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On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

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arXiv 2020年

作者： Chen, Jianyu Zhang, Hong-Kun Zhang, Yiwei School of Mathematical Sciences Center for Dynamical Systems and Differential Equations Soochow University SuzhouJiangsu215006 China Department of Mathematics and Statistics University of Massachusetts Amherst AmherstMA01003 United States School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Hua-Zhong University of Sciences and Technology Wuhan430074 China

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov’s one-step expansion at q-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic Z function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem). For the first time, we are able to make a direct relation between the mixing rates and the Z function, see (2.7). The novelty of our results relies on establishing the regularity of invariant density, as well as verifying the stochastic properties for a large class of unbounded observables. Finally, we verify our assumptions for several well known examples that were previously studied in the literature, and unify results to these examples in our framework. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems

Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows

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arXiv 2019年

作者： Liu, Zhengguang Li, Xiaoli School of Mathematics and Statistics Shandong Normal University Jinan China Fujian Provincial Key Laboratory on Mathematical Modeling High Performance Scientific Computing and School of Mathematical Sciences Xiamen University Xiamen Fujian361005 China

In this paper, we propose several novel numerical techniques to deal with nonlinear terms in gradient ows. These step-by-step solving schemes, termed 3S-SAV and 3S-IEQ schemes, are based on recently popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) approaches. In these constructed numerical methods, the phase function φ and auxiliary variable can be calculated step-by-step. Compared with the traditional SAV/IEQ approaches, there are many advantages for the novel 3S-SAV/3S-IEQ schemes. Firstly, we do not need the restriction of the bounded from below of the nonlinear free energy potential/density function. Secondly, the auxiliary variable combined with nonlinear function can be treated totally explicitly in the 3S-SAV/3S-IEQ approaches. Specially, for solving the discrete scheme based on IEQ approach, the linear system usually involves variable coefficients which change at each time step. However, the discrete scheme based on 3S-IEQ approach leads to linear equation with constant coefficients. Two comparative studies of traditional SAV/IEQ and 3S-SAV/3S-IEQ approaches are considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy. Copyright © 2019, The Authors. All rights reserved.

关键词： Numerical methods

THE EXPONENTIAL SCALAR AUXILIARY VARIABLE (E-SAV) APPROACH FOR PHASE FIELD MODELS AND ITS EXPLICIT computing. ∗

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arXiv 2019年

作者： Zhengguang, L.I.U. Li, Xiaoli School of Mathematics and Statistics Shandong Normal University Jinan China Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing School of Mathematical Sciences Xiamen University XiamenFujian361005 China

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded from below restriction of the nonlinear free energy potential. Then, we prove the unconditional energy stability for the semi-discrete schemes carefully and rigorously. Another contribution is that we can discrete the auxiliary variable combined with the nonlinear term totally explicitly. Such modification is very efficient for fast calculation. Furthermore, the positive property of r can be guaranteed which is very important and reasonable for the models’ equivalence. Besides, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed E-SAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy. Copyright © 2019, The Authors. All rights reserved.

关键词： Numerical models

Spectral Deferred Correction Methods for Fractional Differential Equations

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高等学校计算数学学报（英文版） 2018年第4期11卷 729-751页

作者： Chunwan Lv Mejdi Azaiez Chuanju Xu School of Mathematics and Big Data Foshan University Guangdong 528000China I2M IPB(UMR CNRS 5295) Université de Bordeaux 33607 Pessac France School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University 361005 Xiamen China

In this paper,we propose and analyze a spectral deferred correction method for the fractional differential equation of order *** proposed method is based on a well-known finite difference method of (2-a)-order,see [Sun and Wu,***.,56(2),2006] and [Lin and Xu,***.,225(2),2007],for prediction of the numerical solution,which is then corrected through a spectral deferred correction *** order to derive the convergence rate of the prediction-correction iteration,we first derive an error estimate for the (2-a)-order finite difference method on some non-uniform *** the convergence rate of orders (O) (τ(2-α)(p+1)) and (O) (τ(2-α)+p) of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively,where τ is the maximal time step size and p is the number of correction *** performed numerical test confirms the efficiency of the proposed method.

关键词： Fractional differential equation spectral deferred correction method finite difference method

ERROR ANALYSIS OF THE SAV-MAC SCHEME FOR THE NAVIER-STOKES EQUATIONS

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arXiv 2019年

作者： Li, Xiaoli Shen, Jie School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University XiamenFujian361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell scheme (MAC) is constructed for the Navier-Stokes equations. A particular feature of the scheme is that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried out to show that both velocity and pressure approximations are second-order accurate in time and space. Numerical experiments are presented to verify the theoretical results. Copyright © 2019, The Authors. All rights reserved.

关键词： Viscous flow

On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes phase field model

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arXiv 2019年

作者： Li, Xiaoli Shen, Jie School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University XiamenFujian361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and error analysis. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase field variable, chemical potential, velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the our scheme. Copyright © 2019, The Authors. All rights reserved.

关键词： Navier Stokes equations

STABILITY AND ERROR ESTIMATES OF THE SAV FOURIER-SPECTRAL METHOD FOR THE PHASE FIELD CRYSTAL EQUATION ∗

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arXiv 2019年

作者： Li, Xiaoli Shen, Jie School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling High Performance Scientific Computing Xiamen University XiamenFujian361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAVapproach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(∆t2 + N−m), where ∆t, N and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. Copyright © 2019, The Authors. All rights reserved.

关键词： Spectroscopy

Analysis of the neighborhood pattern similarity measure for the role extraction problem

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arXiv 2020年

作者： Marchand, Melissa Gallivan, Kyle Huang, Wen van Dooren, Paul Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian China Department of Mathematical Engineering Université catholique de Louvain Louvain-La-Neuve Belgium

In this paper we analyze an indirect approach, called the Neighborhood Pattern Similarity approach, to solve the so-called role extraction problem of a large-scale graph. The method is based on the preliminary construction of a node similarity matrix which allows in a second stage to group together, with an appropriate clustering technique, the nodes that are assigned to have the same role. The analysis builds on the notion of ideal graphs where all nodes with the same role, are also structurally equivalent. Copyright © 2020, The Authors. All rights reserved.

关键词： Extraction

Well-posedness of compressible magneto-micropolar fluid equations

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arXiv 2019年

作者： Jia, Cuiman Tan, Zhong Zhou, Jianfeng School of Mathematical Sciences Xiamen University Xiamen Fujian361005 China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing Xiamen University Xiamen Fujian361005 China School of Mathematical Sciences Peking University Beijing100871 China

We are concerned with compressible magneto-micropolar fluid equations (1.1)-(1.2). The global existence and large time behaviour of solutions near a constant state to the magneto-micropolar- Navier-Stokes-Poisson (MMNSP) system is investigated in R3. By a refined energy method, the global existence is established under the assumption that the H3 norm of the initial data is small, but the higher order derivatives can be large. If the initial data belongs to homogeneous Sobolev spaces or homogeneous Besov spaces, we prove the optimal time decay rates of the solution and its higher order spatial derivatives. Meanwhile, we also obtain the usual Lp - L2 (1 ≤ p ≤ 2) type of the decay rates without requiring that the Lp norm of initial data is *** Codes 35Q35, 35M31, 76N15, 35B40, 74A35 Copyright © 2019, The Authors. All rights reserved.

关键词： Banach spaces