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Finite Element Methods for the Navier-Stokes Equations and MHD Equations

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 24951

Special Issue Editors


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Guest Editor
College of Mathematics and Systems Science, Xinjiang University, Urumqi 830049, China
Interests: numerical methods for PDEs; computational fluid dynamics; computational mathematics; uncertainty quantification; machine learning
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
Interests: finite element method; finite difference method; Navier-Stokes equations; MHD equations; primitive equations of the ocean; numerical methods for PDEs
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The incompressible Navier-Stokes equations reflect the basic mechanical law of viscous fluid flow, which ha important implications in fluid mechanics. This problem is one of the main systems studied in pipe flow, flow around airfoils, blood flow, weather, and convective heat transfer inside industrial furnaces. Therefore, solving the 3D steady and unsteady Navier-Stokes equations is of great significance and application value in the field of scientific research and engineering applications. Lots of works studies have been devoted to this problem, and the finite element methods, finite volume methods, and finite difference methods have been the most successful. Furthermore, the incompressible magnetohydrodynamic (MHD) equations mainly describe the behavior of the macroscopic interaction of electrically conducting fluids and magnetic fields. The MHD equations are determined by the Navier-Stokes equations coupled with the pre-Maxwell equations. The resulting system of numerically solving the MHD equations often requires an unrealistic amount of computing power and storage to properly resolve some MHD flow details. Recently, some numerical methods on the first law and second law of thermodynamics and the concepts of entropy (property) and entropy generation (as measure of process irreversibility) have been discussed.

Considering the recent advances in the field of solving the Navier-Stokes equations and the MHD equations, this Special Issue will collect new ideas and numerical methods arising from the field of analysis and modeling of the Navier-Stokes equations and the MHD equations and some numerical methods on the first law and the second law of thermodynamics and the concepts of entropy (property) and entropy generation (as measure of process irreversibility).

This Special Issue will accept unpublished original papers and comprehensive reviews focused (but not limited to) on the following research areas:

  • New mathematical modeling of 3D incompressible Navier-Stokes equations and MHD equations.
  • New numerical methods of 2D/3D incompressible Navier-Stokes equations and MHD equations.
  • Stability and error analysis of new numerical methods of 2D/3D incompressible Navier-Stokes equations and MHD equations.
  • Advanced computational algorithms applied in real problems related to 2D/3D incompressible Navier-Stokes equations and MHD equations.
  • The existence, stability, and uniqueness of 3D incompressible Navier-Stokes equations and MHD equations.
  • Machine learning for 2D/3D incompressible Navier-Stokes equations and MHD equations.
  • Some numerical methods and numerical analysis on the first law and the second law of thermodynamics and the concepts of entropy (property) and entropy generation (as measure of process irreversibility).

Dr. Xinlong Feng
Prof. Dr. Yinnian He
Guest Editors

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Keywords

  • incompressible navier-stokes equations
  • incompressible MHD equations
  • finite element method
  • finite difference method
  • radial basis function
  • discontinuous galerkin method
  • machine learning

Published Papers (16 papers)

