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<p>Magnetohydrodynamics (MHD) describes the interaction between electrically conducting fluids and electromagnetic fields. We propose and analyze a symplectic, second-order algorithm for the evolutionary MHD system in Elsässer variables. We reduce the computational cost of the iterative non-linear solver, at each time step, by partitioning the coupled system into two subproblems of half size, solved in parallel. We prove that the iterations converge linearly, under a time step restriction similar to the one required in the full space-time error analysis. The variable step algorithm unconditionally conserves the energy, cross-helicity and magnetic helicity, and numerical solutions are second-order accurate in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norms. The time adaptive mechanism, based on a local truncation error criterion, helps the variable step algorithm balance accuracy and time efficiency. Several numerical tests support the theoretical findings and verify the advantage of time adaptivity.</p>

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Partitioned conservative, variable step, second-order method for magneto-hydrodynamics in Elsässer Variables

  • 期刊论文

  • 发表日期：

    2026年1月16日

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  • Zhen Yao,
  • Catalin Trenchea,
  • Wenlong Pei

摘要

Magnetohydrodynamics (MHD) describes the interaction between electrically conducting fluids and electromagnetic fields. We propose and analyze a symplectic, second-order algorithm for the evolutionary MHD system in Elsässer variables. We reduce the computational cost of the iterative non-linear solver, at each time step, by partitioning the coupled system into two subproblems of half size, solved in parallel. We prove that the iterations converge linearly, under a time step restriction similar to the one required in the full space-time error analysis. The variable step algorithm unconditionally conserves the energy, cross-helicity and magnetic helicity, and numerical solutions are second-order accurate in the \(L^{2}\) L 2 and \(H^{1}\) H 1 -norms. The time adaptive mechanism, based on a local truncation error criterion, helps the variable step algorithm balance accuracy and time efficiency. Several numerical tests support the theoretical findings and verify the advantage of time adaptivity.