A novel structure-preserving spectral scheme for the incompressible ideal MHD system

We propose an efficient implicit–explicit (IMEX) spectral scheme for the three-dimensional incompressible ideal magnetohydrodynamic (MHD) system that simultaneously preserves several fundamental physical invariants, including the energy, the magnetic Gauss’s law, the magnetic helicity, and the cross helicity. In the proposed scheme, energy conservation is achieved through a Lagrange multiplier formulation, while magnetic helicity and cross helicity are maintained via an error-compensation strategy involving two scalar auxiliary variables satisfying specially constructed ordinary differential equations. The proposed scheme provides the following advantages: (i) it achieves higher computational reliability by simultaneously preserving multiphysics structures at the fully discrete level; (ii) it is nearly linear in the sense that only linear algebraic systems and a single quadratic algebraic equation need to be solved at each time level; (iii) it is highly efficient when integrated with the adaptive time-stepping strategy. Finally, a series of three-dimensional numerical experiments is carried out to demonstrate the accuracy, efficiency, and structure-preserving capability of the proposed method for incompressible ideal MHD flows.