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Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues

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Physical Review E 2023年第1期108卷 015304-015304页

作者： Zhenhua Chai (柴振华) Xiaolei Yuan (袁晓垒) Baochang Shi (施保昌) School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China College of Mathematics and Information Science Hebei University Baoding 071002 China

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter cs is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numeric

关键词： Lattice-Boltzmann methods

Characterizations of stabilizable sets for some parabolic equations in Rn

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arXiv 2020年

作者： Huang, Shanlin Wang, Gengsheng Wang, Ming School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Center for Applied Mathematics Tianjin University Tianjin300072 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China

We consider the parabolic type equation in Rn: (∂t + H)y(t, x) = 0, (t, x) ∈ (0, ∞) × Rn;y(0, x) ∈ L2(Rn), (0.1) where H can be one of the following operators: (i) a shifted fractional Laplacian;(ii) a shifted Hermite operator;(iii) the Schrödinger operator with some general potentials. We call a subset E ⊂ Rn as a stabilizable set for (0.1), if there is a linear bounded operator K on L2(Rn) so that the semigroup {e−t(H−χEK)}t≥0 is exponentially stable. (Here, χE denotes the characteristic function of E, which is treated as a linear operator on L2(Rn).) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a thick set, while when H is a shifted Hermite operator, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1, 18, 24, 32], reveal such phenomena: for some H, the class of stabilizable sets contains strictly the class of observable sets, while for some other H, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials. Copyright © 2020, The Authors. All rights reserved.

关键词： Partial differential equations

OBSERVABLE SETS, POTENTIALS AND SCHRÖDINGER EQUATIONS

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arXiv 2020年

We characterize observable sets for 1-dim Schrödinger equations in R: i∂tu = (−∂2x + x2m)u (with m ∈ N:= {0, 1, . . . }). More precisely, we obtain what follows: First, when m = 0, E ⊂ R is an observable set at some time if and only if it is thick, namely, there is γ > 0 and L > 0 so that E \ [x, x + L] ≥ γL for each x ∈ R;Second, when m = 1 (m ≥ 2 resp.), E is an observable set at some time (at any time resp.) if and only if it is weakly thick, namely lim |E T[−x, x]| > 0. x x→+∞ From these, we see how potentials x2m affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time T > 0 for the above equation with m = 1 if and only if T > π2 .MSC Codes 93B07, 35J10 Copyright © 2020, The Authors. All rights reserved.

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Characterization of the most probable transition paths of stochastic dynamical systems with stable Lévy noise

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arXiv 2018年

作者： Huang, Yuanfei Chao, Ying Yuan, Shenglan Duan, Jinqiao 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 Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is devoted to the investigation of the most probable transition path for stochastic dynamical systems driven by either symmetric α-stable Lévy motion (0 Copyright © 2018, The Authors. All rights r... 详细信息

关键词： Brownian movement

The role of slow manifolds in parameter estimation for a multiscale stochastic system with α-stable Lévy noise

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arXiv 2020年

作者： Chao, Ying Wei, Pingyuan Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences 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

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian α-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by p-moment with p ∈ (1, α), with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system, a prototypical example is present. Copyright © 2020, The Authors. All rights reserved.

关键词： Parameter estimation

The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-Diffusion Process

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arXiv 2018年

作者： Chao, Ying Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences 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

This work is devoted to deriving the Onsager-Machlup function for a class of stochastic dynamical systems under (non-Gaussian) Lévy noise as well as (Gaussian) Brownian noise, and examining the corresponding most probable paths. This Onsager-Machlup function is the Lagrangian giving the most probable path connecting metastable states for jump-diffusion processes. This is done by applying the Girsanov transformation for measures induced by jump-diffusion processes. Moreover, we have found this Lagrangian function is consistent with the result in the special case of diffusion processes. Finally, we apply this new Onsager-Machlup function to investigate dynamical behaviors analytically and numerically in several examples. These include the transitions from one metastable state to another metastable state in a double-well system, with numerical experiments illustrating most probable transition paths for various noise parameters. Copyright © 2018, The Authors. All rights reserved.

