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NON-SPLITTING EULERIAN-LAGRANGIAN WENO SCHEMES FOR TWO-DIMENSIONAL NONLINEAR CONVECTION-DIFFUSION EQUATIONS

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arXiv 2024年

作者： Zheng, Nanyi Cai, Xiaofeng Qiu, Jing-Mei Qiu, Jianxian Department of Mathematical Sciences University of Delaware NewarkDE19716 United States Research Center for Mathematics Advanced Institute of Natural Sciences Beijing Normal University Zhuhai519087 China Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science BNU-HKBU United International College Zhuhai519087 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, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme. © 2024, CC BY.

关键词： Mesh generation

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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arXiv 2021年

作者： Cao, Waixiang Li, Dongfang Zhang, Zhimin School of Mathematical Sciences Beijing Normal University Beijing100875 China 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 Beijing Computational Science Research Center Beijing100193 China Department of Mathematics Wayne State University DetroitMI48202 United States

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without time-step dependent on the spatial mesh size. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence, and optimal error estimates, respectively, of the proposed fully discrete *** Codes 65M60, 65M12, 65N12 © 2021, CC0.

关键词： Wave equations

Homogenization of plasmonic crystals: Seeking the epsilon-near-zero effect

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arXiv 2018年

作者： Maier, Matthias Mattheakis, Marios Kaxiras, Efthimios Luskin, Mitchell Margetis, Dionisios Department of Mathematics Texas A&M University College StationTX77843 United States John A. Paulson School of Engineering and Applied Sciences Harvard University CambridgeMA02138 United States Department of Physics Harvard University CambridgeMA02138 United States School of Mathematics University of Minnesota MinneapolisMN55455 United States Department of Mathematics Institute for Physical Science and Technology Center for Scientific Computation and Mathematical Modeling University of Maryland College ParkMD20910 United States

By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell's equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host. This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. We demonstrate how our homogenization procedure may set the foundation for computational investigations of: effective optical responses of reasonably general geometries, and complicated design problems in the plasmonics of 2D materials.35B27, 35Q60, 74Q10 Copyright © 2018, The Authors. All rights reserved.

关键词： Maxwell equations

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

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arXiv 2021年

作者： Bootland, Niall Dolean, Victorita Graham, Ivan G. Ma, Chupeng Scheichl, Robert Department of Mathematics and Statistics University of Strathclyde GlasgowG1 1XH United Kingdom Laboratoire J.A. Dieudonné CNRS University Côte d’Azur Nice06108 France Department of Mathematical Sciences University of Bath BathBA2 7AY United Kingdom Institute for Applied Mathematics Interdisciplinary Center for Scientific Computing Heidelberg University Heidelberg69120 Germany

GenEO (‘Generalised Eigenvalue problems on the Overlap’) is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers which are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection–diffusion–reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalised eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts which are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the non-self-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus we obtain an iterative solver which is efficient in parallel and very effective for a wide range of convection–diffusion–reaction *** Codes 65N22, 65N55, 65F10 Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method

Glasdi: Parametric Physics-Informed Greedy Latent Space Dynamics Identification

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SSRN

SSRN 2022年

作者： He, Xiaolong Choi, Youngsoo Fries, William D. Belof, Jon Chen, Jiun-Shyan Department of Structural Engineering University of California La Jolla San DiegoCA92093 United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States Applied Mathematics School of Mathematical Sciences University of Arizona TucsonAZ85721 United States Physical and Life Science Lawrence Livermore National Laboratory LivermoreCA94550 United States

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust datadriven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residualbased error indicator and random-subset evaluation is introduced to search for the optimal training samples on-the-fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, an efficient k -nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 66 to 4,417× speed-up with 1 to 5% relative errors. © 2022, The Authors. All rights reserved.

关键词： Nonlinear dynamical systems

High-resolution boundary variation diminishing scheme for two-phase compressible flow with cavitation and evaporation

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Journal of Computational Physics 2024年 513卷

作者： Wakimura, Hiro Li, Tatsuin Shyue, Keh-Ming Aoki, Takayuki Xiao, Feng Department of Mechanical Engineering Tokyo Institute of Technology 2-12-1 i6-29 Ookayama Meguro-ku Tokyo152-8550 Japan Department of Mathematics Institute of Applied Mathematical Sciences National Taiwan University Taipei10617 Taiwan Global Scientific Information and Computing Center Tokyo Institute of Technology 2-12-1 i7-3 Ookayama Meguro-ku Tokyo152-8550 Japan

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. © 2024 Elsevier Inc.

关键词： Finite volume method

Virtual excitations and quantum correlations in ultra-strongly coupled harmonic oscillators under intrinsic decoherence

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arXiv 2022年

作者： Hab-Arrih, Radouan Jellal, Ahmed El Kinani, El Hassan Laboratory of Theoretical Physics Faculty of Sciences Chouaïb Doukkali University PO Box 20 El Jadida24000 Morocco Canadian Quantum Research Center 204-3002 32 Ave VernonBCV1T 2L7 Canada Mathematical Modeling and Scientific Computing Team Department of Mathematics Faculty of Sciences Moulay Ismail University Meknes Morocco

We study the intrinsic decoherence of coupled harmonic oscillators. The Milburn master equation is solved exactly, and the dynamics of virtual ground state excitations are investigated. The interaction of quantum correlations and virtual excitation was then studied. The following is a summary of our major findings. (i) The damped oscillatory profile of all three quantities is the same. (ii) Ultra-strong coupling combined with huge anisotropy values results in the reemergence of entanglement and steering. (iii) To sustain entanglement and steering, virtual excitations are required. (iv) The quantum correlations are amplified in the quantum synchronous regime. (v) Ultra-strong couplings cause inherent decoherence to be avoided. © 2022, CC BY.

关键词： Ground state

Virtual Excitations and Quantum Correlations in Ultra-Strongly Coupled Harmonicoscillators Under Intrinsic Decoherence

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SSRN

SSRN 2022年

作者： Jellal, Ahmed Hab-Arrih, Radouan Chnafa, Hasna El Kinani, El Hassan Laboratory of Theoretical Physics Faculty of Sciences Chouaïb Doukkali University PO Box 20 El Jadida24000 Morocco Canadian Quantum Research Center 204-3002 32 Ave VernonBCV1T 2L7 Canada Mathematical Modeling and Scientific Computing Team Department of Mathematics Faculty of Sciences Moulay Ismail University Meknes Morocco

关键词： Anisotropy

Scalable Multiscale-Spectral Gfem for Composite Aero-Structures

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SSRN

SSRN 2022年

作者： Bénézech, Jean Seelinger, Linus Bastian, Peter Butler, Richard Dodwell, Timothy Ma, Chupeng Scheichl, Robert Material & Structure Research Centre University of Bath BathBA2 7AY United Kingdom Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing Heidelberg University Heidelberg Germany Institute of Data Science and AI University of Exeter DigiLab Exeter United Kingdom Department of Mathematical Sciences University of Bath Bath United Kingdom Institute of Scientific Research Great Bay University Dongguan China

Ill-conditioned and multiscale partial differential equations (PDEs) arise in many fields. It is a very challenging problem to compute a resolved, fine-scale solution or to find a robust low-dimensional approximation. In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. A problem-specific coarse space generated from local eigenproblems yields a spectral element-type method with excellent approximation properties at low basis size, even for challenging multiscale problems. The implementation of the framework in the DUNE software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures. © 2022, The Authors. All rights reserved.

关键词： Finite element method