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DENSITY CONVERGENCE OF A FULLY DISCRETE FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

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arXiv 2022年

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn-Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L1() of the numerical solution. Our results partially answer positively to the open problem emerged in [J. Cui and J. Hong, J. Differential Equations (2020)] on computing the density of the exact solution numerically. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite difference method

A stochastic pitchfork bifurcation in most probable phase portraits

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arXiv 2018年

作者： Wang, Hui Chen, Xiaoli Duan, Jinqiao Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan430074 China

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or none backward pitchfork type bifurcations;(ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations;(iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation. Copyright © 2018, The Authors. All rights reserved.

关键词： Gaussian noise (electronic)

Ergodic optimization theory for axiom a flows

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arXiv 2019年

作者： Huang, Wen Lian, Zeng Ma, Xiao Xu, Leiye Zhang, Yiwei Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Sciences Department of Mathematics University of Science and Technology of China Hefei Anhui China College of Mathematical Sciences Sichuan University Chengdu Sichuan610016 China 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 article, we consider the weighted ergodic optimization problem Axiom A attractors of a C2flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space C0,α(α∈ (0, 1]) or C1the minimizing measure is unique and is supported on a periodic orbit. Copyright © 2019, The Authors. All rights reserved.

关键词： Site selection

THERMODYNAMIC FORMALISM FOR CORRESPONDENCES

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arXiv 2023年

作者： Li, Xiaoran Li, Zhiqiang Zhang, Yiwei School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this work, we investigate the Variational Principle and develop a thermodynamic formalism for correspondences. We define the measure-theoretic entropy for transition probability kernels and topological pressure for correspondences. Based on these two notions, we establish the following results: The Variational Principle holds and equilibrium states exist for continuous potential functions, provided that the correspondence satisfies some expansion property called forward expansiveness. If, in addition, the correspondence satisfies the specification property and the potential function is Bowen summable, then the equilibrium state is unique. On the other hand, for a distance-expanding, open, strongly transitive correspondence and a Hölder continuous potential function, there exists a unique equilibrium state, and the backward orbits are equidistributed. Furthermore, we investigate the Variational Principle for general correspondences. In conformal dynamics, we establish the Variational Principle for the Lee-Lyubich-Markorov-Mazor-Mukherjee anti-holomorphic correspondences, which are matings of some anti-holomorphic rational maps with anti-Hecke groups and not forward expansive. We also show a Ruelle-Perron-Frobenius Theorem for a family of hyperbolic holomorphic correspondences of the form fc(z) = zq/p + *** Codes Primary: 37F05, Secondary: 54C60, 37D35, 37C30, 37D20 Copyright © 2023, The Authors. All rights reserved.

关键词： Thermodynamics

A LOCALLY MASS-CONSERVATIVE ENRICHED PETROV-GALERKIN METHOD WITHOUT PENALTY FOR THE DARCY FLOW IN POROUS MEDIA

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arXiv 2024年

作者： Chen, Huangxin Dong, Piaopiao Sun, Shuyu Wang, Zixuan School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal Saudi Arabia

In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed *** Codes 65M60, 65N30, 76S05 Copyright © 2024, The Authors. All rights reserved.

关键词： Error analysis

A wong-zakai approximation for random slow manifolds with application to parameter estimation

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arXiv 2018年

作者： He, Ziying Zhang, Xinyong Jiang, Tao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Collaborative Innovation Center of China Pilot Reform Exploration and Assessment Hubei Sub-Center Hubei University of Economics Wuhan430205 China

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this *** Codes Primary: 37L55, 35R60, Secondary: 60H15, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Parameter estimation

High-Order Discontinuous Galerkin Schemes for Radiation Hydrodynamics Equations in the Equilibrium-Diffusion Limit

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SSRN

SSRN 2023年

作者： Peng, Gang Luo, Dongmi Qiu, Jianxian Chen, Yibing Command and Control Engineering College Army Engineering University of PLA Nanjing210007 China Institute of Applied Physics and Computational Mathematics National Key Laboratory of Computational Physics Beijing100088 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, the high-order discontinuous Galerkin (DG) schemes are presented for radiation hydrodynamics equations in the equilibrium-diffusion limit. The governor equations contain two parts, one part is a hyperbolic system of conservation laws and another part is a radiative heat transfer term. The discrete schemes are constructed in three parts and possess characteristics of the high-order, oscillation-free and efficient. Firstly, the DG method and local DG (LDG) method are respectively used to discretize the hyperbolic system and radiative heat transfer term to achieve high-order spatial accuracy. Secondly, the combination of the KXRCF shock detector, Hermite WENO limiter, and positivity-preserving limiter is utilized to enhance the robustness of the discretization schemes. Finally, the implicit-explicit schemes are employed for time discretization to enable the schemes suitable for larger time steps. Numerical experiments have verified that the new schemes are high-order accuracy, oscillation-free and efficiency. © 2023, The Authors. All rights reserved.

关键词： Hydrodynamics

Modelling brain-wide neuronal morphology via rooted Cayley trees

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arXiv 2018年

作者： Lin, Congping Huang, Yuanfei Quan, Tingwei Zhang, Yiwei Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan China Hubei Key Lab of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China Britton Chance Center for Biomedical Photonics Wuhan National Laboratory for Optoelectronics-Huazhong University of Science and Technology Wuhan China MoE Key Laboratory for Biomedical Photonics Collaborative Innovation Center for Biomedical Engineering School of Engineering Sciences Huazhong University of Science and Technology Wuhan China

Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent features of brain-wide neuronal morphology. We observe that axonal trees are self-affine while dendritic trees are self-similar. We also show that tree size appear to be random, independent of the number of dendrites within single neurons. Moreover, we consider inhomogeneous branching model which stochastically generates rooted 3-Cayley trees for the brain-wide neuron topology. Based on estimated order-dependent branching probability from actual axonal and dendritic trees, our inhomogeneous model quantitatively captures a number of topological features including size and shape of both axons and dendrites. This sheds lights on a universal mechanism behind the topological formation of brain-wide axonal and dendritic trees. Copyright © 2018, The Authors. All rights reserved.

关键词： Brain

A lattice Boltzmann model for two-phase flow in porous media

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arXiv 2018年

作者： Chai, Zhenhua Liang, Hong Du, Rui Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan430074 China Department of Physics Hangzhou Dianzi University Hangzhou310018 China School of Mathematics Southeast University Nanjing210096 China

In this paper, a lattice Boltzmann (LB) model with double distribution functions is proposed for two-phase flow in porous media where one distribution function is used for pressure governed by the Poisson equation, and the other is applied for saturation evolution described by the convection-diffusion equation with a source term. We first performed a Chapman-Enskog analysis, and show that the macroscopic nonlinear equations for pressure and saturation can be recovered correctly from present LB model. Then in the framework of LB method, we develop a local scheme for pressure gradient or equivalently velocity, which may be more efficient than the nonlocal second-order finite-difference schemes. We also perform some numerical simulations, and the results show that the developed LB model and local scheme for velocity are accurate and also have a second-order convergence rate in space. Finally, compared to the available pore-scale LB models for two-phase flow in porous media, the present LB model has more potential in the study of the large-scale problems. Copyright © 2018, The Authors. All rights reserved.

关键词： Two phase flow