A fourth-order conservation-element and solution-element scheme on curvilinear adaptive mesh refinement grid for magnetohydrodynamics simulations

Abstract

High-precision magnetohydrodynamics (MHD) simulations are essential for understanding the mechanism of solar eruptions and coronal mass ejections (CMEs), where the slow accumulation and impulsive release of magnetic energy are inherent to the corona's extremely low resistivity. However, achieving such simulations with conventional second-order schemes requires extremely high grid resolutions, making routine calculations prohibitively expensive. Here, we present a new code AMR–CESE4–MHD by implementing, for the first time, a fourth-order conservation-element and solution-element (CESE) scheme on general curvilinear geometries with adaptive mesh refinement (AMR) grid for MHD simulations. We propose a smoothness indicator based on second-order partial derivatives to guide the mesh refinement. For solution prolongation and restriction that are required during the grid refinement and coarsening, we use Taylor series expansions based on spatial derivatives available in the CESE solution to preserve solution accuracy during grid changes. We also propose a magnetic divergence control method by correcting the spatial derivatives of the magnetic field so that the magnetic-field divergence and its spatial derivatives evaluated from these auxiliary variables vanish. A series of benchmark tests validate the implementation. The scheme achieves its designed fourth-order accuracy, and the AMR criteria automatically refine near discontinuities. The derivative-control method keeps the ∇·B errors comparable to the combined approach of Powell source term and Marder diffusion term, while showing better conservation properties. The code produces consistent results in both Cartesian and curvilinear coordinates, and is ready for applications in solar eruption and CME modeling.