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Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

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Research Square

Research Square 2021年

作者： Zhang, Yuhong Yang, Xiao-Song School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Science Computing Huazhong University of Science and Technology Wuhan430074 China

We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results. © 2021, CC BY.

关键词： Asymptotic stability

A diffuse-interface lattice Boltzmann method for fluid-particle interaction problems

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arXiv 2021年

作者： Liu, Jiao Huang, Changsheng Chaia, Zhenhua Shia, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, a diffuse-interface lattice Boltzmann method (DI-LBM) is developed for fluid-particle interaction problems. In this method, the sharp interface between the fluid and solid is replaced by a thin but nonzero thickness transition region named diffuse interface, where the physical variables varies continuously. In order to describe the diffuse interface, we introduce a smooth function, which is similar to the order parameter in phase-field model or the volume fraction of solid phase in the partially saturated lattice Boltzmann method (PS-LBM). In addition, to depict the fluid-particle interaction more accurately, a modified force term is also proposed and included in the evolution equation of the DI-LBM. Some classical problems are used to test the DI-LBM, and the results are in good agreement with some available theoretical and numerical works. Finally, it is also found that the DI-LBM is more efficient and accurate than the PS-LBM with the superposition model. © 2021, CC BY.

关键词： Kinetic theory

HYDRODYNAMICS FOR ASYMMETRIC SIMPLE EXCLUSION ON A FINITE SEGMENT WITH GLAUBER-TYPE SOURCE

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arXiv 2024年

作者： Xu, Lu Zhao, Linjie Gran Sasso Science Institute AQ L'Aquila67100 Italy School of Mathematics and Statistics Huazhong University of Science & Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in [3, 20], discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme. Copyright © 2024, The Authors. All rights reserved.

关键词： Hydrodynamics

A multiple-relaxation-time lattice boltzmann model based four-level finite-difference scheme for one-dimensional diffusion equation

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arXiv 2020年

作者： Lin, Yuxin Hong, Ning Shi, Baochang Chai, Zhenhua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of General Education Wuchang University of Technology Wuhan430223 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω0 and the relaxation parameters s1 and s2 corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations, and find that the numerical results are consistent with our theoretical analysis. © 2020, CC BY.

关键词： Finite difference method

A note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations

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arXiv 2017年

作者： Weng, Shangkun Xie, Chunjing School of Mathematics and Statistics Computational Science Hubei Key Laboratory Wuhan University Wuhan Hubei Province430072 China Department of Mathematics Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China

In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in R3. We obtain the optimal decay rate |u(x)| | xC|+1 for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component ua, that is, either O(+11) +1or |ue(r,z)| (C1r)3, where p = Vr2+ z2. In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: |ur(r, z)| + |uz(r, z)| C+1-. We do not require any small assumptions on the forcing *** Codes 76D05 Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

A spectral penalty method for two-sided fractional differential equations with general boundary conditions

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arXiv 2018年

作者： Wang, Nan Mao, Zhiping Huang, Chengming Karniadakis, George Em School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Division of Applied Mathematics Brown University ProvidenceRI02912 United States Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-τ, an extension of [23]) and demonstrate the superior accuracy of SPM for all cases considered. Copyright © 2018, The Authors. All rights reserved.

关键词： Constrained optimization

OBSERVABLE SETS, POTENTIALS AND SCHRÖDINGER EQUATIONS

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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 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.

关键词：

Linearly Implicit and High-Order Energy-Preserving Schemes for Highly Oscillatory Hamiltonian Systems

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SSRN

SSRN 2022年

作者： Li, Dongfang Li, Xiaoxi Zhang, Zhimin 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 Department of Mathematics Wayne State University DetroitMI48202 United States

In this paper, a family of novel energy-preserving schemes are presented for numerically solving highly oscillatory Hamiltonian systems. These schemes are constructed by using the relaxation idea in the extrapolated Runge–Kutta (ERK) methods. After obtaining the relaxation parameter, it is shown that the methods can be arbitrarily high order accurate, linearly implicit and keep the original discrete energy conserved, while the previous energy-preserving schemes for highly oscillatory Hamiltonian systems are usually fully implicit. Numerical comparisons with various typical energy-preserving schemes are presented. The numerical results show that the proposed schemes are highly competitive and effective. © 2022, The Authors. All rights reserved.

关键词： Runge Kutta methods

A finite-difference lattice Boltzmann model with second-order accuracy of time and space for incompressible flow

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arXiv 2019年

作者： Chen, Xinmeng Chai, Zhenhua Wang, Huili 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 School of Mathematics and Computer Science Wuhan Textile University Wuhan430073 China

In this paper, a kind of finite-difference lattice Boltzmann method with the second-order accuracy of time and space (T2S2-FDLBM) is proposed. In this method, a new simplified two-stage fourth order time-accurate discretization approach is applied to construct time marching scheme, and the spatial gradient operator is discretized by a mixed difference scheme to maintain a second-order accuracy both in time and space. It is shown that the previous finite-difference lattice Boltzmann method (FDLBM) proposed by Guo [1] is a special case of the T2S2-FDLBM. Through the von Neumann analysis, the stability of the method is analyzed and two specific T2S2-FDLBMs are discussed. The two T2S2-FDLBMs are applied to simulate some incompressible flows with the nonuniform grids. Compared with the previous FDLBM and SLBM, the T2S2-FDLBM is more accurate and more stable. The value of the Courant-Friedrichs-Lewy condition number in our method can be up to 0.9, which also significantly improves the computational efficiency. Copyright © 2019, The Authors. All rights reserved.

关键词： Incompressible flow