<p>In this article, we propose and analyze a fully discrete implicit-explicit (IMEX) variable time-step scheme for the incompressible magnetohydrodynamic (MHD) system, in which nonlinear terms are treated fully explicitly while linear terms are handled implicitly. The method combines the variable time-step second-order backward differentiation formula (BDF2) for the temporal with the spatial discretization facilitated by the Taylor-Hood finite element method. Under a mild restriction <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<r_k<r_{\max }\approx 4.8645$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo><</mo> <msub> <mi>r</mi> <mo movablelimits="true">max</mo> </msub> <mo>≈</mo> <mn>4.8645</mn> </mrow> </math></EquationSource> </InlineEquation> on the ratio of adjacent time steps <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$r_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>, we derive rigorous local optimal <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-error estimates for both velocity and magnetic induction in the three-dimensional MHD system. The key points in our analysis include the construction and a modified discrete Gr<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ddot{\text {o}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mtext>o</mtext> <mo>¨</mo> </mover> </math></EquationSource> </InlineEquation>nwall’s inequality specifically tailored to the MHD system, together with the establishment of the unconditional stability of the numerical solutions. Finally, numerical experiments are performed to validate the theoretical findings.</p>

A variable time-step IMEX BDF2 scheme for the incompressible MHD system

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发表日期：

2026年5月4日

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Qianqian Ding,
Yifan Luo,
Shipeng Mao

摘要

In this article, we propose and analyze a fully discrete implicit-explicit (IMEX) variable time-step scheme for the incompressible magnetohydrodynamic (MHD) system, in which nonlinear terms are treated fully explicitly while linear terms are handled implicitly. The method combines the variable time-step second-order backward differentiation formula (BDF2) for the temporal with the spatial discretization facilitated by the Taylor-Hood finite element method. Under a mild restriction $0<r_k<r_{\max }\approx 4.8645$ $0 < r_{k} < r_{max} \approx 4.8645$ on the ratio of adjacent time steps $r_k$ $r_{k}$ , we derive rigorous local optimal $H^1$ $H^{1}$ -error estimates for both velocity and magnetic induction in the three-dimensional MHD system. The key points in our analysis include the construction and a modified discrete Gr $\ddot{\text {o}}$ $\ddot{o}$ nwall’s inequality specifically tailored to the MHD system, together with the establishment of the unconditional stability of the numerical solutions. Finally, numerical experiments are performed to validate the theoretical findings.