<p>A Voigt regularization is considered for the incompressible MHD equations in Elsässer variables. Then based on the BDF2, we propose and analyze a linearized and unconditionally stable finite element algorithm for this problem, which can be decoupled when the regularization parameters are equal. With the help of the Voigt regularization, the present algorithm removes the restriction involving the kinematic viscosity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2849_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> and magnetic permeability <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2849_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2849_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{2}< \nu /\nu _m< 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>ν</mi> <mo stretchy="false">/</mo> <msub> <mi>ν</mi> <mi>m</mi> </msub> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which comes from the BDF2 for the MHD equations in Elsässer variables. Furthermore, the unconditionally stability and convergence of this algorithm for the Voigt regularization of MHD equations in Elsässer variables are proved. Finally, several numerical simulations are provided to confirm the numerical theory, and show that the proposed algorithm outperforms the usual BDF2 for the problem outside the interval.</p>

A Voigt Regularization for Incompressible MHD Equations in Elsässer Variables

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发表日期：

2025年3月8日

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Xingwei Yang,
Pengzhan Huang,
Yinnian He

摘要

A Voigt regularization is considered for the incompressible MHD equations in Elsässer variables. Then based on the BDF2, we propose and analyze a linearized and unconditionally stable finite element algorithm for this problem, which can be decoupled when the regularization parameters are equal. With the help of the Voigt regularization, the present algorithm removes the restriction involving the kinematic viscosity $\nu $ $ν$ and magnetic permeability $\nu _m$ $ν_{m}$ , $\frac{1}{2}< \nu /\nu _m< 2$ $\frac{1}{2} < ν / ν_{m} < 2$ , which comes from the BDF2 for the MHD equations in Elsässer variables. Furthermore, the unconditionally stability and convergence of this algorithm for the Voigt regularization of MHD equations in Elsässer variables are proved. Finally, several numerical simulations are provided to confirm the numerical theory, and show that the proposed algorithm outperforms the usual BDF2 for the problem outside the interval.