Pub Date : 2026-02-01Epub Date: 2025-12-04DOI: 10.1016/j.camwa.2025.11.015
Chenyang Wang, Yan Xu
We propose a kernel compensation mimetic difference (MD) scheme to solve the grad-div eigenvalue problem. This method utilizes a curl-curl type compensation operator along with carefully selected boundary conditions to effectively manage the infinite-dimensional kernel of the grad-div operator. To ensure high accuracy, we apply stencil-based MD operators to discretize the grad-div operator under Dirichlet boundary conditions. This results in a numerical scheme characterized by a sparse stiff matrix with a narrow bandwidth while achieving high-order accuracy. We construct the compensation operator with a proper boundary condition that is orthogonal to the discrete grad-div operator. A generalized identification method for spurious eigenvalues is presented. The resulting scheme offers several advantages, including high-order accuracy, enhanced computational efficiency with reduced memory usage, and excellent scalability for parallel computation. Numerical tests demonstrate that our approach not only converges at the expected rates but also performs satisfactorily in terms of speed.
{"title":"A kernel compensation mimetic difference scheme for the grad-div eigenvalue problem","authors":"Chenyang Wang, Yan Xu","doi":"10.1016/j.camwa.2025.11.015","DOIUrl":"10.1016/j.camwa.2025.11.015","url":null,"abstract":"<div><div>We propose a kernel compensation mimetic difference (MD) scheme to solve the grad-div eigenvalue problem. This method utilizes a curl-curl type compensation operator along with carefully selected boundary conditions to effectively manage the infinite-dimensional kernel of the grad-div operator. To ensure high accuracy, we apply stencil-based MD operators to discretize the grad-div operator under Dirichlet boundary conditions. This results in a numerical scheme characterized by a sparse stiff matrix with a narrow bandwidth while achieving high-order accuracy. We construct the compensation operator with a proper boundary condition that is orthogonal to the discrete grad-div operator. A generalized identification method for spurious eigenvalues is presented. The resulting scheme offers several advantages, including high-order accuracy, enhanced computational efficiency with reduced memory usage, and excellent scalability for parallel computation. Numerical tests demonstrate that our approach not only converges at the expected rates but also performs satisfactorily in terms of speed.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 20-40"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685731","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 : 2026-02-01Epub Date: 2025-12-24DOI: 10.1016/j.camwa.2025.12.014
Jiaxing Chen, Lei Wang, Jiawei Xiang
In research on the integration of computer-aided design (CAD) and numerical simulations of boundary types, commonly used methods for constructing approximate functions include non-uniform rational B-splines (NURBS) interpolation and moving least-squares (MLS) interpolation. These methods struggle to precisely enforce boundary conditions due to the lack of Kronecker delta properties in their shape functions. To address this limitation, this paper proposes a geometry-independent spline boundary element method (GISBEM) that introduces a transformation matrix to construct a spline interpolation function as the shape function, enabling direct application of boundary conditions akin to the boundary element method (BEM). First, the concept of geometry-independent field approximation (GIFT) is introduced, where the geometry is accurately described by NURBS, and the field variables of the elements are approximated using B-spline interpolation and transformation matrices. Second, the computation formats for the 3D potential and elasticity problems are derived using parameter mapping. Third, the calculation of variables at the boundary points is performed on the element using the relationship between the variables, with subsequent processing similar to that of BEM. Finally, the effectiveness and accuracy of the proposed method are verified through numerical examples.
