A constrained-gradient method to control divergence errors in numerical MHD

In numerical magnetohydrodynamics (MHD), a major challenge is maintaining ∇⋅B=0. Constrained transport (CT) schemes achieve this but have been restricted to specific methods. For more general (meshless, moving-mesh, ALE) methods, ‘divergence-cleaning’ schemes reduce the ∇⋅B errors; however they can still be significant and can lead to systematic errors which converge away slowly. We propose a new constrained gradient (CG) scheme which augments these with a projection step, and can be applied to any numerical scheme with a reconstruction. This iteratively approximates the least-squares minimizing, globally divergence-free reconstruction of the fluid. Unlike ‘locally divergence free’ methods, this actually minimizes the numerically unstable ∇⋅B terms, without affecting the convergence order of the method. We implement this in the mesh-free code gizmo and compare various test problems. Compared to cleaning schemes, our CG method reduces the maximum ∇⋅B errors by ∼1–3 orders of magnitude (∼2–5 dex below typical errors if no ∇⋅B cleaning is used). By preventing large ∇⋅B at discontinuities, this eliminates systematic errors at jumps. Our CG results are comparable to CT methods; for practical purposes, the ∇⋅B errors are eliminated. The cost is modest, ∼30 per cent of the hydro algorithm, and the CG correction can be implemented in a range of numerical MHD methods. While for many problems, we find Dedner-type cleaning schemes are sufficient for good results, we identify a range of problems where using only Powell or ‘8-wave’ cleaning can produce order-of-magnitude errors.