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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-07-25 DOI: 10.1002/num.23129
Xinhui Wu, Nathaniel Trask, Jesse Chan
High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.
众所周知,高阶方案在出现冲击不连续或未充分解析的解特征时是不稳定的,传统上需要额外的过滤、限制或人工粘度来避免解炸裂。熵稳定方案通过确保物理相关解满足半离散熵不等式来解决这种不稳定性,而与离散化参数无关。然而,还必须采取额外措施,确保解满足正相关性等物理约束。在这项研究中,我们针对二维(2D)三角网格上的非线性浅水方程(SWE)提出了一种高阶熵稳定非连续伽勒金(ESDG)方法,该方法保留了水高的正定性。该方案结合了低阶实在性保留方法和使用凸极限的高阶熵稳定方法。对于具有连续水深剖面的拟合网格,该方法具有熵稳定性和良好的平衡性。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-07-01 DOI: 10.1002/num.23125
Jiayi Duan, Xiao Li, Zhonghua Qiao
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.
该模型由两个卡恩-希利亚德(Cahn-Hilliard)型方程组成,自由能包含金兹堡-朗道(Ginzburg-Landau)双井势、对数弗洛里-哈金斯(Flory-Huggins)势和非线性耦合熵。数值方案是解耦的线性方案,通过中心差分空间近似与基于自由能凸分裂的一阶和二阶指数时差法相结合而建立。为了保证方案的线性,对非线性项,尤其是对数项进行了显式近似,这就要求数值解的约束保持,从而使算法具有鲁棒性。我们对一阶和二阶方案进行了收敛分析,并详细证明了边界保持特性,其中高阶一致性分析适用于一阶情况。此外,还通过凸分裂的性质获得了能量稳定性。通过数值实验验证了方案的准确性和稳定性,并模拟了相分离和表面活性剂吸附的动力学过程。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-06-29 DOI: 10.1002/num.23123
Yue Luo, Lei Zhang, Pingwen Zhang, Zhiyi Zhang, Xiangcheng Zheng
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.
我们分析了高指数鞍点动力学的半隐式方案,它为寻找任意指数鞍点和构建解景观提供了一种强大的数值方法。与鞍动力学的显式方案相比,半隐式离散化放宽了步长并加速了收敛,但相应的数值分析与显式方案相比遇到了新的困难。具体来说,由于采用半隐式处理方法,每个时间步长的特征向量的正交属性无法得到充分利用,而特征向量的计算又与正交化过程相耦合,这使得数值分析更加复杂。针对这些问题,我们通过技术分割和多变量循环归纳程序等方法证明了半隐式方案的误差估计。我们进一步分析了求解半隐式系统的广义最小残差求解器的收敛速率。我们进行了大量的数值实验,以证实半隐式方案在构建复杂系统解景观方面的效率和准确性。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-06-28 DOI: 10.1002/num.23126
Jianfang Lu, Yan Jiang, Chi‐Wang Shu, Mengping Zhang
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.
本文在 Petrov-Galerkin 框架下研究了具有一类细分点的标量双曲守恒定律的谱体积 (SV) 方法。由于 DG 方法与 SV 方法在细分点的适当选择上存在紧密联系,因此很自然地可以用 Galerkin 形式分析 SV 方法,并推导出与 DG 方法类似的理论结果。本文研究了一类 SV 方法,其细分点是特定多项式的零点,其中包含一个参数。本文提供了这种细分下的片常数函数的性质,包括试解空间和测试函数空间之间的正交性。借助这些性质,我们能够推导出 SV 方法的能量稳定性、最优先验误差估计值以及任意高阶精度。我们还利用修正函数技术研究了数值解的超收敛性,并证明了细分点的不同选择会产生不同的超收敛阶次。在数值实验中,通过在 SV 方法中选择不同的参数,数值结果证实了理论结论。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-06-27 DOI: 10.1002/num.23127
Shibin Dai, Abba Ramadan
In this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.
在本文中,我们利用 Nehari 流形和具有 Dirichlet 边界条件的负拉普拉奇的特征值问题,分析研究了有界域上具有四元双阱势和 Dirichlet 边界条件的 de Gennes-Cahn-Hilliard 能量的最小值。我们的分析揭示了一种由边界值和分岔参数决定的分岔现象,分岔参数描述了隔离二元混合物两相的过渡层的厚度。具体来说,当边界值与纯相的平均值精确一致,且分岔参数超过或等于临界阈值时,最小化器就会呈现唯一的形式,代表均相状态。反之,当分岔参数低于该临界值时,就会出现两个对称的最小化子。如果边界值大于或小于纯相的平均值,对称性就会被打破,从而产生一个独特的最小化器。此外,我们还结合边界条件和 de Gennes-Cahn-Hilliard 能量的特征,推导出了这些最小化器的边界。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-06-27 DOI: 10.1002/num.23119
K. Poochinapan, P. Manorot, T. Mouktonglang, B. Wongsaijai
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.
