A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

Greif, Chen; Li, Dan; Schötzau, Dominik; Wei, Xiaoxi

doi:10.1016/j.cma.2010.05.007

Cited by 132 publications

(105 citation statements)

References 30 publications

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“…Gerbeau, C. Le Bris and T. Lelièvre presented some mathematical theories and numerical methods for the steady MHD equations in the book [13]. For more extensive investigation of the steady MHD equations, please see [14][15][16][17][18][19] and their references.…”

Section: Introductionmentioning

confidence: 98%

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Zhang

2015

Computers & Mathematics with Applications

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a b s t r a c tIn this article, a decoupled fully discrete scheme for solving 2D magnetohydrodynamics (MHD) equations is proposed. The decoupled scheme is used for time discretization, and the finite element method is used for spatial discretization. Firstly, the almost unconditional stability ( t ≤ C ) of this scheme is established. Then optimal L 2 and H 1 error estimates of numerical solution are provided. Finally, a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme.

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Section: Introductionmentioning

confidence: 98%

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Zhang

2015

Computers & Mathematics with Applications

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“…The external magnetic field is given as B =(0,1) ⊤ and the magnetic boundary conditions are set to the electrically insulating boundary condition on all boundaries since the magnetic Reynolds number is very low. The present boundary conditions are very similar to the boundary conditions employed by Greif et al for an electrically conducting fluid inside a channel with a backward facing step at a lower Reynolds number. The computational mesh is generated with the CUBIT mesh generation environment using paving algorithm and then two levels of template‐based two‐refinement algorithm are applied next to the solid walls.…”

Section: Numerical Resultsmentioning

confidence: 54%

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

Ata

Şahin

2019

Numerical Methods in Fluids

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Summary A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face‐centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence‐free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large‐scale algebraic linear equations are solved in a fully coupled manner using the one‐ and two‐level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid‐driven cavity and backward facing step benchmark problems in two and three dimensions. The lid‐driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.

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“…If exactly divergence free elements (e.g., [6,8,14,36]) are used, this implies unconditional stability if C ν T > 1. The constant C s in the second condition of (4.2) comes from the Sobolev embedding inequality and thus only depends on the domain.…”

Section: Stability Of the Methods With Eddy Viscositymentioning

confidence: 99%

A Higher Order Ensemble Simulation Algorithm for Fluid Flows

Jiang

2014

J Sci Comput

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This report presents an efficient, higher order method for fast calculation of an ensemble of solutions of the Navier-Stokes equations. We give a complete stability and convergence analysis of the method for laminar flows and an extension to turbulent flows. For high Reynolds number flows, we propose and analyze an eddy viscosity model with a recent reparameterization of the mixing length. This turbulence model depends on an ensemble mean compatible with the higher order method. We show the turbulence model has superior stability, also demonstrated in numerical tests. We also give tests showing the potential of the new method for exploring flow problems to compute turbulence intensities, effective Lyapunov exponents, windows of predictability and to verify the selective decay principle.

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A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

Cited by 132 publications

References 30 publications

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

A Higher Order Ensemble Simulation Algorithm for Fluid Flows

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