Magnetohydrodynamics (MHD) in RAMSES-CPP

Status: The MHD module is fully integrated with the AmrGrid system and is the primary focus of the Phase 2 multi-dimensional expansion. Core physics, Constrained Transport, and adaptive refinement for magnetic gradients are fully operational and verified against standard benchmarks.

The MHD module in RAMSES-CPP provides a robust, divergence-free implementation of ideal magnetohydrodynamics on adaptive grids.

Key Features

• Riemann Solvers: Supports HLLD (standard for MHD) and LLF (Local Lax-Friedrichs) for high robustness.
• Constrained Transport (CT): Uses a staggered grid arrangement for magnetic fields to maintain $\nabla \cdot B = 0$ to machine precision.
• 2nd-Order Predictor-Corrector: Implements a MUSCL-Hancock scheme with a $dt/2$ time prediction step for 2nd-order accuracy in both space and time.
• Adaptive Magnetic Refinement: Supports automatic mesh refinement based on magnetic energy gradients (err_grad_b2), essential for capturing MHD shocks and vortices.
• Dimensional Rotation: Features a robust state-rotation framework in MhdSolver::cmpflxm to ensure physical consistency in multi-dimensional sweeps.

Implementation Details

Staggered Grid and Variable Layout

Magnetic fields are stored at the faces of the cells (indices 6-8 for left faces, and $nvar-2$ to $nvar$ for right faces), while hydro variables are stored at cell centers.

• $nvar = 11 + nener$ for MHD simulations.
• Fluxes are computed at cell interfaces and correctly back-rotated to the global coordinate system after the Riemann solver step.

EMF and CT Update

The Electromotive Forces (EMFs) are computed at cell edges using electric field components from the Riemann fluxes. The face-centered magnetic fields are then updated using the curl of these EMFs: \(\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}\) This staggered update ensures that the divergence $\nabla \cdot B$ remains zero to machine precision throughout the simulation.

Verified Benchmarks

• 1D MHD Shock Tube: Verified correct wave propagation and jump conditions.
• 2D Orszag-Tang Vortex: Successfully executed high-resolution adaptive runs (~35,000 leaf cells at Level 9). Verified symmetric density evolution and machine-precision divergence maintenance ($\nabla \cdot B \approx 10^{-16}$).

Configuration

To enable MHD, use the following CMake flag:

cmake .. -DRAMSES_USE_MHD=ON

In the namelist, ensure hydro=.true. and appropriate MHD parameters are set in the &HYDRO_PARAMS and &PHYSICS_PARAMS blocks.