We construct splitting methods for the solution of the equations of magnetohydrodynamics (MHD). Due to the physical significance of the involved operators, splittings into three or even four operators with positive coefficients are appropriate for a physically correct and efficient solution of the equations. To efficiently obtain an accurate solution approximation, adaptive choice of the time-steps is important particularly in the light of the unsmooth dynamics of the system. Thus, we construct new method coefficients in conjunction with associated error estimators by optimizing the leading local error term. As a proof of concept, we demonstrate that adaptive splitting faithfully reflects the solution behavior also in the presence of a shock for the viscous Burgers equation, which serves as a simplified model problem displaying several features of the Navier-Stokes equation for incompressible flow.

翻译：本文构造了针对磁流体动力学（MHD）方程求解的分裂方法。鉴于所涉及算子的物理意义，采用三算子甚至四算子（均具有正系数形式）分裂法可实现对上述方程的物理正确且高效求解。为高效获得精确近似解，特别是在系统动力学不光滑的情况下，时间步长的自适应选择至关重要。为此，我们通过优化主导局部误差项，结合相应误差估计器构建了新的方法系数。作为概念验证，我们证明自适应分裂方法能够准确反映含激波情形下的解行为——以粘性Burgers方程为例（该方程作为简化模型问题，展现了不可压缩流动Navier-Stokes方程的若干特征）。