关键词： Lagrange multipliers

Topology optimization including a model of the layer-by-layer additive manufacturing process

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arXiv 2022年

作者： Haveroth, Geovane A. Thore, C.-J. Correa, M.R. Ausas, R.F. Jakobsson, S. Cuminato, J.A. Klarbring, A. Department of Applied Mathematics and Statistics Institute of Mathematics and Computer Sciences University of São Paulo SP São Carlos Brazil Division of Solid Mechanics Department of Management and Engineering Institute of Technology Linköping University Linköping581 83 Sweden Arcam EBM Mölndal Sweden Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computing University of Campinas Campinas-SP Brazil

A topology optimization formulation including a model of the layer-by-layer additive manufacturing (AM) process is considered. Defined as a multi-objective minimization problem, the formulation accounts for the performance and cost of both the final and partially manufactured designs and allows for considering AM-related issues such as overhang and residual stresses in the optimization. The formulation is exemplified by stiffness optimization in which the overhang is limited by adding mechanical or thermal compliance as a measure of the cost of partially manufactured designs. Convergence of the model as the approximate layer-by-layer model is refined is shown theoretically, and an extensive numerical study indicates that this convergence can be fast, thus making it a computationally viable approach useful for including AM-related issues into topology optimization. The examples also show that drips and sharp corners associated with some geometry-based formulations for overhang limitation can be avoided. The codes used in this article are written in Python using only open sources libraries and are available for reference. Copyright © 2022, The Authors. All rights reserved.

关键词： Stiffness

A parameter estimator based on Smoluchowski-Kramers approximation 1

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arXiv 2018年

作者： He, Ziying Duan, Jinqiao Cheng, Xiujun Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski-Kramers approximation. We establish the consistency of the estimator by using Γ-convergence theory. We further illustrate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Vaccine efficacy and SARS-CoV-2 control in California and *** the session 2020e2026:A modeling study

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Infectious Disease Modelling 2022年第1期7卷 62-81页

作者： Md Shahriar Mahmud Md Kamrujjaman Md Mashih Ibn Yasin Adan Md Alamgir Hossain Md Mizanur Rahman Md Shahidul Islam Muhammad Mohebujjaman Md Mamun Molla Department of Computer Science and Engineering State University of BangladeshDhaka1205Bangladesh Department of Mathematics University of DhakaDhaka1000Bangladesh Department of Mathematics and Statistics University of CalgaryCalgaryABCanada Computational Biology Research Lab(CBRL) Department of PharmacyJagannath UniversityDhaka1100Bangladesh Hitotsubashi Institute for Advanced Study Hitotsubashi UniversityNaka KunitachiTokyo186-8601Japan Department of Mathematics and Physics Texas A&M International UniversityLaredoTX78041USA Department of Mathematics&Physics North South UniversityDhaka1229Bangladesh Center for Applied Scientific Computing(CASC) North South UniversityDhaka1229Bangladesh

Background:Besides maintaining health precautions,vaccination has been the only prevention from SARS-CoV-2,though no clinically proved 100%effective vaccine has been developed till *** this stage,to withhold the debris of this pandemic-experts need to know the impact of the vaccine efficacy rates,the threshold level of vaccine effectiveness and how long this pandemic may extent with vaccines that have different efficacy *** this article,a mathematical model study has been done on the importance of vaccination and vaccine efficiency rate during an ongoing ***:We simulated a five compartment mathematical model to analyze the pandemic scenario in both California,and whole *** considered four vaccines,Pfizer(95%),Moderna(94%),AstraZeneca(79%),and Johnson&Johnson(72%),which are being used rigorously to control the SARS-CoV-2 pandemic,in addition with two special cases:a vaccine with 100%efficacy rate and no vaccine under ***-CoV-2 related data of California,and *** used in this ***:Both the infection and death rates are very high in *** model suggests that the pandemic situation in California will be under control in the last quartile of the year 2023 if vaccination program is continued with the Pfizer *** this time,six waves may happen from the beginning of the immunization where the case fatality and recovery rates will be 1.697%and 98.30%,***,according to the considered model,this period might be extended to the mid of 2024 when vaccines with lower efficacy rates are *** the other hand,the daily cases and deaths in the *** be under control at the end of 2026 with multiple *** the number of susceptible people will fall down to none in the beginning of 2027,there is less chance to stop the vaccination program if vaccinated with a vaccine other than a 100%effective vaccine or Pfizer,and at that case vaccination program must run till the mid of *** to this s

关键词： Vaccine Model California Control measurement SARS-CoV-2