Structure-preservation, accuracy, and robustness are critical aims for magnetohydro dynamic (MHD) simulations. In this paper, we propose a high-order, magnetic helicity conservmethod for compressible ideal MHD systems. Notably, helicity conservation can be achieved using arbitrary high-order time integration methods, overcoming the limitations of Crank--Nicolson methods commonly used in existing literature. To our knowledge, this is the first Lagrangian and ALE scheme that addresses magnetic helicity conservation in MHD. These structure-preserving properties arise from a new Lagrangian MHD formulation with differential forms and an extended definition of material derivatives through Lie derivatives. By utilizing high-order finite element exterior calculus (FEEC) discretization, these properties are maintained at negligible cost during Lagrangian steps. To overcome mesh distortion issues, we introduce a mesh optimization strategy that automatically smooths the mesh based on quality indicators. After mesh smoothing steps, we propose structurepreserving remapping methods, including a positivity-preserving remap for density and a high-order vector potential-based remap for magnetic fields that preserves both divergence-free condition and magnetic helicity.

Publication:

SIAM JOURNAL ON SCIENTIFIC COMPUTING

http://dx.doi.org/10.1137/25M1796230

Author:

SHIPENG MAO

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences,University of Chinese Academy of Sciences, Beijing 100049, China

Email：maosp@lsec.cc.ac.cn

RUIJIE XI

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences,University of Chinese Academy of Sciences, Beijing 100049, China

Email：xiruijie@amss.ac.cn