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Elastic wave propagation in curvilinear coordinates with mesh refinement interfaces by a fourth order finite difference method

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arXiv 2020年

作者： Zhang, Lu Wang, Siyang Petersson, N. Anders Department of Applied Physics and Applied Mathematics Columbia University New YorkNY10027 United States Department of Mathematics and Mathematical Statistics Umeå University Umeå Sweden Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94551 United States

We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form on curvilinear meshes by using a fourth order finite difference operator satisfying a summation-by-parts property. The method is energy stable and high order accurate. The highlight is that mesh sizes can be chosen according to the velocity structure of the material so that computational efficiency is improved. At the mesh refinement interfaces with hanging nodes, physical interface conditions are imposed by using ghost points and interpolation. With a fourth order predictor-corrector time integrator, the fully discrete scheme is energy conserving. Numerical experiments are presented to verify the fourth order convergence rate and the energy conserving *** Codes 65M06, 65M12 Copyright © 2020, The Authors. All rights reserved.

关键词： Finite difference method

Island-dynamics model for mound formation: Effect of a step-edge barrier

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Physical Review E 2014年第2期90卷 022404-022404页

作者： Joe Papac Dionisios Margetis Frederic Gibou Christian Ratsch Department of Mathematics University of California Los Angeles California 90095 USA Department of Mathematics and Institute for Physical Science and Technology and Center for Scientific Computation and Mathematical Modeling University of Maryland College Park Maryland 20742 USA Department of Mechanical Engineering and Department of Computer Science and Department of Mathematics University of California Santa Barbara California 93106 USA Department of Mathematics and Institute for Pure and Applied Mathematics University of California Los Angeles California 90095 USA

We formulate and implement a generalized island-dynamics model of epitaxial growth based on the level-set technique to include the effect of an additional energy barrier for the attachment and detachment of atoms at step edges. For this purpose, we invoke a mixed, Robin-type, boundary condition for the flux of adsorbed atoms (adatoms) at each step edge. In addition, we provide an analytic expression for the requisite equilibrium adatom concentration at the island boundary. The only inputs are atomistic kinetic rates. We present a numerical scheme for solving the adatom diffusion equation with such a mixed boundary condition. Our simulation results demonstrate that mounds form when the step-edge barrier is included, and that these mounds steepen as the step-edge barrier increases.

关键词： HEAT equation OPTOELECTRONIC devices EPITAXY CRYSTAL morphology CRYSTAL lattices

Slope selection in unstable multilayer growth in 1+1 dimensions: Step flow models with downward funneling

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Physical Review E 2019年第5期100卷 052802-052802页

作者： Ian Johnson Christian Ratsch Frederic Gibou Dionisios Margetis Department of Mathematics University of Maryland College Park Maryland 20742 USA Institute for Pure and Applied Mathematics and Department of Mathematics University of California Los Angeles California 90095 USA Department of Mechanical Engineering Department of Computer Science and Department of Mathematics University of California Santa Barbara California 93106 USA Institute for Physical Science and Technology Department of Mathematics and Center for Scientific Computation and Mathematical Modeling University of Maryland College Park Maryland 20742 USA

We study analytically and numerically aspects of the dynamics of slope selection for one-dimensional models describing the motion of line defects, steps, in homoepitaxial crystal growth. The kinetic processes include diffusion of adsorbed atoms (adatoms) on terraces, attachment and detachment of atoms at steps with large yet finite, positive Ehrlich-Schwoebel step-edge barriers, material deposition on the surface from above, and the mechanism of downward funneling (DF) via a phenomenological parameter. In this context, we account for the influence of boundary conditions at extremal steps on the dynamics of slope selection. Furthermore, we consider the effect of repulsive, nearest-neighbor force-dipole step-step interactions. For geometries with straight steps, we carry out numerical simulations of step flow, which demonstrate that slope selection eventually occurs. We apply perturbation theory to characterize time-periodic solutions of step flow for slope-selected profiles. By this method, we show how a simplified step flow theory with constant probabilities for the motion of deposited atoms can serve as an effective model of slope selection in the presence of DF. Our analytical findings compare favorably to step simulations.

关键词： Growth Interface & surface thermodynamics Surface & interfacial phenomena Surface diffusion Surface instabilities

A quasi-conservative DG-ALE method for multi-component flows using the non-oscillatory kinetic flux

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arXiv 2021年

作者： Luo, Dongmi Li, Shiyi Huang, Weizhang Qiu, Jianxian Chen, Yibing Institute of Applied Physics and Computational Mathematics Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China Department of Mathematics University of Kansas LawrenceKS66045 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China

A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces. Copyright © 2021, The Authors. All rights reserved.

