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<p>This paper focuses on the unconditionally optimal error estimates of a fully discrete decoupled scheme for two-phase magnetohydrodynamic (MHD) model with different viscosities and electric conductivities, by using the zero-energy-contribution (ZEC) method for the temporal discretization and mixed finite elements for the spatial discretization. Based on the ZEC property of the nonlinear and coupled terms of the model, an ordinary differential equation is designed to introduce a nonlocal scalar auxiliary variable which will play a key role in the design and the energy stability of the decoupled scheme. Combining fully explicit treatment on the nonlinear and coupled terms with the stabilization method for nonlinear potential, a decoupled temporal discrete scheme is proposed. Utilizing mixed finite elements for the spatial discretization in this temporal discrete scheme, a fully discrete scheme is proposed. Both schemes are proved to be mass-conservative and unconditionally energy stable. The unconditionally optimal error estimates of the temporal discrete scheme are derived for two dimensional and three dimensional (2D/3D) cases. Utilizing a modified Maxwell projection with variable electric conductivities, the superconvergence of its negative norm estimates, mathematical induction, and the unconditional stability of the numerical scheme, we also derive the optimal error estimates in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2773_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm for the fully discrete scheme in 2D/3D cases, without any restriction on the time step size and mesh size. Finally, numerical experiments are provided to verify the theoretical results.</p>

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Unconditionally Optimal Convergent Zero-Energy-Contribution Scheme for Two Phase MHD Model

  • 期刊论文

  • 发表日期：

    2025年1月18日

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  • Jinjin Yang,
  • Shipeng Mao,
  • Xiaoming He

摘要

This paper focuses on the unconditionally optimal error estimates of a fully discrete decoupled scheme for two-phase magnetohydrodynamic (MHD) model with different viscosities and electric conductivities, by using the zero-energy-contribution (ZEC) method for the temporal discretization and mixed finite elements for the spatial discretization. Based on the ZEC property of the nonlinear and coupled terms of the model, an ordinary differential equation is designed to introduce a nonlocal scalar auxiliary variable which will play a key role in the design and the energy stability of the decoupled scheme. Combining fully explicit treatment on the nonlinear and coupled terms with the stabilization method for nonlinear potential, a decoupled temporal discrete scheme is proposed. Utilizing mixed finite elements for the spatial discretization in this temporal discrete scheme, a fully discrete scheme is proposed. Both schemes are proved to be mass-conservative and unconditionally energy stable. The unconditionally optimal error estimates of the temporal discrete scheme are derived for two dimensional and three dimensional (2D/3D) cases. Utilizing a modified Maxwell projection with variable electric conductivities, the superconvergence of its negative norm estimates, mathematical induction, and the unconditional stability of the numerical scheme, we also derive the optimal error estimates in \(L^2\) L 2 -norm for the fully discrete scheme in 2D/3D cases, without any restriction on the time step size and mesh size. Finally, numerical experiments are provided to verify the theoretical results.