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Research

18 pages, 4385 KiB  
Article
A Second-Order Crank–Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction
by Chunya Wu, Xinlong Feng and Lingzhi Qian
Entropy 2022, 24(11), 1512; https://doi.org/10.3390/e24111512 - 23 Oct 2022
Viewed by 1212
Abstract
In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms [...] Read more.
In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method. Full article
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26 pages, 2629 KiB  
Article
Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations
by Kaiwen Shi, Haiyan Su and Xinlong Feng
Entropy 2022, 24(10), 1395; https://doi.org/10.3390/e24101395 - 30 Sep 2022
Cited by 1 | Viewed by 1099
Abstract
In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “·u=0”, which allows us [...] Read more.
In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “·u=0”, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter ϵ, the time-step size τ, and the mesh size h. Finally, two numerical tests show that our scheme is effective. Full article
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11 pages, 706 KiB  
Article
Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods
by Yanzi Zhao and Xinlong Feng
Entropy 2022, 24(10), 1338; https://doi.org/10.3390/e24101338 - 23 Sep 2022
Viewed by 1368
Abstract
In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the [...] Read more.
In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify the accuracy and effectiveness of the proposed method. Full article
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14 pages, 1281 KiB  
Article
Dynamic Weight Strategy of Physics-Informed Neural Networks for the 2D Navier–Stokes Equations
by Shirong Li and Xinlong Feng
Entropy 2022, 24(9), 1254; https://doi.org/10.3390/e24091254 - 6 Sep 2022
Cited by 13 | Viewed by 2756
Abstract
When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, [...] Read more.
When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, which can balance the gradients of each loss term during training. The weight is updated by the idea of the minmax algorithm. The neural network identifies which types of training data are harder to train and forces it to focus on those data before training the next step. Specifically, it adjusts the weight of the data that are difficult to train to maximize the objective function. On this basis, one can adjust the network parameters to minimize the objective function and do this alternately until the objective function converges. We demonstrate that the dynamic weights are monotonically non-decreasing and convergent during training. This method can not only accelerate the convergence of the loss, but also reduce the generalization error, and the computational efficiency outperformed other state-of-the-art PINNs algorithms. The validity of the method is verified by solving the forward and inverse problems of the Navier–Stokes equation. Full article
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21 pages, 8919 KiB  
Article
Linear Full Decoupling, Velocity Correction Method for Unsteady Thermally Coupled Incompressible Magneto-Hydrodynamic Equations
by Zhe Zhang, Haiyan Su and Xinlong Feng
Entropy 2022, 24(8), 1159; https://doi.org/10.3390/e24081159 - 19 Aug 2022
Cited by 2 | Viewed by 1339
Abstract
We propose and analyze an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations in this paper. The proposed method is a first-order velocity correction projection algorithms in time marching, including standard velocity correction and rotation velocity correction, which can completely decouple all [...] Read more.
We propose and analyze an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations in this paper. The proposed method is a first-order velocity correction projection algorithms in time marching, including standard velocity correction and rotation velocity correction, which can completely decouple all variables in the model. Meanwhile, the schemes are not only linear and only need to solve a series of linear partial differential equations with constant coefficients at each time step, but also the standard velocity correction algorithm can produce the Neumann boundary condition for the pressure field, but the rotational velocity correction algorithm can produce the consistent boundary which improve the accuracy of the pressure field. Thus, improving our computational efficiency. Then, we give the energy stability of the algorithms and give a detailed proofs. The key idea to establish the stability results of the rotation velocity correction algorithm is to transform the rotation term into a telescopic symmetric form by means of the Gauge–Uzawa formula. Finally, numerical experiments show that the rotation velocity correction projection algorithm is efficient to solve the thermally coupled magneto-hydrodynamic equations. Full article
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16 pages, 777 KiB  
Article
Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh
by Lulu Li, Haiyan Su and Yinnian He
Entropy 2022, 24(8), 1129; https://doi.org/10.3390/e24081129 - 15 Aug 2022
Viewed by 1329
Abstract
In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby [...] Read more.
In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby bypassing the obstacle of standard finite element discretizations on arbitrary mesh. The velocity and pressure are approximated by the practical significative lowest equal-order virtual element space pair (Xh,Qh) which does not satisfy the discrete inf-sup condition. Combined with the penalty method, the error estimation is proved rigorously. Numerical results on the 3D polygonal mesh illustrate the theoretical results and effectiveness of the proposed method. Full article
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20 pages, 925 KiB  
Article
Analysis of the Element-Free Galerkin Method with Penalty for Stokes Problems
by Tao Zhang and Xiaolin Li
Entropy 2022, 24(8), 1072; https://doi.org/10.3390/e24081072 - 3 Aug 2022
Cited by 1 | Viewed by 1293
Abstract
The element-free Galerkin (EFG) method with penalty for Stokes problems is proposed and analyzed in this work. A priori error estimates of the penalty method, which is used to deal with Dirichlet boundary conditions, are derived to illustrate its validity in a continuous [...] Read more.
The element-free Galerkin (EFG) method with penalty for Stokes problems is proposed and analyzed in this work. A priori error estimates of the penalty method, which is used to deal with Dirichlet boundary conditions, are derived to illustrate its validity in a continuous sense. Based on a feasible assumption, it is proved that there is a unique weak solution in the modified weak form of penalized Stokes problems. Then, the error bounds with the penalty factor for the EFG discretization are derived, which provide a rationale for choosing an efficient penalty factor. Numerical examples are given to confirm the theoretical results. Full article
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21 pages, 735 KiB  
Article
Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations
by Cong Xie and Kun Wang
Entropy 2022, 24(7), 948; https://doi.org/10.3390/e24070948 - 7 Jul 2022
