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A radial basis function (RBF)-finite difference method for solving improved Boussinesq model with error estimation and description of solitary waves 径向基函数-有限差分法求解具有误差估计和孤立波描述的改进Boussinesq模型
IF 3.9 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-11-15 DOI: 10.1002/num.23077
Mostafa Abbaszadeh, AliReza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej
The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi-discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function-finite difference method. The convergence rate and stability of the fully-discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme.
Boussinesq方程在流体力学、水科学等方面有一定的应用。本文研究了一种改进的Boussinesq模型。首先,采用有限差分近似对时间变量的导数进行离散。然后,根据不动点定理,研究了半离散格式解的存在唯一性。此外,还给出了半离散格式的无条件稳定性和收敛性。然后,基于径向基函数-有限差分法构造了全离散公式。分析了全离散格式的收敛速度和稳定性。最后,通过一维和二维实例验证了所提方案的有效性。
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引用次数: 0
Efficient and accurate temporal second-order numerical methods for multidimensional multi-term integrodifferential equations with the Abel kernels 具有阿贝尔核的多维多项积分微分方程的有效和精确的时间二阶数值方法
IF 3.9 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-11-14 DOI: 10.1002/num.23082
Mingchao Zhao, Hao Chen, Kexin Li
This work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three-dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high-dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order τ2+h14+h24�$$ {tau}^2+{h}_1^4+{h}_2^4 $$�, where τ�$$ tau $$�, h1�$$ {h}_1 $$�, and h2�$$ {h}_2 $$� denote the time-space step sizes, respectively, and the second scheme converges to the space-time second-order accuracy. Finally, the numerical results verify the correctness of the theoretical analysis and show that the method of this article is competitive with the existing research work.
本文给出了求解具有多项弱奇异阿贝尔核的多维抛物型积分微分方程的两种时间二阶交替方向隐式数值格式。对于二维(2D)情况,分别采用Crank-Nicolson方法和积积分规则对时间导数项和积分项进行离散化,并采用紧凑差分公式结合ADI算法对空间进行离散化;对于三维情况,采用与二维情况相同的时间离散化方法,然后利用空间上的标准有限差分构造一个完全离散的ADI有限差分格式。采用ADI技术降低了高维问题的计算成本。此外,通过能量论证严格证明了两种ADI格式的稳定性和收敛性,其中第一种格式收敛于τ2+h14+h24 $$ {tau}^2+{h}_1^4+{h}_2^4 $$阶,其中τ $$ tau $$、h1 $$ {h}_1 $$和h2 $$ {h}_2 $$分别表示时空步长,第二种格式收敛于时空二阶精度。最后,数值结果验证了理论分析的正确性,表明本文方法与已有的研究工作相比具有一定的竞争力。
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引用次数: 0
Unfitted generalized finite element methods for Dirichlet problems without penalty or stabilization 无惩罚或稳定Dirichlet问题的非拟合广义有限元方法
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-11-09 DOI: 10.1002/num.23081
Qinghui Zhang
Abstract Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM.
非拟合有限元法(FEM)对于具有演化边界或几何复杂边界的问题具有吸引人的优点。传统的非拟合fem采用惩罚项、参数或拉格朗日乘子来弱地施加狄利克雷边界条件。这在一定程度上增加了实现中的计算复杂性。本文针对Dirichlet问题,提出了一种不存在任何惩罚和镇定的非拟合广义有限元(GFEM)。这是通过划分GFEM的统一框架和设计一组新的Dirichlet边界富集来实现的。富集可分为两组:一组用于强施加Dirichlet边界条件,另一组用作变分公式的能量空间。能量空间的形状函数在边界处消失,可以采用传统的拟合有限元中的标准变分公式,从而不需要惩罚和稳定化。严格证明了能量范数下的最优收敛速度。通过数值实验和与其他方法的比较,验证了该算法的理论结果和有效性。数值结果表明,新方法的条件与标准有限元法的条件相同。
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引用次数: 0
Numerical approximation for hybrid‐dimensional flow and transport in fractured porous media 裂隙多孔介质中混合维流动和输运的数值近似
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-11-05 DOI: 10.1002/num.23080
Jijing Zhao, Hongxing Rui
Abstract This article presents the stable miscible displacement problem in fractured porous media, and finite element discretization is constructed for this reduced model. The transmission interface conditions presented in this article enable us to derive a stability result and conduct the case where the pressure and concentration are both discontinuous across the fracture. The error estimates for and norm are established under the assumption of regular solutions. We perform some numerical examples to verify the theoretical analysis. Last, some unsteady physical experiments, more realistic test cases, are presented to prove the validity of the model and method.
