# Browsing by Author "Yue, X"

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Item Open Access Behavior of different numerical schemes for random genetic drift(BIT Numerical Mathematics, 2019-09-01) Xu, S; Chen, M; Liu, C; Zhang, R; Yue, XIn the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.Item Open Access Homogenization for chemical vapor infiltration process(Communications in Mathematical Sciences, 2017-01-01) Zhang, C; Bai, Y; Xu, S; Yue, XMulti-scale modeling and numerical simulations of the isothermal chemical vapor infiltration (CVI) process for the fabrication of carbon fiber reinforced silicon carbide (C/SiC) composites were presented in [Bai, Yue and Zeng, Commun. Comput. Phys., 7(3):597-612, 2010]. The homogenization theory, which played a fundamental role in the multi-scale algorithm, will be rigorously established in this paper. The governing system, which is a multi-scale reaction-diffusion equation, is different in the two stages of CVI process, so we will consider the homogenization for the two stages respectively. One of the main features is that the reaction only occurs on the surface of fiber, so it behaves as a singular surface source. The other feature is that in the second stage of the process when the micro pores inside the fiber bundles are all closed, the diffusion only occurs in the macro pores between fiber bundles and we face up with a problem in a locally periodic perforated domain.Item Open Access Homogenization of thermal-hydro-mass transfer processes(Discrete and Continuous Dynamical Systems - Series S, 2015-02-01) Xu, S; Yue, XIn the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coefficients. In this paper we first derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the first order expansions.Item Open Access Homogenization theory of ion transportation in multicellular tissue(Discrete and Continuous Dynamical Systems - B, 2023) Xiao, C; Huang, H; Xu, S; Yu, T; Yue, XItem Open Access Homogenization: In mathematics or physics?(Discrete and Continuous Dynamical Systems - Series S, 2016-10-01) Xu, S; Yue, X; Zhang, CIn mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in axed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium isxed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to innity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in H1, while in standard homogenization theory, the source term is assumed to be at least compacted in H1. A real example is also given to show the validation of our observation and results.Item Open Access Modeling and Simulating Asymmetrical Conductance Changes in Gramicidin Pores(Computational and Mathematical Biophysics, 2014-01-01) Xu, S; Chen, M; Majd, S; Yue, X; Liu, CGramicidin A is a small and well characterized peptide that forms an ion channel in lipid membranes. An important feature of gramicidin A (gA) pore is that its conductance is affected by the electric charges near the its entrance. This property has led to the application of gramicidin A as a biochemical sensor for monitoring and quantifying a number of chemical and enzymatic reactions. Here, a mathematical model of conductance changes of gramicidin A pores in response to the presence of electrical charges near its entrance, either on membrane surface or attached to gramicidin A itself, is presented. In this numerical simulation, a two dimensional computational domain is set to mimic the structure of a gramicidin A channel in the bilayer surrounded by electrolyte. The transport of ions through the channel is modeled by the Poisson-Nernst-Planck (PNP) equations that are solved by Finite Element Method (FEM). Preliminary numerical simulations of this mathematical model are in qualitative agreement with the experimental results in the literature. In addition to the model and simulations, we also present the analysis of the stability of the solution to the boundary conditions and the convergence of FEM method for the two dimensional PNP equations in our model.Item Open Access Numerical method for multi-alleles genetic drift problem(SIAM Journal on Numerical Analysis, 2019-01-01) Xu, S; Chen, X; Liu, C; Yue, XGenetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright-Fisher model. The diffusion limit of this model is a degenerate diffusion-convection equation. Due to the degeneration and convection, Dirac singularities will always develop at the boundaries as time evolves, i.e., the fixation phenomenon occurs. Theoretical analysis has proven that the weak solution of this equation, regarded as measure, conserves total probability and expectations. In the current work, we propose a scheme for 3-alleles model with absolute stability and generalize it to N-alleles case (N > 3). Our method can conserve not only total probability and expectations, but also positivity. We also prove that the discrete solution converges to a measure as the mesh size tends to zero, which is the exact measure solution of the original problem. The simulations illustrate that the probability density decays to zero first on the inner nodes, then also on the edge nodes except at the three vertex nodes, on which the density finally concentrates. The results correctly predict the fixation probability and are consistent with theoretical ones and with direct Monte Carlo simulations.