Cited by 1 | Viewed by 1122
Abstract
Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded [...] Read more.
Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time t>0. Finally, some numerical examples are shown to verify the theoretical predictions. Full article
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16 pages, 338 KiB  
Article
A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients
by Qianqian Ding, Xiaonian Long and Shipeng Mao
Entropy 2022, 24(7), 912; https://doi.org/10.3390/e24070912 - 30 Jun 2022
Viewed by 1064
Abstract
In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. [...] Read more.
In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme. Full article
22 pages, 700 KiB  
Article
The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem
by Guoliang He and Yong Zhang
Entropy 2022, 24(6), 768; https://doi.org/10.3390/e24060768 - 30 May 2022
Cited by 2 | Viewed by 1427
Abstract
This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space [...] Read more.
This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle P1P0 trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results. Full article
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25 pages, 1199 KiB  
Article
Numerical Analysis and Comparison of Three Iterative Methods Based on Finite Element for the 2D/3D Stationary Micropolar Fluid Equations
by Xin Xing and Demin Liu
Entropy 2022, 24(5), 628; https://doi.org/10.3390/e24050628 - 29 Apr 2022
Cited by 6 | Viewed by 1625
Abstract
In this paper, three iterative methods (Stokes, Newton and Oseen iterative methods) based on finite element discretization for the stationary micropolar fluid equations are proposed, analyzed and compared. The stability and error estimation for the Stokes and Newton iterative methods are obtained under [...] Read more.
In this paper, three iterative methods (Stokes, Newton and Oseen iterative methods) based on finite element discretization for the stationary micropolar fluid equations are proposed, analyzed and compared. The stability and error estimation for the Stokes and Newton iterative methods are obtained under the strong uniqueness conditions. In addition, the stability and error estimation for the Oseen iterative method are derived under the uniqueness condition of the weak solution. Finally, numerical examples test the applicability and the effectiveness of the three iterative methods. Full article
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23 pages, 10344 KiB  
Article
Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations
by Haiyan Su, Jiali Xu and Xinlong Feng
Entropy 2022, 24(5), 587; https://doi.org/10.3390/e24050587 - 22 Apr 2022
Cited by 2 | Viewed by 1614
Abstract
Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on [...] Read more.
Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods. Full article
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13 pages, 4143 KiB  
Article
A Uzawa-Type Iterative Algorithm for the Stationary Natural Convection Model
by Aytura Keram and Pengzhan Huang
Entropy 2022, 24(4), 543; https://doi.org/10.3390/e24040543 - 13 Apr 2022
Cited by 1 | Viewed by 1483
Abstract
In this study, a Uzawa-type iterative algorithm is introduced and analyzed for solving the stationary natural convection model, where physical variables are discretized by utilizing a mixed finite element method. Compared with the common Uzawa iterative algorithm, the main finding is that the [...] Read more.
In this study, a Uzawa-type iterative algorithm is introduced and analyzed for solving the stationary natural convection model, where physical variables are discretized by utilizing a mixed finite element method. Compared with the common Uzawa iterative algorithm, the main finding is that the proposed algorithm produces weakly divergence-free velocity approximation. In addition, the convergence results of the proposed algorithm are provided, and numerical tests supporting the theory are presented. Full article
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25 pages, 1371 KiB  
Article
Numerical Analysis and Comparison of Four Stabilized Finite Element Methods for the Steady Micropolar Equations
by Jingnan Liu and Demin Liu
Entropy 2022, 24(4), 454; https://doi.org/10.3390/e24040454 - 25 Mar 2022
Cited by 2 | Viewed by 1537
Abstract
In this paper, four stabilized methods based on the lowest equal-order finite element pair for the steady micropolar Navier–Stokes equations (MNSE) are presented, which are penalty, regular, multiscale enrichment, and local Gauss integration methods. A priori properties, existence, uniqueness, stability, and error estimation [...] Read more.
In this paper, four stabilized methods based on the lowest equal-order finite element pair for the steady micropolar Navier–Stokes equations (MNSE) are presented, which are penalty, regular, multiscale enrichment, and local Gauss integration methods. A priori properties, existence, uniqueness, stability, and error estimation based on Fem approximation of all the methods are proven for the physical variables. Finally, some numerical examples are displayed to show the numerical characteristics of these methods. Full article
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17 pages, 1867 KiB  
Article
A Positivity-Preserving Finite Volume Scheme for Nonequilibrium Radiation Diffusion Equations on Distorted Meshes
by Di Yang, Gang Peng and Zhiming Gao
Entropy 2022, 24(3), 382; https://doi.org/10.3390/e24030382 - 9 Mar 2022
Viewed by 1584
Abstract
In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns [...] Read more.
In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme. Full article
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23 pages, 16988 KiB  
Article
Finite Element Iterative Methods for the Stationary Double-Diffusive Natural Convection Model
by Yaxin Wei and Pengzhan Huang
Entropy 2022, 24(2), 236; https://doi.org/10.3390/e24020236 - 3 Feb 2022
Cited by 2 | Viewed by 1292
Abstract
In this paper, we consider the stationary double-diffusive natural convection model, which can model heat and mass transfer phenomena. Based on the fixed point theorem, the existence and uniqueness of the considered model are proved. Moreover, we design three finite element iterative methods [...] Read more.
In this paper, we consider the stationary double-diffusive natural convection model, which can model heat and mass transfer phenomena. Based on the fixed point theorem, the existence and uniqueness of the considered model are proved. Moreover, we design three finite element iterative methods for the considered problem. Under the uniqueness condition of a weak solution, iterative method I is stable. Compared with iterative method I, iterative method II is stable with a stronger condition. Moreover, iterative method III is stable with the strongest condition. From the perspective of viscosity, iterative method I displays well in the case of a low viscosity number, iterative method II runs well with slightly low viscosity, and iterative method III can deal with high viscosity. Finally, some numerical experiments are presented for testing the correctness of the theoretic analysis. Full article
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