Numerical Methods for Partial Differential Equations最新文献_第3页

Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation 浅水方程的熵稳定非连续伽勒金方法与子单元实在性保持

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-07-25 DOI: 10.1002/num.23129

Xinhui Wu, Nathaniel Trask, Jesse Chan

引用次数: 0

Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations 二元流体-表面活性相场方程的能量稳定和保界数值方案的收敛性和稳定性分析

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-07-01 DOI: 10.1002/num.23125

Jiayi Duan, Xiao Li, Zhonghua Qiao

{"title":"Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations","authors":"Jiayi Duan, Xiao Li, Zhonghua Qiao","doi":"10.1002/num.23125","DOIUrl":"https://doi.org/10.1002/num.23125","url":null,"abstract":"In this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"67 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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

Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape 高指数鞍动力学的半隐式方法及其在构建解景观中的应用

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-06-29 DOI: 10.1002/num.23123

Yue Luo, Lei Zhang, Pingwen Zhang, Zhiyi Zhang, Xiangcheng Zheng

{"title":"Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape","authors":"Yue Luo, Lei Zhang, Pingwen Zhang, Zhiyi Zhang, Xiangcheng Zheng","doi":"10.1002/num.23123","DOIUrl":"https://doi.org/10.1002/num.23123","url":null,"abstract":"We analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"14 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws 线性标量双曲守恒定律的一类谱量法分析

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-06-28 DOI: 10.1002/num.23126

Jianfang Lu, Yan Jiang, Chi‐Wang Shu, Mengping Zhang

{"title":"Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws","authors":"Jianfang Lu, Yan Jiang, Chi‐Wang Shu, Mengping Zhang","doi":"10.1002/num.23126","DOIUrl":"https://doi.org/10.1002/num.23126","url":null,"abstract":"In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"44 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions 强锚定条件下的 de Gennes-Cahn-Hilliard 能量最小值

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-06-27 DOI: 10.1002/num.23127

Shibin Dai, Abba Ramadan

引用次数: 0

A linear finite difference scheme with error analysis designed to preserve the structure of the 2D Boussinesq paradigm equation 带误差分析的线性有限差分方案，旨在保留二维布森斯克范式方程的结构

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-06-27 DOI: 10.1002/num.23119

K. Poochinapan, P. Manorot, T. Mouktonglang, B. Wongsaijai

{"title":"A linear finite difference scheme with error analysis designed to preserve the structure of the 2D Boussinesq paradigm equation","authors":"K. Poochinapan, P. Manorot, T. Mouktonglang, B. Wongsaijai","doi":"10.1002/num.23119","DOIUrl":"https://doi.org/10.1002/num.23119","url":null,"abstract":"Use of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an ‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"107 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 flux‐based HDG method 基于通量的 HDG 方法

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-05-31 DOI: 10.1002/num.23117

Issei Oikawa

引用次数: 0

Extensions and investigations of space‐time generalized Riemann problems numerical schemes for linear systems of conservation laws with source terms 带源项线性守恒定律系统的时空广义黎曼问题数值方案的扩展与研究

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-05-31 DOI: 10.1002/num.23118

Rodolphe Turpault

引用次数: 0

A parareal exponential integrator finite element method for semilinear parabolic equations 半线性抛物方程的准指数积分有限元法

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-05-29 DOI: 10.1002/num.23116

Jianguo Huang, Lili Ju, Yuejin Xu

引用次数: 0

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type 幂律型动态粘弹性模型的非连续 Galerkin 有限元方法

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-05-06 DOI: 10.1002/num.23107

Yongseok Jang, Simon Shaw

{"title":"Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type","authors":"Yongseok Jang, Simon Shaw","doi":"10.1002/num.23107","DOIUrl":"https://doi.org/10.1002/num.23107","url":null,"abstract":"Linear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"11 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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