{"title":"Geometry-independent spline boundary element method to analyze three-dimensional potential and elasticity problems","authors":"Jiaxing Chen, Lei Wang, Jiawei Xiang","doi":"10.1016/j.camwa.2025.12.014","DOIUrl":"10.1016/j.camwa.2025.12.014","url":null,"abstract":"<div><div>In research on the integration of computer-aided design (CAD) and numerical simulations of boundary types, commonly used methods for constructing approximate functions include non-uniform rational B-splines (NURBS) interpolation and moving least-squares (MLS) interpolation. These methods struggle to precisely enforce boundary conditions due to the lack of Kronecker delta properties in their shape functions. To address this limitation, this paper proposes a geometry-independent spline boundary element method (GISBEM) that introduces a transformation matrix to construct a spline interpolation function as the shape function, enabling direct application of boundary conditions akin to the boundary element method (BEM). First, the concept of geometry-independent field approximation (GIFT) is introduced, where the geometry is accurately described by NURBS, and the field variables of the elements are approximated using B-spline interpolation and transformation matrices. Second, the computation formats for the 3D potential and elasticity problems are derived using parameter mapping. Third, the calculation of variables at the boundary points is performed on the element using the relationship between the variables, with subsequent processing similar to that of BEM. Finally, the effectiveness and accuracy of the proposed method are verified through numerical examples.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 192-208"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145822951","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 : 2026-02-01Epub Date: 2025-12-24DOI: 10.1016/j.camwa.2025.12.012
Sitao Zhang , Lin Liu , Yu Liu , Hongqing Song , Libo Feng
The paper contributes to investigating the quantum transport on a comb structure, whose feature of quantum motion is geometrically constrained such that the quantum motion is exclusively restricted to the backbone along the -direction. Instead of substituting with the fractional derivative directly, the one-dimensional (1D) time fractional Schrödinger equation (TFSE) is formulated, which is derived by mathematical derivation from the 2D Schrödinger equation with the wave operator (SEWO). The infinite boundaries are replaced with the absorbing boundary conditions (ABCs). The finite difference method (FDM) is formulated, followed by theoretical analysis to establish stability and convergence. A fast scheme is employed to enhance computational efficiency. The contrast of the numerical and exact solutions is analyzed by innovating a source term. In addition, the contrast of the distributions for the ABCs and zero boundary conditions is analyzed. Results show that ABCs provide a more accurate numerical solution compared to zero boundary conditions. Finally, the evolutions of the modules and the phasic pictures with different parameters are given.
{"title":"The mechanism analysis for the fractional quantum dynamics on a comb structure with the absorbing boundary conditions","authors":"Sitao Zhang , Lin Liu , Yu Liu , Hongqing Song , Libo Feng","doi":"10.1016/j.camwa.2025.12.012","DOIUrl":"10.1016/j.camwa.2025.12.012","url":null,"abstract":"<div><div>The paper contributes to investigating the quantum transport on a comb structure, whose feature of quantum motion is geometrically constrained such that the quantum motion is exclusively restricted to the backbone along the <span><math><mi>x</mi></math></span>-direction. Instead of substituting with the fractional derivative directly, the one-dimensional (1D) time fractional Schrödinger equation (TFSE) is formulated, which is derived by mathematical derivation from the 2D Schrödinger equation with the wave operator (SEWO). The infinite boundaries are replaced with the absorbing boundary conditions (ABCs). The finite difference method (FDM) is formulated, followed by theoretical analysis to establish stability and convergence. A fast scheme is employed to enhance computational efficiency. The contrast of the numerical and exact solutions is analyzed by innovating a source term. In addition, the contrast of the distributions for the ABCs and zero boundary conditions is analyzed. Results show that ABCs provide a more accurate numerical solution compared to zero boundary conditions. Finally, the evolutions of the modules and the phasic pictures with different parameters are given.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 209-229"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145822952","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 : 2026-02-01Epub Date: 2025-12-04DOI: 10.1016/j.camwa.2025.11.013
Jinxiu Zhang , Xuehua Yang , Song Wang
This paper develops and analyzes a linearized compact three-levelfinite difference (CTLFD) scheme to solve the two-dimensional (2-D) Kuramoto-Tsuzuki dynamics dominated by strong nonlinear characteristics. This method combines the compact difference method (CDM) in space with the Crank-Nicolson (C-N) scheme in time, achieving an overall convergence rate of , where τ, hx, and hy represent the time and space step sizes, respectively. Nonlinear terms are linearized in a semi-implicit manner to enhance stability and computational efficiency. A rigorous stability and error analysis is carried out using an energy technique together with mathematical induction, confirming boundedness, uniqueness, and the optimal convergence for the numerical solution under two discrete norms. Finally, three sets of numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and robustness of the proposed scheme.