摘要本文提出了裂缝性多孔介质中稳定混相驱替问题,并对该简化模型进行了有限元离散化处理。本文提出的传输界面条件使我们能够得出稳定性结果,并进行压力和浓度在裂缝上均不连续的情况。在正则解的假设下,建立了误差估计和范数。通过算例验证了理论分析的正确性。最后,给出了一些非定常物理实验和较为实际的测试用例,验证了模型和方法的有效性。
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引用次数: 0
Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem 线性边值问题轴对称拉普拉斯方程的五阶精度数值算法
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-30 DOI: 10.1002/num.23079
Hu Li, Jin Huang
Abstract In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.
摘要为了在三维空间中得到具有线性边界问题的轴对称拉普拉斯方程的数值解,提出了求解该问题的正交法。首先,利用格林公式将问题转化为具有弱奇异算子的积分方程。其次,将中矩形公式与奇异积分公式相结合,构造了求解积分方程的正交法,该方法具有精度高、计算量小的优点。第三,利用紧算子理论证明了数值解的收敛性,得到了误差的奇次单参数渐近展开式。在此基础上,构造了外推算法(EA),进一步提高了数值解的精度。外推一次后,数值解的精度可达到。最后给出了两个数值算例,验证了该方法的有效性。
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引用次数: 0
Error analyses on block‐centered finite difference schemes for distributed‐order non‐Fickian flow 分布阶非菲克流块中心有限差分格式的误差分析
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-24 DOI: 10.1002/num.23078
Xuan Zhao, Ziyan Li
Abstract In this article, two numerical schemes are designed and analyzed for the distributed‐order non‐Fickian flow. Two different processing techniques are applied to deal with the time distributed‐order derivative for the constructed two schemes, while the classical block‐centered finite difference method is used in spatial discretization. To be precise, one adopts the standard numerical scheme called SD scheme in the temporal direction, and the other utilizes an efficient method called EF scheme. We derive the stabilities of the two schemes rigorously. The convergence result of the SD scheme for pressure and velocity is . However, to get a faster computing speed, the super parameter is needed for the EF scheme, which leads to the accuracy is . Finally, some numerical experiments are carried out to verify the theoretical analysis.
摘要本文设计并分析了分布阶非菲克流的两种数值格式。对于所构建的两种格式,采用了两种不同的处理技术来处理时间分布阶导数,而在空间离散化中则使用了经典的块中心有限差分方法。准确地说,一种是在时间方向上采用标准的数值格式SD格式,另一种是采用一种高效的方法EF格式。我们严格地推导了这两种方案的稳定性。压力和速度SD格式的收敛结果为。然而,为了获得更快的计算速度,EF格式需要超参数,导致精度为。最后通过数值实验验证了理论分析的正确性。
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引用次数: 0
Richardson extrapolation method for solving the Riesz space fractional diffusion problem 求解Riesz空间分数扩散问题的Richardson外推法
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-19 DOI: 10.1002/num.23076
Ren‐jun Qi, Zhi‐zhong Sun
Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.
