established with second-order accuracy in both space and time. Numerical experiments support the theoretical results and demonstrate the efficiency of our proposed approach. Extensive numerical tests demonstrate that the positivity preserving and mass conserving properties are crucial in simulating the Patlak-Keller-Segel equation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616690","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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