The second order projection method in time for the time-dependent natural convection problem

Qian, Yanxia; Zhang, Tong

doi:10.1007/s10492-016-0133-y

Cited by 16 publications

(3 citation statements)

References 22 publications

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“…The process of solving the Navier-Stokes equation follows the projection method [32,33], where n represents a certain moment, n + 1 is the next moment of n moment, and ∆t is the time step. The central difference scheme is used for the discretization of the diffusion term on the right side of the above equation, and the QUICK (Quadratic Upstream Interpolation for Convective Kinetics) format is used for the convection term to obtain the temporary velocity u*, which is then brought into the pressure term to obtain the actual velocity u n+1 .…”

Section: Governing Equations Of the Front-tracking Methodsmentioning

confidence: 99%

A Coupled Machine Learning and Lattice Boltzmann Method Approach for Immiscible Two-Phase Flows

Li,

Zhou,

et al. 2023

Mathematics

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An innovative coupling numerical algorithm is proposed in the current paper, the front-tracking method–lattice Boltzmann method–machine learning (FTM-LBM-ML) method, to precisely capture fluid flow phase interfaces at the mesoscale and accurately simulate dynamic processes. This method combines the distinctive abilities of the FTM to accurately capture phase interfaces and the advantages of the LBM for easy handling of mesoscopic multi-component flow fields. Taking a single vacuole rising as an example, the input and output sets of the machine learning model are constructed using the FTM’s flow field, such as the velocity and position data from phase interface markers. Such datasets are used to train the Bayesian-Regularized Back Propagation Neural Network (BRBPNN) machine learning model to establish the corresponding relationship between the phase interface velocity and the position. Finally, the trained BRBPNN neural network is utilized within the multi-relaxation LBM pseudo potential model flow field to predict the phase interface position, which is compared with the FTM simulation. It was observed that the BRBPNN-predicted interface within the LBM exhibits a high degree of consistency with the FTM-predicted interface position, showing that the BRBPNN model is feasible and satisfies the accuracy requirements of the FT-LB coupling model.

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Section: Governing Equations Of the Front-tracking Methodsmentioning

confidence: 99%

A Coupled Machine Learning and Lattice Boltzmann Method Approach for Immiscible Two-Phase Flows

Li,

Zhou,

et al. 2023

Mathematics

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“…Due to the coupling of three variables among the natural convection equations, finding the numerical solutions becomes a difficult task, much attention has been attracted in recent years. For example, we can refer to Du et al, Wu et al, Zhang et al, and Zhang et al for the variational multiscale methods, for the projection method, for the decoupled method, for iteration methods, and the references therein, all above mentioned works are one order schemes and 2D case.…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

Liang

Zhang

2019

Math Methods in App Sciences

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In this paper, we consider the Crank-Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank-Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two-grid Crank-Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two-grid Crank-Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H 1 -norm and for pressure in L 2 -norm. However, the two-grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes. KEYWORDSCrank-Nicolson extrapolation scheme, error estimates, stability, the natural convection equations, two-grid method MSC CLASSIFICATION 65N15; 65N30; 76D07Math Meth Appl Sci. 2019;42:6165-6191.wileyonlinelibrary.com/journal/mma

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“…On the contrary, inspired by the higher order projection methods for the Navier-Stokes equations proposed by Shen [30] and for the natural convection problem proposed by Qian and Zhang [24], we propose a higher order pressure segregation scheme for the time-dependent MHD equations in this article. This scheme allows us to decouple the MHD system into two sub-problems at each time step.…”

Section: Introductionmentioning

confidence: 99%

A higher order pressure segregation scheme for the time-dependent magnetohydrodynamics equations

2019

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The second order projection method in time for the time-dependent natural convection problem

Abstract: The second order projection method in time for the time-dependent natural convection problem

Cited by 16 publications

References 22 publications

A Coupled Machine Learning and Lattice Boltzmann Method Approach for Immiscible Two-Phase Flows

A Coupled Machine Learning and Lattice Boltzmann Method Approach for Immiscible Two-Phase Flows

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

A higher order pressure segregation scheme for the time-dependent magnetohydrodynamics equations

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