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A Low-rank solver for the stochastic unsteady navier-stokes problem

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arXiv 2019年

作者： Elman, Howard Su, Tengfei Department of Computer Science Institute for Advanced Computer Studies University of Maryland College ParkMD20742 United States Applied Mathematics & Statistics and Scientific Computation Program University of Maryland College ParkMD20742 United States

We study a low-rank iterative solver for the unsteady Navier–Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard’s method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Combined with effective mean-based preconditioners and the idea of inexact solve, we show that only a small number of linear iterations are needed at each Picard step. The proposed algorithm is tested with a model of flow in a two-dimensional symmetric step domain with different settings to demonstrate the computational efficiency. Copyright © 2019, The Authors. All rights reserved.

关键词： Navier Stokes equations

A unified fourth-order Bhatnagar-Gross-Krook lattice Boltzmann model for high-dimensional linear hyperbolic equations

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arXiv 2024年

作者： Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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

In this work, we first develop a unified Bhatnagar-Gross-Krook lattice Boltzmann (BGK-LB) model for the d(d ≥ 1)-dimensional linear hyperbolic equation (L-HE), where the natural moments and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure are considered. Subsequently, at the acoustic scaling, we conduct an accuracy analysis on the developed BGK-LB model by the direct Taylor expansion (DTE) method, and present the second- and third-order moments of the equilibrium distribution functions (EDFs) to ensure that the BGK-LB model can be fourth-order consistent with the L-HE. And on this basis, when considering the Dirichlet boundary condition, the fourth-order full-way and half-way boundary schemes are proposed to approximate the unknown distribution functions to ensure that the BGK-LB model can be overall fourth-order accurate. Thereafter, based on the kinetic entropy theory, we derive the conditions that the fourth-order moments of the EDFs should satisfy to ensure the microscopic entropy stability of the BGK-LB model. In addition, with the aid of the von Neumann stability analysis, we also discuss the L2 stability of the BGK-LB model and numerically plot the stability regions. In particular, from a numerical perspective, we find that the region of microscopic entropy stability is identical to that of L2 stability. Finally, we carry out some numerical experiments to test the accuracy and stability of the BGK-LB model, and the numerical results are in agreement with our theoretical analysis. In addition, we compare the developed full-way and half-way boundary schemes for the Dirichlet boundary condition, which shows that the full-way boundary scheme is more stable. Copyright © 2024, The Authors. All rights reserved.

关键词： Entropy

A consistent and conservative diffuse-domain lattice Boltzmann method for multiphase flows in complex geometries

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arXiv 2023年

作者： Liu, Xi Zhan, Chengjie Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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

Modeling and simulation of multiphase flows in complex geomerties are challenging due to the complexity in describing the interface topology changes among different phases and the difficulty in implementing the boundary conditions on the irregular solid surface. In this work, we first developed a diffuse-domain (DD) based phase-field model for multiphase flows in complex geometries. In this model, the irregular fluid region is embedded into a larger and regular domain by introducing a smooth characteristic function. Then, the reduction-consistent and conservative phase-field equation for the multiphase field and the consistent and conservative Navier-Stokes equations for the flow field are reformulated as the diffuse-domain based consistent and conservative (DD-CC) equations where some additional source terms are added to reflect the effects of boundary conditions. In this case, there is no need to directly treat the complex boundary conditions on the irregular solid surface, and additionally, based on a matched asymptotic analysis, it is also shown that the DD-CC equations can converge to the original governing equations as the interface width parameter tends to zero. Furthermore, to solve the DD-CC equations, we proposed a novel and simple lattice Boltzmann (LB) method with a Hermite-moment-based collision matrix which can not only keep consistent and conservation properties, but also improve the numerical stability with a flexible parameter. With the help of the direct Taylor expansion, the macroscopic DD-CC equations can be recovered correctly from the present LB method. Finally, to test the capacity of LB method, several benchmarks and complex problems are considered, and the numerical results show that the present LB method is accurate and efficient for the multiphase flows in complex geomerties. © 2023, CC BY.

关键词： Boundary conditions

A phase-field-based lattice Boltzmann method for two-phase flows with the interfacial mass/heat transfer

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arXiv 2024年

作者： Chen, Baihui Zhan, Chengjie Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass/heat transfer. Through the Chapman–Enskog analysis, we show that the present LB method can correctly recover the governing equations for phase field, flow field and concentration/temperature field. In particular, to derive the two-scalar equations for the mass/heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided. The accuracy and efficiency of the present method is also tested through several benchmark problems, and the influence of mass/heat transfer on the fluid viscosity is further considered by introducing an exponential relation. The numerical results show that the present LB method is suitable for the two-phase flows with interfacial mass/heat transfer. © 2024, CC BY.

