Journal of Computational and Applied Mathematics最新文献

英文中文

An efficient temporal multiscale algorithm for simulating a long-term plaque growth problem in relation to power-law blood flows 模拟与幂律血流有关的长期斑块生长问题的高效时间多尺度算法

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-04 DOI: 10.1016/j.cam.2025.116666

Xinyu Li , Ping Lin , Weifeng Zhao

This paper discusses the problem of non-Newtonian fluids with time multiscale characteristics, especially considering the type of power-law blood flow in a narrowed blood vessel due to plaque growth. In the vessel, the blood flow is considered as a fast-scale periodic motion, while the vessel wall grows on a slow scale. We use an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem. The approximation error is analyzed only for a largely simplified linear system, where the simple front-tracking technique is used to update the slow vessel wall growth. An effective multiscale method is then designed based on the approximation problem. The front-tracking technique also makes the implementation of the multiscale algorithm easier. Compared with the traditional direct solving process, this method shows a strong acceleration effect. Finally, we present a concrete numerical example. Through comparison, the relative error between the results of the multi-scale algorithm and the direct solving process is small, which is consistent with the theoretical analysis.

{"title":"An efficient temporal multiscale algorithm for simulating a long-term plaque growth problem in relation to power-law blood flows","authors":"Xinyu Li , Ping Lin , Weifeng Zhao","doi":"10.1016/j.cam.2025.116666","DOIUrl":"10.1016/j.cam.2025.116666","url":null,"abstract":"<div><div>This paper discusses the problem of non-Newtonian fluids with time multiscale characteristics, especially considering the type of power-law blood flow in a narrowed blood vessel due to plaque growth. In the vessel, the blood flow is considered as a fast-scale periodic motion, while the vessel wall grows on a slow scale. We use an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem. The approximation error is analyzed only for a largely simplified linear system, where the simple front-tracking technique is used to update the slow vessel wall growth. An effective multiscale method is then designed based on the approximation problem. The front-tracking technique also makes the implementation of the multiscale algorithm easier. Compared with the traditional direct solving process, this method shows a strong acceleration effect. Finally, we present a concrete numerical example. Through comparison, the relative error between the results of the multi-scale algorithm and the direct solving process is small, which is consistent with the theoretical analysis.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"469 ","pages":"Article 116666"},"PeriodicalIF":2.1,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143783945","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}

引用次数: 0

Crank–Nicolson alternative direction implicit method for two-dimensional variable-order space-fractional diffusion equations with nonseparable coefficients

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-02 DOI: 10.1016/j.cam.2025.116655

Qiu-Ya Wang , Cui-Yun Lin , Cheng-Xue Lao

In this paper, we propose alternative direction implicit (ADI) schemes to address the initial boundary value problem for two-dimensional variable-order fractional diffusion equations (VOFDEs). The Crank–Nicolson (CN) method and various ADI schemes employing different finite difference methods are utilized to approximate the temporal derivative and the spatial variable-order (VO) fractional derivatives, respectively, resulting in CN-ADI schemes. We present and prove theoretical results concerning the stability and convergence of these ADI schemes. Since the order of the VO derivatives depends on spatial and temporal variables, the resulting coefficient matrices from the discretization of VOFDEs are dense and lack a Toeplitz-like structure. We propose banded preconditioners to accelerate PGMRES methods for solving the resulting discretized linear systems. Numerical results demonstrate the high efficiency of the proposed ADI schemes.

引用次数: 0

On adaptive anisotropic mesh optimization for convection–diffusion problems

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-02 DOI: 10.1016/j.cam.2025.116661

Petr Knobloch , René Schneider

Numerical solution of convection-dominated problems requires the use of layer-adapted anisotropic meshes. Since a priori construction of such meshes is difficult for complex problems, it is proposed to generate them in an adaptive way by moving the node positions in the mesh such that an a posteriori error estimator of the overall error of the approximate solution is reduced. This approach is formulated for a SUPG finite element discretization of a stationary convection–diffusion problem defined in a two-dimensional polygonal domain. The optimization procedure is based on the discrete adjoint technique and a SQP method using the BFGS update. The optimization of node positions is applied to a coarse grid only and the resulting anisotropic mesh is then refined by standard adaptive red-green refinement. Four error estimators based on the solution of local Dirichlet problems are tested and it is demonstrated that an

L^{2}

norm based error estimator is the most robust one. The efficiency of the proposed approach is demonstrated on several model problems whose solutions contain typical boundary and interior layers.

