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

Zhang, Guodong; He, Ya‐Ling

doi:10.1016/j.camwa.2015.03.019

Cited by 47 publications

(17 citation statements)

References 44 publications

(48 reference statements)

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“…Remark 3.3. In our stability results, Theorem 3.1 and Theorem 3.2, they both need a time step condition t ≤ C. This time step restriction is verified in our article [25]. We observe the H 1 norm of numerical solutions have a uniform bound when t is less than a constant.…”

Section: Theorem 32 Under the Conditions Of Theorem 31 Then The Fsupporting

confidence: 61%

“…We prove this scheme has

H^{2}

stability provided the time step size

Δ t \leq C .

We also give the optimal

L^{2} - H^{1}

error estimates for velocity and magnetic field, and the optimal

L^{2}

convergence rate for pressure under the stability condition. The fully discrete scheme and numerical implementation are considered in the article .…”

Section: Discussionmentioning

confidence: 99%

“…In order to deduce the error estimates of spatial discretization in Ref. , we also need the discrete version of (2.12). Theorem Under the conditions of Theorem 3.1, then the following estimate holds

τ (t_{m}) (‖ curl d_{t} {boldb}^{m} ‖_{0}^{2} + ‖ \nabla d_{t} {boldu}^{m} ‖_{0}^{2}) + Δ t \sum_{n = 1}^{m} τ (t_{n}) (‖ d_{t} {boldb}^{n} ‖_{2}^{2} + ‖ d_{t} {boldu}^{n} ‖_{2}^{2} + ‖ d_{t} p^{n} ‖_{1}^{2}) \leq C, 1 \leq m \leq N,

where

τ (t_{n}) = \min {1, t_{n}} .

Proof This proof is divided into (

n \geq 2

) and (

n = 1

) cases.…”

Section: The Decoupled Scheme and H 2 Stabilitymentioning

confidence: 99%

“…In order to deduce the error estimates of spatial discretization in Ref. [25], we also need the discrete version of (2.12).…”

Section: )mentioning

confidence: 99%

“…In this article, we present this scheme has almost unconditional (

Δ t \leq C

) stability and convergence, and has the optimal

L^{2} - H^{1}

convergence rates. The finite element spatial discretization and numerical implementation are considered in the article .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Decoupled schemes for unsteady MHD equations. I. time discretization

Zhang

2017

Numerical Methods Partial

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In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional ( Δ t ≤ C ) H 2 stability and convergence of this scheme are provided. The optimal L 2 − H 1 error estimates for velocity and magnet are provided, and the optimal L 2 error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017

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Section: Theorem 32 Under the Conditions Of Theorem 31 Then The Fsupporting

confidence: 61%

“…We prove this scheme has

H^{2}

stability provided the time step size

Δ t \leq C .

We also give the optimal

L^{2} - H^{1}

error estimates for velocity and magnetic field, and the optimal

L^{2}

convergence rate for pressure under the stability condition. The fully discrete scheme and numerical implementation are considered in the article .…”

Section: Discussionmentioning

confidence: 99%

τ (t_{m}) (‖ curl d_{t} {boldb}^{m} ‖_{0}^{2} + ‖ \nabla d_{t} {boldu}^{m} ‖_{0}^{2}) + Δ t \sum_{n = 1}^{m} τ (t_{n}) (‖ d_{t} {boldb}^{n} ‖_{2}^{2} + ‖ d_{t} {boldu}^{n} ‖_{2}^{2} + ‖ d_{t} p^{n} ‖_{1}^{2}) \leq C, 1 \leq m \leq N,

where

τ (t_{n}) = \min {1, t_{n}} .

Proof This proof is divided into (

n \geq 2

) and (

n = 1

) cases.…”

Section: The Decoupled Scheme and H 2 Stabilitymentioning

confidence: 99%

“…In order to deduce the error estimates of spatial discretization in Ref. [25], we also need the discrete version of (2.12).…”

Section: )mentioning

confidence: 99%

“…In this article, we present this scheme has almost unconditional (

Δ t \leq C

) stability and convergence, and has the optimal

L^{2} - H^{1}

convergence rates. The finite element spatial discretization and numerical implementation are considered in the article .…”

Section: Introductionmentioning

confidence: 99%

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Decoupled schemes for unsteady MHD equations. I. time discretization

Zhang

2017

Numerical Methods Partial

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A modified characteristics projection finite element method for the non‐stationary incompressible thermally coupled MHD equations

Guo

2023

Math Methods in App Sciences

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In this paper, a modified characteristics projection finite element method is proposed for the non‐stationary incompressible thermally couple MHD equations. The modified characteristics projection finite element method combines the modified characteristics finite element method with the projection method to verify the unconditional stability and optimal error estimation of velocity, pressure, magnetic field and temperature in the non‐stationary incompressible thermally couple MHD equations. In order to show the effectiveness of the proposed method, some numerical results are presented. The numerical results are consistent with our theoretical analysis, and our method is effective.

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A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

Sedaghatjoo

Dehghan

Hosseinzadeh

2017

Numerical Methods Partial

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The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 106. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.

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

Cited by 47 publications

References 44 publications

Decoupled schemes for unsteady MHD equations. I. time discretization

Decoupled schemes for unsteady MHD equations. I. time discretization

A modified characteristics projection finite element method for the non‐stationary incompressible thermally coupled MHD equations

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

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