关键词： Galerkin methods

Universal behavior of a dispersive Dirac cone in gradient-index plasmonic metamaterials

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Physical Review B 2018年第3期97卷 035307-035307页

作者： Matthias Maier Marios Mattheakis Efthimios Kaxiras Mitchell Luskin Dionisios Margetis School of Mathematics University of Minnesota Minneapolis Minnesota 55455 USA School of Engineering and Applied Sciences Harvard University Cambridge Massachusetts 02138 USA Department of Physics Harvard University Cambridge Massachusetts 02138 USA Department of Mathematics and Institute for Physical Science and Technology and Center for Scientific Computation and Mathematical Modeling University of Maryland College Park Maryland 20742 USA

We demonstrate analytically and numerically that the dispersive Dirac cone emulating an epsilon-near-zero (ENZ) behavior is a universal property within a family of plasmonic crystals consisting of two-dimensional (2D) metals. Our starting point is a periodic array of 2D metallic sheets embedded in an inhomogeneous and anisotropic dielectric host that allows for propagation of transverse-magnetic (TM) polarized waves. By invoking a systematic bifurcation argument for arbitrary dielectric profiles in one spatial dimension, we show how TM Bloch waves experience an effective dielectric function that averages out microscopic details of the host medium. The corresponding effective dispersion relation reduces to a Dirac cone when the conductivity of the metallic sheet and the period of the array satisfy a critical condition for ENZ behavior. Our analytical findings are in excellent agreement with numerical simulations.

关键词： Plasmonics Graphene Metamaterials Bloch wave theory Numerical techniques

Stability and Error Analysis of the Linear, Decoupled Imex Schemes for Sixth-Order Cahn-Hilliard-Type Equations

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SSRN

SSRN 2024年

作者： Zheng, Nan Shen, Jie Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China School of Mathematical Science Eastern Institute of Technology Ningbo China

In this paper, we develop efficient implicit-explicit (IMEX) schemes for solving sixth-order Cahn-Hilliard-type equations based on the generalized scalar auxiliary variable (GSAV) approach. These novel schemes provide several remarkable advantages: (i) they are linear and only require solving one elliptic equation with constant coefficients at each time step;(ii) they are unconditionally energy stable and yield a uniform bound for the numerical solution. We also establish rigorous error estimates of up to fifth-order for these schemes, and present various numerical experiments to validate the stability and accuracy of the proposed schemes. © 2024, The Authors. All rights reserved.

关键词： Error analysis

An adaptive moving mesh discontinuous Galerkin method for the radiative transfer equation

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arXiv 2018年

作者： Zhang, Min Cheng, Juan Huang, Weizhang Qiu, Jianxian School of Mathematical Sciences Xiamen University XiamenFujian361005 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Department of Mathematics University of Kansas LawrenceKS66045 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical *** Codes 65M50, 65M60, 65M70, 65R05, 65.75 Copyright © 2018, The Authors. All rights reserved.

关键词： Radiative transfer

Most Probable Transition Pathways and Maximal Likely Trajectories in a Genetic Regulatory System

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arXiv 2018年

作者： Cheng, Xiujun Wang, Hui Wang, Xiao Duan, Jinqiao Li, Xiaofan Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Statistics Henan University Kaifeng475000 China Department of Applied Mathematics Illinois Institute of Technology Chicago60616 United States School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator’s concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period);(ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise;(b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid sp

关键词： Trajectories

Wavelets based feature extraction with PCA for predicting autism in neonates using Navïe bayes classifier

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Research Square

Research Square 2021年

作者： Mensah, Isabel Amoako-Yirenkyi, Peter Frempong, Nana K. Lamptey, George P. National Institute for Mathematical Sciences Kumasi Ghana Department of Mathematics KNUST Kumasi Ghana Department of Statistics and actuarial Science KNUST Kumasi Ghana Center for Scientific Computing and Industrial modeling NIMS GHANA Kumasi Ghana Department of Biochemistry and Biotechnology KNUST Kumasi Ghana

Background: Current studies show early interventions of autism increase significant long-term positive effects symptoms and, later skills. Currently, These interventions are based on the use of an early diagnostic test. Existing methods for diagnosing Autism Spectrum Disorders (ASDs) such as cognitive tests, Intelligence Quotient, and standardized tests like the Autism Diagnostic Observation Schedule (ADOS) are functionally limited since they rely on child development for diagnoses. The standard is that a child must be at least three(3) years to undergo these tests. Accurate diagnosis is only possible after this period, and this may contribute to delayed diagnosis with an overall effect on the health system. In this era of increasing genetic data, it is possible to infer the genetic patterns of the disorder. This study introduces a novel and rigorous approach for predicting ASDs in neonates and their subsequent severity by identifying significant genes that contribute to the disorder. Methods: We used a wavelet transform and t-test to identify the significant genes that contribute to the disease. We subsequently employed the Naive Bayes classifier in the prediction of the autistic status of the neonate. Additionally, Principal Component Analysis (PCA) was employed to remove all the dependencies among the genes to enhance classification. Finally, we used the K-means clustering method to predict the severity level of the disease in the neonate. Results: Up to 200 differentially expressed genes were identified and used for predicting the ASD status of th child with a classification accuracy of 95.91%. Also, the results of the K-means demonstrated that the higher the mean of the cluster, the more severe the disease would be among that corresponding group. Optimizing and implementing these models in clinical settings may significantly reduce the health burden of ASDs. © 2021, CC BY.

关键词： Wavelet transforms