{"title":"A compact difference method for the 2-D Kuramoto-Tsuzuki complex equation with Neumann boundary characterized by strong nonlinear effects","authors":"Jinxiu Zhang , Xuehua Yang , Song Wang","doi":"10.1016/j.camwa.2025.11.013","DOIUrl":"10.1016/j.camwa.2025.11.013","url":null,"abstract":"<div><div>This paper develops and analyzes a linearized compact three-levelfinite difference (CTLFD) scheme to solve the two-dimensional (2-D) Kuramoto-Tsuzuki dynamics dominated by strong nonlinear characteristics. This method combines the compact difference method (CDM) in space with the Crank-Nicolson (C-N) scheme in time, achieving an overall convergence rate of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>h</mi><mi>x</mi><mn>4</mn></msubsup><mo>+</mo><msubsup><mi>h</mi><mi>y</mi><mn>4</mn></msubsup><mo>)</mo></mrow></mrow></math></span>, where <em>τ, h<sub>x</sub></em>, and <em>h<sub>y</sub></em> represent the time and space step sizes, respectively. Nonlinear terms are linearized in a semi-implicit manner to enhance stability and computational efficiency. A rigorous stability and error analysis is carried out using an energy technique together with mathematical induction, confirming boundedness, uniqueness, and the optimal convergence for the numerical solution under two discrete norms. Finally, three sets of numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and robustness of the proposed scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 1-19"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658796","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 : 2026-02-01Epub Date: 2025-12-08DOI: 10.1016/j.camwa.2025.11.014
Boyang Yu , Yonghai Li , Jiangguo Liu
This paper develops two novel numerical solvers on quadrilateral meshes for coupled flow and transport problems. The focus is placed on preserving physical properties such as mass conservation and concentration positivity. Quadrilateral meshes are used due to their flexibility in accommodation of domain geometry. A weak Galerkin (WG) finite element scheme with linear shape functions is utilized to solve the Darcy equation. Mapped bilinear finite volumes on the same mesh are then used to solve the time-dependent convection-diffusion equation, which rely on the numerical velocity obtained from the Darcy scheme. Techniques for positivity-correction are applied to both diffusive and convective fluxes. Global mass conservation, positivity-preserving, and optimal order convergence are carefully examined under appropriate conditions. Numerical tests demonstrate robustness of our new solvers in handling convection dominance and anisotropy/heterogeneity in permeability and/or diffusion.
{"title":"New solvers for coupled flow and transport on quadrilateral meshes: Property-preserving and optimal-order convergence","authors":"Boyang Yu , Yonghai Li , Jiangguo Liu","doi":"10.1016/j.camwa.2025.11.014","DOIUrl":"10.1016/j.camwa.2025.11.014","url":null,"abstract":"<div><div>This paper develops two novel numerical solvers on quadrilateral meshes for coupled flow and transport problems. The focus is placed on preserving physical properties such as mass conservation and concentration positivity. Quadrilateral meshes are used due to their flexibility in accommodation of domain geometry. A weak Galerkin (WG) finite element scheme with linear shape functions is utilized to solve the Darcy equation. Mapped bilinear finite volumes on the same mesh are then used to solve the time-dependent convection-diffusion equation, which rely on the numerical velocity obtained from the Darcy scheme. Techniques for positivity-correction are applied to both diffusive and convective fluxes. Global mass conservation, positivity-preserving, and optimal order convergence are carefully examined under appropriate conditions. Numerical tests demonstrate robustness of our new solvers in handling convection dominance and anisotropy/heterogeneity in permeability and/or diffusion.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 73-90"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731495","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}