{"title":"On adaptive anisotropic mesh optimization for convection–diffusion problems","authors":"Petr Knobloch , René Schneider","doi":"10.1016/j.cam.2025.116661","DOIUrl":"10.1016/j.cam.2025.116661","url":null,"abstract":"<div><div>Numerical solution of convection-dominated problems requires the use of layer-adapted anisotropic meshes. Since a priori construction of such meshes is difficult for complex problems, it is proposed to generate them in an adaptive way by moving the node positions in the mesh such that an a posteriori error estimator of the overall error of the approximate solution is reduced. This approach is formulated for a SUPG finite element discretization of a stationary convection–diffusion problem defined in a two-dimensional polygonal domain. The optimization procedure is based on the discrete adjoint technique and a SQP method using the BFGS update. The optimization of node positions is applied to a coarse grid only and the resulting anisotropic mesh is then refined by standard adaptive red-green refinement. Four error estimators based on the solution of local Dirichlet problems are tested and it is demonstrated that an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm based error estimator is the most robust one. The efficiency of the proposed approach is demonstrated on several model problems whose solutions contain typical boundary and interior layers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"469 ","pages":"Article 116661"},"PeriodicalIF":2.1,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143783944","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}

引用次数: 0

A recipe for learning Variably Scaled Kernels via Discontinuous Neural Networks

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-01 DOI: 10.1016/j.cam.2025.116653

G. Audone, F. Della Santa, E. Perracchione, S. Pieraccini

The efficacy of interpolating via Variably Scaled Kernels (VSKs) is known to be dependent on the definition of a proper scaling function, but no numerical recipes to construct it are available. Previous works suggest that such a function should mimic the target one, but no theoretical evidence is provided. This paper fills both the gaps: it proves that a scaling function reflecting the target one may lead to enhanced approximation accuracy, and it provides a user-independent tool for learning the scaling function by means of Discontinuous Neural Networks (

δ

NN), i.e., NNs able to deal with possible discontinuities. Numerical evidence supports our claims, as it shows that the key features of the target function can be clearly recovered in the learned scaling function.

引用次数: 0

A fast sequentially-decoupled matrix-decomposed algorithm for variable-order time-fractional optimal control and error estimate

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-01 DOI: 10.1016/j.cam.2025.116667

Jinhong Jia , Hong Wang , Zhaojie Zhou , Xiangcheng Zheng

We introduce a fast sequentially decoupled matrix-decomposed approach tailored for optimal control problems constrained by a Caputo time-fractional diffusion equation featuring hidden memory and space-dependent order. Our method unveils a quasi translation-invariant structure adept at managing spatio-temporal dependencies. This structure not only slashes the computational burden of coefficients from

O (M N^{2})

O (M N ln N)

, where

M

and

N

denote the spatial degrees of freedom and temporal steps in discretization, respectively, but also untangles the coupling between the space-dependent order and the inner product of the finite element method. Furthermore, we derive a swift matrix-decomposed algorithm designed to tackle the first-order optimality system, yielding a marked improvement in computational cost from

O (M N^{2})

O (M N {ln}^{3} N)

in each iteration. We substantiate our approach through rigorous numerical analysis and present numerical experiments to validate the theoretical underpinnings.

{"title":"A fast sequentially-decoupled matrix-decomposed algorithm for variable-order time-fractional optimal control and error estimate","authors":"Jinhong Jia , Hong Wang , Zhaojie Zhou , Xiangcheng Zheng","doi":"10.1016/j.cam.2025.116667","DOIUrl":"10.1016/j.cam.2025.116667","url":null,"abstract":"<div><div>We introduce a fast sequentially decoupled matrix-decomposed approach tailored for optimal control problems constrained by a Caputo time-fractional diffusion equation featuring hidden memory and space-dependent order. Our method unveils a quasi translation-invariant structure adept at managing spatio-temporal dependencies. This structure not only slashes the computational burden of coefficients from <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>M</mi><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>M</mi><mi>N</mi><mo>ln</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>M</mi></math></span> and <span><math><mi>N</mi></math></span> denote the spatial degrees of freedom and temporal steps in discretization, respectively, but also untangles the coupling between the space-dependent order and the inner product of the finite element method. Furthermore, we derive a swift matrix-decomposed algorithm designed to tackle the first-order optimality system, yielding a marked improvement in computational cost from <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>M</mi><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>M</mi><mi>N</mi><msup><mrow><mo>ln</mo></mrow><mrow><mn>3</mn></mrow></msup><mi>N</mi><mo>)</mo></mrow></mrow></math></span> in each iteration. We substantiate our approach through rigorous numerical analysis and present numerical experiments to validate the theoretical underpinnings.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"469 ","pages":"Article 116667"},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760013","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}

