Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows

Yüksel, Gamze; Işık, Osman Raşit

doi:10.1016/j.apm.2014.10.007

Cited by 24 publications

(17 citation statements)

References 8 publications

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“…The stability analysis of partitioned methods for MHD at small magnetic Reynolds number is presented by Layton et al in [3]. Numerical analysis of finite element method with Crank-Nicolson discretization [4] and Backward-Euler discretization [5] are performed by Yuksel et. al.…”

Section: Introductionmentioning

confidence: 99%

Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

Yüksel

Yaman

2021

Mugla Journal of Science and Technology

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In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter 𝜅(|𝑢 − 𝑢 ̅|(𝑢 − 𝑢 ̅)) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.

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Section: Introductionmentioning

confidence: 99%

Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

Yüksel

Yaman

2021

Mugla Journal of Science and Technology

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“…By applying the idea of Lagrange multiplier to magnetic subproblem, semi-implicit Euler scheme and nodal-based FEMs were studied by Prohl (2008) and Wacker et al (2016), respectively. The backward-Euler scheme was investigated by Yuksel and Isik (2014). We refer the reader to Layton et al (2014) and Ravindran (2008) for more efficient fully discrete schemes for evolutionary MHD equations.…”

Section: Introductionmentioning

confidence: 99%

A parallel finite element algorithm for nonstationary incompressible magnetohydrodynamics equations

Tang

2018

HFF

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Purpose The purpose of this paper is to design a parallel finite element (FE) algorithm based on fully overlapping domain decomposition for solving the nonstationary incompressible magnetohydrodynamics (MHD). Design/methodology/approach The fully discrete Euler implicit/explicit FE subproblems, which are defined in the whole domain with vast majority of the degrees of freedom associated with the particular subdomain, are solved in parallel. In each subproblem, the linear term is treated by implicit scheme and the nonlinear term is solved by explicit one. Findings For the algorithm, the almost unconditional convergence with optimal orders is validated by numerical tests. Some interesting phenomena are presented. Originality/value The proposed algorithm is effective, easy to realize with low communication costs and preferred for solving the strong nonlinear MHD system.

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“…In most terrestrial applications, such as the liquid metal, the magnetic Reynolds number of MHD flows is small. In this article, we study the following MHD equations at small magnetic Reynolds numbers, which are also called simplified MHD equations or reduced MHD equations, see, for example, [2,3,[14][15][16]…”

Section: Introductionmentioning

confidence: 99%

Ω \subset ℝ^{d} (d = 2 or 3)

, body force f , magnetic B , and time

T > 0

, find fluid velocity

u : Ω \times [0, T] \to ℝ^{d}

, pressure

p : Ω \times [0, T] \to ℝ

, and electric potential

ϕ : Ω \times [0, T] \to ℝ

satisfy

\begin{array}{l} left & \frac{1}{N} ({boldu}_{t} + u \cdot \nabla u) - \frac{1}{M^{2}} Δ u + \nabla p = f + B \times \nabla ϕ + (u \times B) \times B, \\ left & \nabla \cdot u = 0, \\ left & Δ ϕ = \nabla \cdot (u \times B), \end{array}

with the homogeneous Dirichlet boundary conditions and the initial condition

\begin{array}{l} left & u = 0 on \partial Ω \times [0, T], \\ left \end{array}

…”

Section: Introductionmentioning

confidence: 99%

“…In , Yuksel and Ingram gave a comprehensive error analysis for both the semidiscrete and a fully discrete approximate of time‐depended simplified MHD flow. Yuksel and Isik provided a stability and convergence analysis of a finite element discretization for time‐dependent simplified MHD flows with Backward‐Euler discretization. In these coupled methods, the coupling system is assembled at each time step and then solved iteratively.…”

Section: Introductionmentioning

confidence: 99%

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A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers

Rong

Hou

2017

Numerical Methods Partial

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This article aims to study the partitioned method for magnetohydrodynamic flows at small magnetic Reynolds numbers. We design a partitioned second‐order method and show that this method is stable under a time step ( Δ t ) restrict condition. Our method can decouple the magnetohydrodynamic equations so that we can solve two relatively simple subproblems separately at each time step, which is computationally economic. A complete theoretical analysis of error estimates is also given. Finally, we present numerical experiments to support our theory.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1966–1986, 2017

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Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows

Cited by 24 publications

References 8 publications

Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

A parallel finite element algorithm for nonstationary incompressible magnetohydrodynamics equations

A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers

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