We propose a two-compartment mathematical model of myocardial perfusion, representing myocardial tissue at the arteriolar level. The model comprises a simplified two-dimensional geometrical analog of the complex three-dimensional myocardial vasculature. In the advective compartment, consisting of a 2D vasculature analog, fluid flow and contrast agent transport are governed by Navier-Stokes and system of advection-diffusion equations, respectively. The surrounding myocardium, included in porous capillary compartment and modeled as a porous medium, assumes purely diffusive transport without fluid flow. Contrast agent exchange occurs through the interface between the two compartments. The model is numerically solved using the lattice Boltzmann method, with GPU implementation enabling massive parallelization. Sample contrast agent profiles are analyzed for both healthy and defective tissues. The model’s capability to interpret actual MRI perfusion curves is evaluated using mathematical optimization techniques. Furthermore, the model is employed for a binary classification test to evaluate its agreement with the expert opinion of a qualified clinician. Myocardial blood flow approximations from the proposed model compare favorably to results from established medical software utilizing signal-deconvolution methods. Despite its simplifications, the 2D model accurately represents essential perfusion dynamics, matching or exceeding clinical software in agreement with expert evaluations. Although tested on a small number of patients, this proof of concept shows potential for direct application during perfusion exams or generating synthetic data for machine learning.
{"title":"Mathematical modeling of myocardial perfusion using lattice Boltzmann method","authors":"Jan Kovář , Radek Fučík , Tarique Hussain , Munes Fares , Radomír Chabiniok","doi":"10.1016/j.camwa.2025.12.005","DOIUrl":"10.1016/j.camwa.2025.12.005","url":null,"abstract":"<div><div>We propose a two-compartment mathematical model of myocardial perfusion, representing myocardial tissue at the arteriolar level. The model comprises a simplified two-dimensional geometrical analog of the complex three-dimensional myocardial vasculature. In the advective compartment, consisting of a 2D vasculature analog, fluid flow and contrast agent transport are governed by Navier-Stokes and system of advection-diffusion equations, respectively. The surrounding myocardium, included in porous capillary compartment and modeled as a porous medium, assumes purely diffusive transport without fluid flow. Contrast agent exchange occurs through the interface between the two compartments. The model is numerically solved using the lattice Boltzmann method, with GPU implementation enabling massive parallelization. Sample contrast agent profiles are analyzed for both healthy and defective tissues. The model’s capability to interpret actual MRI perfusion curves is evaluated using mathematical optimization techniques. Furthermore, the model is employed for a binary classification test to evaluate its agreement with the expert opinion of a qualified clinician. Myocardial blood flow approximations from the proposed model compare favorably to results from established medical software utilizing signal-deconvolution methods. Despite its simplifications, the 2D model accurately represents essential perfusion dynamics, matching or exceeding clinical software in agreement with expert evaluations. Although tested on a small number of patients, this proof of concept shows potential for direct application during perfusion exams or generating synthetic data for machine learning.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 230-253"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925930","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 : 2026-02-01Epub Date: 2025-12-08DOI: 10.1016/j.camwa.2025.11.022
Ismail Labaali , Maria Rosaria Lancia , Chiara Sorgentone
Boundary integral methods are a powerful tool to solve partial differential equations by reformulating them as integral equations over the boundary of the domain. When dealing with boundary integral methods, and in particular with the numerical integration of layer potentials, it is essential to estimate the magnitude of the error associated with the underlying quadrature rule. As the evaluation point approaches the boundary, the integral becomes nearly-singular and the associated quadrature error increases rapidly. Being able to estimate such error is needed to identify when the accuracy becomes inadequate, and the use of a specialized quadrature method is required. In this work we provide accurate quadrature error estimates for the Gauss-Legendre and the trapezoidal rules in computing two-dimensional layer potentials with logarithmic singularities.