引用次数: 0

A discontinuous Galerkin method for a coupled Brinkman–Biot problem

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-04-01 DOI: 10.1016/j.cam.2025.116659

Jialiang Bian , Rui Li , Zhangxin Chen

In this paper, the Brinkman–Biot model is used to simulate the coupling problem of the Brinkman flow and deformable poroelastic media flow, which need to interact through the interface. By introducing the total pressure to rewrite the poroelastic equations, the possible locking phenomenon of the Biot system is overcome. By taking into account complex permeability coefficient, the strong stiffness caused by the Biot system is solved. A discontinuous Galerkin finite element method is used to solve the problem of complex poroelastic media caused by coupling. Firstly, the space is discretised by the discontinuous Galerkin finite element method, and the time is discretised by the backward Euler method. Then the semi-discretisation scheme and the full discretisation scheme are given. Secondly, in the framework of the Galerkin approximation, the existence and uniqueness of solutions and error estimates of semi-discrete and fully discrete schemes are analysed by means of differential algebraic equation theory and weak compactness demonstration. Finally, through numerical experiments, the theoretical convergence rate of the numerical solution of the model and whether the interface conditions coincide are verified. The channel filtration and the actual hydraulic fracturing fluid flow situation are simulated, and the effectiveness and accuracy of the method are verified.

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引用次数: 0

Error estimate of high order Runge–Kutta local discontinuous Galerkin method for nonlinear convection-dominated Sobolev equation

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-03-31 DOI: 10.1016/j.cam.2025.116657

Caiyue Du , Di Zhao , Qiang Zhang

In this paper we consider an efficient fully-discrete scheme for solving the nonlinear convection-dominated Sobolev equation, which adopts the local discontinuous Galerkin method with generalized numerical fluxes and high order explicit Runge–Kutta time-marching. By the generalized Gauss-Radau projection and the matrix transferring process, we obtain the optimal

L^{2}

-norm error estimate in both space and time. It is worth mentioning that the bounding constant in error estimate is independent of the reciprocals of diffusion and dispersion coefficients. Finally, numerical experiments are presented to support theoretical results.

引用次数: 0

Stability and uniqueness of coupled nonlinear finite element solution for anisotropic diffusion equation with nonlinear capacity term

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-03-31 DOI: 10.1016/j.cam.2025.116664

Jun Fang, Zhijun Shen, Xia Cui

This paper presents the stability of a two-layer coupled discretization fully implicit finite element scheme as well as the uniqueness of its solution. The scheme has been proposed for solving multi-dimensional anisotropic diffusion equations with nonlinear capacity term, and the existence and convergence of its solution have been proved in [Fang et al., J. Comput. Appl. Math. 438 (2024) 115512]. However, the basic theoretical analysis is incomplete, for example, the stability and uniqueness have not been solved yet, which are very important for the application of numerical methods in engineering. In this paper, we further develop the discrete functional analysis techniques to establish a framework with relatively comprehensive theoretical results. Wherein by introducing Ritz projection, rewriting the error equations into equivalent forms, and choosing appropriate test functions, we propose a new inductive argument to overcome the difficulties arising from the coupled nonlinear discretization of the diffusion operator and capacity term. Consequently, on the basis of the existence and convergence properties, we prove for the first time that the nonlinear finite element method is stable, thereby its solution is unique. Numerical examples show that the scheme is stable and has no numerical oscillations compared with the classical Crank–Nicolson scheme.

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引用次数: 0

Finite element method analysis of flutter: Comparing Scott–Vogelius and Taylor–Hood elements

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-03-31 DOI: 10.1016/j.cam.2025.116662

Karel Vacek , Petr Sváček

This paper focuses on the numerical simulation of the fluid–structure interaction (FSI) problem of an incompressible flow and a vibrating airfoil. The fluid flow is governed by the incompressible Navier–Stokes equations. The finite element method (FEM) is employed for the discretization of the weak form of equations. The main attention is paid to comparison of performance for different choices of finite element spaces together with a proper stabilization method. Two choices of the couple of finite element spaces are considered for velocity–pressure approximations. The first one is the standard Taylor–Hood finite element, the second one is the Scott–Vogelius element consisting of continuous piecewise quadratic velocities combined with discontinuous piecewise linear pressures. The barycentric refined mesh is used for the case of the Scott–Vogelius element in order to satisfy the Babuška–Brezzi inf-sup condition. The finite element approximations further require additional stabilization of the dominating convection. Here, the performance of the stream-line upwind Petrov–Galerkin (SUPG) stabilization, the SUPG together with the grad-div stabilization, the streamline-diffusion/local-projection stabilization approach is tested. The numerical results are presented and compared with the available data.