有限差分法因其效率高、效果好、应用广泛,已成功解决了一般偏微分方程的问题。例如,利用基于 Crank-Nicolson/Adams-Bashforth 技术的线性-隐式有限差分方案,可以对二维布森斯克范式方程进行数值研究。首先,通过数值方案推导并保留保守量。然后,进行收敛性和稳定性分析,模拟数值解,并根据数值解的有界性证明其存在性和唯一性。分析发现,在均匀网格上,空间精度为二阶。模拟的数值结果表明,所提出的方案在时间和空间上都提供了令人满意的二阶精度,并保留了离散不变式。此外,以往的科学文献综述几乎没有提供关于二维布森斯克范式方程中具有分散效应的研究项的证据。目前的研究通过应用所提出的方案对初始高斯条件进行数值分析来探索求解行为。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-05-31 DOI: 10.1002/num.23117
Issei Oikawa
In this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.
本文针对稳态扩散问题提出了基于通量的可杂化非连续伽勒金(HDG)方法,并通过让稳定参数趋于无穷大提出了一种新方法。假设存在 inf-sup 条件,我们证明了该方法的拟合优度和最优阶误差估计。我们证明了一些三角形元素满足 inf-sup 条件。我们还提供了数值结果来支持我们的理论结果。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-05-31 DOI: 10.1002/num.23118
Rodolphe Turpault
The space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied.
时空广义黎曼问题法可以获得任意高阶的数值方案,可用于具有源项的线性双曲守恒定律系统的超大时间步长。Berthon 等人(J. Sci. Comput.55 (2013), 268-308.) 中介绍过在一维和二维三角形非结构网格上的应用。本文的目的是对它们进行补充,以回答涉及它们时出现的一些重要问题。本文详细介绍了四边形网格的计算方法,讨论了近似基础的选择,并在实际案例中使用了 Legendre 多项式。此外,还考虑了在不影响精度的情况下添加限幅器以保持某些特性的问题。最后,研究了该方案在扩散机制中的渐近行为。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-05-29 DOI: 10.1002/num.23116
Jianguo Huang, Lili Ju, Yuejin Xu
In this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.
在本文中,我们借助变分公式和抛物线框架,提出了一种用于求解矩形域中半线性抛物方程的抛物线指数有限元方法。首先使用有限元法,利用连续片断多线性矩形基函数对模型方程进行空间离散化,得到半离散系统。然后,我们使用显式指数 Runge-Kutta 方法和 Parareal 框架对时间方向进行离散,从而得到全离散数值方案。为了进一步提高计算速度,我们设计了一种基于张量乘谱分解和快速傅立叶变换的快速求解器。在一定的正则性假设下,我们成功地推导出了基于并行方法的-正则最优误差估计值。我们还进行了大量二维和三维数值实验,以验证理论结果并证明我们方法的性能。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2024-05-06 DOI: 10.1002/num.23107
Yongseok Jang, Simon Shaw
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.
线性粘弹性可以用应力松弛函数来表征。我们考虑用幂律型应力松弛来产生分数阶粘弹性模型。控制方程是一个具有弱奇异内核的第二类 Volterra 积分问题。我们采用了空间不连续 Galerkin 方法、对称内部惩罚 Galerkin 方法(SIPG)进行空间离散化,以及时间隐式有限差分方案、Crank-Nicolson 方法。此外,为了处理 Volterra 核中的弱奇异性,我们使用了线性插值技术。我们提出了先验稳定性和误差分析,而不依赖于格伦沃尔不等式,因此提供了不会随时间呈指数增长的高质量边界。这表明我们的数值方案非常适合长时间模拟。尽管时间上的正则性有限,我们还是确定了 SIPG 在时间上的次优分数阶精度以及最佳收敛性。我们对精确解的不同规律性进行了数值实验,以验证我们的误差估计。最后,我们介绍了基于真实材料数据的数值模拟。
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引用次数: 0
期刊
Numerical Methods for Partial Differential Equations
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