{"title":"Quadrature error estimates for kernels with logarithmic singularity","authors":"Ismail Labaali , Maria Rosaria Lancia , Chiara Sorgentone","doi":"10.1016/j.camwa.2025.11.022","DOIUrl":"10.1016/j.camwa.2025.11.022","url":null,"abstract":"<div><div>Boundary integral methods are a powerful tool to solve partial differential equations by reformulating them as integral equations over the boundary of the domain. When dealing with boundary integral methods, and in particular with the numerical integration of layer potentials, it is essential to estimate the magnitude of the error associated with the underlying quadrature rule. As the evaluation point approaches the boundary, the integral becomes nearly-singular and the associated quadrature error increases rapidly. Being able to estimate such error is needed to identify when the accuracy becomes inadequate, and the use of a specialized quadrature method is required. In this work we provide accurate quadrature error estimates for the Gauss-Legendre and the trapezoidal rules in computing two-dimensional layer potentials with logarithmic singularities.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 91-99"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731496","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 : 2026-02-01Epub Date: 2025-12-05DOI: 10.1016/j.camwa.2025.11.019
Xiao Qi , Yubin Yan
In this paper, we propose and analyze an efficient numerical method for solving the stochastic Allen-Cahn equation with additive noise. The method combines a stabilized semi-implicit time discretization scheme with a reduced-order finite element spatial discretization method. The core idea is to approximate the original high-dimensional solution space via a low-dimensional subspace, constructed by the Proper Orthogonal Decomposition (POD) method based on an ensemble of snapshots from the full-order model at selected time instances. First, we rigorously establish the spatio-temporal strong convergence rates between the mild solution and the reduced-order solution. Second, in large-sample simulations, the reduced-order basis exhibits a certain generalization capability in capturing the average behavior of the numerical solutions. Numerical experiments are provided to verify the theoretical error estimates and to demonstrate the effectiveness of the proposed method.
{"title":"An efficient reduced-order approximation for the stochastic Allen-Cahn equation","authors":"Xiao Qi , Yubin Yan","doi":"10.1016/j.camwa.2025.11.019","DOIUrl":"10.1016/j.camwa.2025.11.019","url":null,"abstract":"<div><div>In this paper, we propose and analyze an efficient numerical method for solving the stochastic Allen-Cahn equation with additive noise. The method combines a stabilized semi-implicit time discretization scheme with a reduced-order finite element spatial discretization method. The core idea is to approximate the original high-dimensional solution space via a low-dimensional subspace, constructed by the Proper Orthogonal Decomposition (POD) method based on an ensemble of snapshots from the full-order model at selected time instances. First, we rigorously establish the spatio-temporal strong convergence rates between the mild solution and the reduced-order solution. Second, in large-sample simulations, the reduced-order basis exhibits a certain generalization capability in capturing the average behavior of the numerical solutions. Numerical experiments are provided to verify the theoretical error estimates and to demonstrate the effectiveness of the proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 56-72"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685729","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 : 2026-02-01Epub Date: 2025-12-10DOI: 10.1016/j.camwa.2025.11.026
Weihao Xie , Zhifang Du , Guangxue Wang , Huaibao Zhang
In this work, we propose a fifth-order Hermite weighted compact nonlinear scheme (HWCNS) within the framework of the two-stage fourth-order Lax-Wendroff-type time discretization by Li and Du (SIAM J Sci Comput 38 (5):A3046-A3069, 2016). Unlike the traditional weighted compact nonlinear scheme (WCNS), which uses the Runge-Kutta method for high-order temporal integration, and solves only for nodal values, this new HWCNS simultaneously evolves both nodal and midpoint values within the same time discretization. These solutions are then used to compute first-order spatial derivatives at the nodes. By incorporating both nodal values and their derivatives, the scheme enables a high-order nonlinear Hermite interpolation. Furthermore, we introduce a “polynomial stencil selection procedure” derived from the targeted essentially non-oscillatory (TENO) scheme to improve the performance of the nonlinear interpolation. A variety of benchmark cases are addressed in one- and two-dimensional dimensions. The proposed scheme, based on the Lax-Wendroff time discretization, exhibits promising characteristics of minimal dissipation and dispersion errors for fine-scale features in smooth flow regions, and demonstrates robust shock-capturing capabilities with high resolutions, benefiting from its compact stencil in both time and space. Moreover, integration of the TENO technique into the nonlinear interpolation yields a further reduction in numerical dissipation, as shown in the numerical tests.