关键词： Distribution functions

Enhancing Electrocardiography Data Classification Confidence: A Robust Gaussian Process Approach (MuyGPs)

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arXiv 2024年

作者： Nnyaba, Ukamaka V. Shemtaga, Hewan M. Collins, David W. Muyskens, Amanda L. Priest, Benjamin W. Billor, Nedret Department of Mathematics and Statistics Auburn University AuburnAL36849 United States Department of Mathematics and Statistics Auburn University AuburnAL36849 United States Computational Engineering Division Lawrence Livermore National Laboratory LivermoreCA94550 United States Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States

Analyzing electrocardiography (ECG) data is essential for diagnosing and monitoring various heart diseases. The clinical adoption of automated methods requires accurate confidence measurements, which are largely absent from existing classification methods. In this paper, we present a robust Gaussian Process classification hyperparameter training model (MuyGPs) for discerning normal heartbeat signals from the signals affected by different arrhythmias and myocardial infarction. We compare the performance of MuyGPs with traditional Gaussian process classifier as well as conventional machine learning models, such as, Random Forest, Extra Trees, k-Nearest Neighbors and Convolutional Neural Network. Comparing these models reveals MuyGPs as the most performant model for making confident predictions on individual patient ECGs. Furthermore, we explore the posterior distribution obtained from the Gaussian process to interpret the prediction and quantify uncertainty. In addition, we provide a guideline on obtaining the prediction confidence of the machine learning models and quantitatively compare the uncertainty measures of these models. Particularly, we identify a class of less-accurate (ambiguous) signals for further diagnosis by an expert. © 2024, CC BY.

关键词： Convolutional neural networks

A General Fourth-Order Mesoscopic Multiple-Relaxation-Time Lattice Boltzmann Model and Equivalent Macroscopic Finite-Difference Scheme for Two-Dimensional Diffusion Equations

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SSRN

SSRN 2023年

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the equivalent macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain an equivalent macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis of the finite-difference scheme and the mesoscopic MRT-LB model. It is found that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis. © 2023, The Authors. All rights reserved.

关键词： Finite difference method

Regularized Lattice Boltzmann Method Based Maximum Principle and Energy Stability Preserving Finite-Difference Scheme for the Allen-Cahn Equation

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SSRN

SSRN 2024年

作者： Chen, Ying Liu, Xi Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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

The Allen-Cahn equation (ACE) inherently possesses two crucial properties: the maximum principle and the energy dissipation law. Preserving these two properties at the discrete level is also necessary in the numerical methods for the ACE. In this paper, unlike the traditionally top-down macroscopic numerical schemes which discretize the ACE directly, we first propose a novel bottom-up mesoscopic regularized lattice Boltzmann method based macroscopic numerical scheme for d (=1, 2, 3)-dimensional ACE, where the DdQ(2d+1) [(2d+1) discrete velocities in d-dimensional space] lattice structure is adopted. In particular, the proposed macroscopic numerical scheme has a second-order accuracy in space, and can also be viewed as an implicit-explicit finite-difference scheme for the ACE, in which the nonlinear term is discretized semi-implicitly, the temporal derivative and dissipation term of the ACE are discretized by using the explicit Euler method and second-order central difference method, respectively. Then we demonstrate that the proposed scheme can preserve the maximum bound principle and the original energy dissipation law at the discrete level under some conditions. Finally, some numerical experiments are conducted to validate our theoretical analysis. © 2024, The Authors. All rights reserved.

关键词： Maximum principle

A comparative study of two Allen-Cahn models for immiscible N-phase flows by using a consistent and conservative lattice Boltzmann method

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arXiv 2023年

作者： Zhan, Chengjie Liu, Xi Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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

In this work, we conduct a detailed comparison between two second-order conservative Allen-Cahn (AC) models [Model A: Zheng et al., Phys. Rev. E 101, 0433202 (2020) and Model B: Mirjalili and Mani, (2023)] for the immiscible N-phase flows. Mathematically, these two AC equations can be proved to be equivalent under some approximate conditions. However, the effects of these approximations are unclear from the theoretical point of view, and would be considered numerically. To this end, we propose a consistent and conservative lattice Boltzmann method for the AC models for N-phase flows, and present some numerical comparisons of accuracy and stability between these two AC models. The results show that both two AC models have good performances in accuracy, but the Model B is more stable for the realistic complex N-phase flows, although there is an adjustable parameter in the Model A. © 2023, CC BY-NC-ND.

关键词： Kinetic theory

A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and equivalent macroscopic finite-difference scheme for two-dimensional diffusion equations

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arXiv 2023年

关键词： Finite difference method