摘要Richardson外推法具有精度高、易于实现等优点,在微分方程数值求解中得到了广泛的应用。对于Riesz空间分数阶扩散方程,采用分数中心差分算子逼近分数阶导数,利用离散解的渐近展开式构造了两种差分格式的Richardson外推方法。具体而言,对于Crank-Nicolson差分格式,外推方法包含两个外推公式,分别在时间和空间方向上达到四阶和六阶。紧致差分格式的外推方法包含一个外推公式,当时间步长与空间步长平方成正比时,可以得到六阶。利用离散分数Sobolev嵌入不等式证明了外推解的最大范数误差估计。对高维和非线性情况也进行了推广。数值结果验证了该方法的理论收敛阶和有效性。
{"title":"Richardson extrapolation method for solving the Riesz space fractional diffusion problem","authors":"Ren‐jun Qi, Zhi‐zhong Sun","doi":"10.1002/num.23076","DOIUrl":"https://doi.org/10.1002/num.23076","url":null,"abstract":"Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730881","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 semi‐Lagrangian ε$$ varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate 随机利率跳跃扩散下连续提取GMWBs的半拉格朗日ε $$ varepsilon $$单调傅立叶方法
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-19 DOI: 10.1002/num.23075
Yaowen Lu, Duy‐Minh Dang
Abstract We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.
在具有跳跃扩散和随机利率的现实建模设置下,我们开发了一种有效的连续提取保证最小提取收益(GMWBs)的定价方法。利用脉冲随机控制框架,我们将无套利GMWB定价问题表述为具有三维空间的具有交叉导数项的时变Hamilton - Jacobi - Bellman (HJB)拟变分不等式(QVI)。通过建立在半拉格朗日方法和相关线性偏积分微分方程的格林函数的组合上的一种新的数值方法,我们开发了一种单调傅里叶定价方法,其中是单调容忍的。结合HJB - QVI的一个可证明的强比较结果,我们从数学上证明了所提出的格式对HJB - QVI的粘度解的收敛性。我们全面研究了同时考虑子账户过程和随机利率的跳跃对GMWBs的无套利价格和公平保险费以及持有人的最优提现行为的影响。
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引用次数: 0
Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian 用谱伽辽金隐式Runge - Kutta方法求解带谱分数阶拉普拉斯的分数阶扩散方程
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-12 DOI: 10.1002/num.23074
Yanming Zhang, Yu Li, Yuexin Yu, Wansheng Wang
Abstract An efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.
摘要提出了一种求解具有谱分数阶拉普拉斯算子的分数阶扩散方程的高效、高精度的时间和空间数值方法。主要思想是用一阶隐式Runge - Kutta方法对时间进行离散化,用类傅里叶基函数的谱伽辽金方法对空间进行逼近。鉴于谱伽辽金方法的正交性,该方法的质量矩阵是恒等矩阵,刚度矩阵是对角矩阵,这使得该方法对高维问题是有效的。当经典阶()的隐式Runge‐Kutta方法是代数稳定的时,该方法是稳定且收敛的。作为本文的另一个重要贡献,我们导出了具有最优收敛阶的空间误差估计,该估计依赖于精确解的正则性而不依赖于分数参数。这改进了先前依赖于分数参数的结果。数值实验验证和补充了我们的理论结果。
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引用次数: 0
Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation 时间分数阶非线性Schrödinger方程的双网格有限元方法
3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-09 DOI: 10.1002/num.23073
Hanzhang Hu, Yanping Chen, Jianwei Zhou
Abstract A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.
摘要提出了求解时间分数阶非线性Schrödinger方程的非均匀L1格式双网格有限元方法。证明了-范数和-范数的有限元解在没有任何时间步长条件(取决于空间步长)的情况下是有界的。然后,在没有任何时间步长条件的情况下,证明了两网格解在范数下的最优阶误差估计。最后通过数值实验对理论结果进行了验证。
{"title":"Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation","authors":"Hanzhang Hu, Yanping Chen, Jianwei Zhou","doi":"10.1002/num.23073","DOIUrl":"https://doi.org/10.1002/num.23073","url":null,"abstract":"Abstract A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093531","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
期刊
Numerical Methods for Partial Differential Equations
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