{"title":"Two-stage fourth-order Hermite weighted compact nonlinear scheme for hyperbolic conservation laws","authors":"Weihao Xie , Zhifang Du , Guangxue Wang , Huaibao Zhang","doi":"10.1016/j.camwa.2025.11.026","DOIUrl":"10.1016/j.camwa.2025.11.026","url":null,"abstract":"<div><div>In this work, we propose a fifth-order Hermite weighted compact nonlinear scheme (HWCNS) within the framework of the two-stage fourth-order Lax-Wendroff-type time discretization by Li and Du (SIAM J Sci Comput 38 (5):A3046-A3069, 2016). Unlike the traditional weighted compact nonlinear scheme (WCNS), which uses the Runge-Kutta method for high-order temporal integration, and solves only for nodal values, this new HWCNS simultaneously evolves both nodal and midpoint values within the same time discretization. These solutions are then used to compute first-order spatial derivatives at the nodes. By incorporating both nodal values and their derivatives, the scheme enables a high-order nonlinear Hermite interpolation. Furthermore, we introduce a “polynomial stencil selection procedure” derived from the targeted essentially non-oscillatory (TENO) scheme to improve the performance of the nonlinear interpolation. A variety of benchmark cases are addressed in one- and two-dimensional dimensions. The proposed scheme, based on the Lax-Wendroff time discretization, exhibits promising characteristics of minimal dissipation and dispersion errors for fine-scale features in smooth flow regions, and demonstrates robust shock-capturing capabilities with high resolutions, benefiting from its compact stencil in both time and space. Moreover, integration of the TENO technique into the nonlinear interpolation yields a further reduction in numerical dissipation, as shown in the numerical tests.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 171-191"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731488","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}
In this paper, the Crank-Nicolson and BDF2 schemes, based on the serendipity virtual element method, are applied to solve parabolic problems. The main content of this article is to analyze the fully discrete error in zero and energy norm for these two second-order fully discrete schemes and to conduct numerical experiments. We derive optimal zero and energy norm error estimates for the Crank-Nicolson and BDF2 schemes. In addition, the errors in zero norm and energy norm of the semi-discrete scheme of serendipity virtual elements are estimated. Through numerical experiments conducted on rectangular, convex polygonal, and non-convex polygonal meshes, discrete errors are presented in the spatial and temporal directions, respectively. Finally, we demonstrated the computational complexity advantage of serendipity virtual elements by calculating the total degrees of freedom.
{"title":"Numerical analysis of the second-order fully discrete schemes for parabolic problem based on serendipity virtual element method","authors":"Jianjun Wan, Yuanjiang Xu, Jiaxin Wei, Shilei Xu, Chunyan Niu","doi":"10.1016/j.camwa.2025.11.016","DOIUrl":"10.1016/j.camwa.2025.11.016","url":null,"abstract":"<div><div>In this paper, the Crank-Nicolson and BDF2 schemes, based on the serendipity virtual element method, are applied to solve parabolic problems. The main content of this article is to analyze the fully discrete error in zero and energy norm for these two second-order fully discrete schemes and to conduct numerical experiments. We derive optimal zero and energy norm error estimates for the Crank-Nicolson and BDF2 schemes. In addition, the errors in zero norm and energy norm of the semi-discrete scheme of serendipity virtual elements are estimated. Through numerical experiments conducted on rectangular, convex polygonal, and non-convex polygonal meshes, discrete errors are presented in the spatial and temporal directions, respectively. Finally, we demonstrated the computational complexity advantage of serendipity virtual elements by calculating the total degrees of freedom.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 155-170"},"PeriodicalIF":2.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731487","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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