{"title":"Finite element method analysis of flutter: Comparing Scott–Vogelius and Taylor–Hood elements","authors":"Karel Vacek , Petr Sváček","doi":"10.1016/j.cam.2025.116662","DOIUrl":"10.1016/j.cam.2025.116662","url":null,"abstract":"<div><div>This paper focuses on the numerical simulation of the fluid–structure interaction (FSI) problem of an incompressible flow and a vibrating airfoil. The fluid flow is governed by the incompressible Navier–Stokes equations. The finite element method (FEM) is employed for the discretization of the weak form of equations. The main attention is paid to comparison of performance for different choices of finite element spaces together with a proper stabilization method. Two choices of the couple of finite element spaces are considered for velocity–pressure approximations. The first one is the standard Taylor–Hood finite element, the second one is the Scott–Vogelius element consisting of continuous piecewise quadratic velocities combined with discontinuous piecewise linear pressures. The barycentric refined mesh is used for the case of the Scott–Vogelius element in order to satisfy the Babuška–Brezzi inf-sup condition. The finite element approximations further require additional stabilization of the dominating convection. Here, the performance of the stream-line upwind Petrov–Galerkin (SUPG) stabilization, the SUPG together with the grad-div stabilization, the streamline-diffusion/local-projection stabilization approach is tested. The numerical results are presented and compared with the available data.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"469 ","pages":"Article 116662"},"PeriodicalIF":2.1,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746897","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}

引用次数: 0

Optimal error estimations and superconvergence analysis of anisotropic FEMs with variable time steps for reaction–diffusion equations

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics

Pub Date : 2025-03-31 DOI: 10.1016/j.cam.2025.116656

Lifang Pei , Chao Xu , Jiwei Zhang , Yanmin Zhao

By combining variable-time-step two-step backward differentiation formula (VSBDF2) with anisotropic finite element methods (FEMs), a fully discrete scheme with non-uniform meshes both in time and space is constructed for the reaction–diffusion equations. Two approaches are provided to prove the optimal error estimates in

L^{2}

-norm and global superconvergence result in

H^{1}

-norm under a mild adjacent time-step ratio restriction

0 < r_{k} < r_{max} \approx 4.8645

. The first approach is based on the use of anisotropic properties of the interpolation operators, but needs a higher regularity of the solution and is only valid for some special elements. The second approach is based on the combination technique of interpolation and energy projection operators, and needs a lower regularity of the solution, which can be regarded as a unified framework of convergence analysis. Numerical experiments are provided to demonstrate our theoretical analysis.

{"title":"Optimal error estimations and superconvergence analysis of anisotropic FEMs with variable time steps for reaction–diffusion equations","authors":"Lifang Pei , Chao Xu , Jiwei Zhang , Yanmin Zhao","doi":"10.1016/j.cam.2025.116656","DOIUrl":"10.1016/j.cam.2025.116656","url":null,"abstract":"<div><div>By combining variable-time-step two-step backward differentiation formula (VSBDF2) with anisotropic finite element methods (FEMs), a fully discrete scheme with non-uniform meshes both in time and space is constructed for the reaction–diffusion equations. Two approaches are provided to prove the optimal error estimates in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and global superconvergence result in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm under a mild adjacent time-step ratio restriction <span><math><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo><</mo><msub><mrow><mi>r</mi></mrow><mrow><mo>max</mo></mrow></msub><mo>≈</mo><mn>4</mn><mo>.</mo><mn>8645</mn></mrow></math></span>. The first approach is based on the use of anisotropic properties of the interpolation operators, but needs a higher regularity of the solution and is only valid for some special elements. The second approach is based on the combination technique of interpolation and energy projection operators, and needs a lower regularity of the solution, which can be regarded as a unified framework of convergence analysis. Numerical experiments are provided to demonstrate our theoretical analysis.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"469 ","pages":"Article 116656"},"PeriodicalIF":2.1,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768722","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}

引用次数: 0

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